Babylonian Mathematics

Introduction'Babylonian' is a general word to describe the people living in

Mesopotamia, a fertile plain between the Tigris and Euphrates rivers
(present day Turkey and Syria).The Babylonian civilisation (dating from
around 2000-600BC) replaced that of the Sumerians and Akkadians, and so
inherited their sexagesimal (i.e. base 60) number system. The Sumerians
had created a form of writing based on cuneiform (wedge-shaped) symbols,
which the Babylonians also adopted. This is how most of their texts have
come down to us: as symbols written on wet clay tablets which were then
baked in the hot sun so the clay set and the symbols were permanent.
Thousands of these tablets have survived to the present day.

Babylonian mathematics was, in many ways, more advanced than Egyptian

maths. They could extract square and cube roots, work with Pythagorean
triples 1200 years before Pythagoras, had a knowledge of pi and possibly e
(the exponential function), could solve some quadratics and even
polynomials of degree 8, solved linear equations and could also deal with
circular measurement. Babylonian mathematics was based much more on
algebra and less on geometry, in contrast to the Greeks.

Babylonian Numerals

Cuneiform numbers could be written using a combination of two symbols: a

vertical wedge for '1' and a corner wedge for '10'. The Babylonians had a
sexagesimal system and used the concept of place value to write numbers
larger than 60. So they had 59 symbols for the numbers 1-59, and then the
symbols were repeated in different columns for larger numbers. For
example, a '2' in the second column from the right meant (2 x 60)=120, and
a '2' in the column third from the right meant (2 x 602)=7200.
The numbers 1-59 are written below:
 To use the sexagesimal notation in modern language we separate the
'columns' by commas, so that the number
7267 = 2(602) + 1(60) + 7
would be written as 2,1,7.

There are some problems with this system. The first is that in practice there
is no way of separating the 'columns' except by a gap in the numbers, so '2'
looks very similar to 61 (=1,1). A more serious problem is that there was no
symbol for zero to put into an empty column, so '1' is indistinguishable from
'60 (=1,0). Generally we can work out what the numbers were from the
context of the probltaem, but this isn't exactly a very satisfactory way of
doing things. Later Babylonian civilisations did eventually invent a symbol for
zero, so obviously they were aware of this deficiency in their system too.

The base 60 number system of the Babylonians was successful enough to

have worked its way through time to appear in our present day modern
world. We still have 60 minutes in an hour, 60 seconds in a minute, 360
degrees in a circle and 60 minutes in a degree. Even our 24 hour clock is a
legacy from the ancient Babylonians.

Babylonian Number Tables

One aspect of Babylonian mathematics shared with the Egyptians is that of

making tables to ease the effort of calculations. They made tables of many
things which allowed them to develop their maths further than previous
civilisations, and to calculate things like square roots with as much accuracy
as mathematicians in the times of the Renaissance.

Reciprocal Tables

The Babylonians had no special algorithm for long division, and instead used
the fact that
a/b
= a x (1/b)
They created tables of reciprocals converted to sexagesimal notation. In the
notation introduced earlier, we can use a semi-colon to indicate a decimal
point. Then the number 1/2 would be written as (0;30)= 0(1)+30(60-1). Thus
division was a lot easier than the rather messy duplation method of the
Egyptians and made arithmetical calculations much easier to carry out.

60 is a useful base here because many numbers have finite base 60

fractions, e.g. 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12, 1/15 and 1/20. However,
some numbers (such as 1/7, 1/13) were infinite fractions, and only their
approximations were given. It is a shame the Babylonians did not consider
these numbers further, since they would have yielded periodically repeating
sexagesimal fractions that could have provoked investigation into infinite
series.

Tables of Squares

The Babylonian method of multiplication is quite ingenious and only relies on

knowing the squares of numbers. They used the formulas
ab = [(a+b)2 - a2 - b2]/2

ab = [(a+b)2 - (a-b)2]/4

for easy multiplication of two numbers. They didn't always use this method
though; sometimes it was just as simple to multiply and add, e.g. to multiply
by 39 you multiply by 30 and 9 and add the results together.

Square and Cube Roots

It is quite amazing to find, on an ancient stone tablet, a highly accurate

approximation to the square root of 2. In sexagesimal notation this is
(1;24,52,10) which in decimal is 1.41421296, and differs by about 0.0000006
from the true value. Accuracy in these kinds of computations was quite easy
with the fractional notation they had, and approximations to other square
roots were also given.
There are two possible methods of approximating square roots. [Here I shall
call the square root of x sqrt(x) because I can't find a way of getting the
square root symbol to work on these web pages!] The first of these uses the
approximation
sqrt(a2 + b) approx= a + b/2a
which is derived from the first few terms of the expansion of the binomial
series.

The second method uses an algorithm which was later ascribed to the
Greeks.
Let a = a1 be an initial approximation. If a1 < sqrt(2) then 2/a1 > sqrt(2). So
as a better approximation take a2 = (a1 + 2/a1)/2. Repeat the process until
you have an answer as accurate as you want.

Quadratic Equations and the n3 + n2 table

One important table for Babylonian algebra was that of the values of n3 +
n2 for integer values of n from 1 to 30. These tables could be used to solve
cubic equations of the form
ax3 + bx2 = c
although note that the Babylonians would not have had this algebraic
notation.
Multiplying through by a2/b3 gives:
(ax/b)3 + (ax/b)2 = ca2/b3
Putting y = ax/b gives us the equation
y3 + y2 = ca2/b3
which can be solved by looking up in the table to find the value of y and then
substituting back.
It is amazing that without the use of modern notation for these equations the
Babylonians could recognise equations of a certain type and the methods for
solving them.

It is hardly surprising then to find that the Babylonians were also proficient at
solving quadratic equations. If linear problems are found in their texts then
the answers are simply given without any working; these problems were
obviously thought too elementary for much attention. To solve quadratic
equations the Babylonians used a method equivalent to using our quadratic
formula. Many quadratics are arrived at from considering simultaneous
equations such as x+y=p, xy=q, which yields the quadratic x2 + q = px.
The Babylonians could even reduce equations of the form ax2 + bx = c to
the normal form y2 + by = ac using the substitution y = ax, which is
quite astounding given that they had no formal algebraic system.

Exponentials and Logarithms

By now, the mathematical achievements of the Babylonians should have
impressed you enough so that you won't be surprised by yet more
remarkable things. Ancient tablets have been found listing successive
powers of numbers. The question was then asked in a problem text, to what
power must a certain number be raised to yield a given number, i.e. the
logarithm to a certain base. However, 'logarithm tables' were not used for
general calculation but were only used to solve specific problems.

Pythagorean Triples

The Plimpton Tablet pictured to the left dates from about 1700BC, and
although a large chip has been broken off it the numbers contained in the
table are still recognisable as Pythagorean Triples. A Pythagorean triple
consists of three integers which satisfy the equation a2 + b2 = c2.
http://www.bath.ac.uk/~ma2jc/babylonian.html

OTHER LINK

<- Virtual Exhibitions in Informatics

The Quadratic Formula x = −b ± √ b2 - 4ac ____________ 2a is quite complicated. You may wonder how
people used to solve quadratic equations before they had this formula, and how they discovered the
Quadratic Formula in the fi rst place. Here is some of the history. What Problem First Led to Quadratics?
Our knowledge of ancient civilizations is based only on what survives today. The earliest known
problems that led to quadratic equations are on Babylonian tablets dating from 1700 BCE. In these
problems, the Babylonians were trying to fi nd two numbers x and y that satisfy the system ⎧ ⎨ ⎩ x + y =
b xy = c . This suggests that some Babylonians were interested in fi nding the dimensions x and y of a
rectangle with a given area c and a given perimeter 2b. The historian Victor Katz suggests that maybe
there were some people who believed that if you knew the area of a rectangle, then you knew its
perimeter. In solving these problems, these Babylonians may have been trying to show that many
rectangles with different dimensions have the same area. Example Find the dimensions of a rectangular
fi eld whose perimeter is 300 meters and whose area is 4,400 square meters. Solution Let L and W be
the length and width of this rectangle. Then ⎧ ⎨ ⎩ ? + ? = 300 ? · ? = 4,400 . This system can be solved
by substitution. First solve the top equation for W. ? = 300 - ? W = 150 - ? BIG IDEA The Quadratic
Formula can be proved using the properties of numbers and operations. (continued on next page)
GUIDED Use the graph below to fi nd each length. 1 ļ2 ļ1 ļ3 ļ4 2 3 4 ļ4 1 2 3 4 ļ3 ļ2 ļ1 y x A E D B C a. ED b.
CD c. BC d. AD Mental Math SMP08ALG_NA_SE2_C13_L04.indd 795 6/5/07 9:34:33 AM 796 Using
Algebra to Prove Now, substitute ? for W in the second equation. L( ? ) = 4,400 This is a quadratic
equation and so it can be solved by using either the Quadratic Formula or factoring to get L = ? or L = ? .
Now substitute these values for L in either of the original equations to get W = ? or W = ? . So, the
dimensions of the fi eld are ? m by ? m. How the Babylonians Solved Quadratics The Babylonians, like
the Greeks who came after them, used a geometric approach to solve problems like these. Using today’s
algebraic language and notation, here is what they did. It is a sneaky way to solve this sort of problem.
Look back at the Example. Because L + W = 150, the average of L and W is 75. This means that L is as
much greater than 75 as W is less than 75. So let L = 75 + x and W = 75 - x. Substitute these values into
the second equation. L · W = 4,400 (75 + x)(75 - x) = 4,400 5,625 - x2 = 4,400 x2 = 1,225 Taking the
square root, x = 35 or x = −35. If x = 35: L = 75 + x, so L = 75 + 35 = 110 W = 75 - x, so W = 75 - 35 = 40 If x
= −35: L = 75 + −35 = 40 W = 75 - −35 = 110 Either solution tells us that the fi eld is 40 meters by 110
meters. QY Notice what the Babylonians did. They took a complicated quadratic equation and, with a
clever substitution, reduced it to an equation of the form x2 = k. That equation is easy to solve. Then
they substituted the solution back into the original equation. Chapter 13 This tablet contains 14 lines of
a mathematical text in cuneiform script and a geometric design. Source: Iraq Museum QY Use the
Babylonian method to fi nd two numbers whose sum is 72 and whose product is 1,007. (Hint: Let one of
the numbers be 36 + x, the other 36 - x.) SMP08ALG_NA_SE2_C13_L04.indd 796 6/5/07 9:34:52 AM A
History and Proof of the Quadratic Formula 797 The Work of Al-Khwarizmi The work of the Babylonians
was lost for many years. In 825 CE, about 2,500 years after the Babylonian tablets were created, a
general method that is similar to today’s Quadratic Formula was authored by the Arab mathematician
Muhammad bin Musa al-Khwarizmi in a book titled Hisab al-jabr w’al-muqabala. Al-Khwarizmi’s
techniques were more general than those of the Babylonians. He gave a method to solve any equation
of the form ax2 + bx = c, where a, b, and c are positive numbers. His book was very infl uential. The word
“al-jabr” in the title of his book led to our modern word “algebra.” Our word “algorithm” comes from al-
Khwarizmi’s name. A Proof of the Quadratic Formula Neither the Babylonians nor al-Khwarizmi worked
with an equation of the form ax2 + bx + c = 0, because they considered only positive numbers, and if a,
b, and c are positive, this equation has no positive solutions. In 1545, a Renaissance scientist, Girolamo
Cardano, blended al-Khwarizmi’s solution with geometry to solve quadratic equations. He allowed
negative solutions and even square roots of negative numbers that gave rise to complex numbers, a
topic you will study in Advanced Algebra. In 1637, René Descartes published La Géometrie that
contained the Quadratic Formula in the form we use today. Now we prove why the formula works.
Examine the argument in the following steps closely. See how each equation follows from the preceding
equation. The idea is quite similar to the one used by the Babylonians, but a little more general. We
work with the equation ax 2 + bx + c = 0 until the left side is a perfect square. Then the equation has the
form t 2 = k, which you know how to solve for t. Given the quadratic equation: ax 2 + bx + c = 0 with a ≠
0. We know a ≠ 0 because otherwise the equation is not a quadratic equation. Step 1 Multiply both sides
of the equation by __ 1 a . This makes the left term equal to x2 + __b a x + _c a . The right side remains 0
because 0 · __ 1 a = 0. x2 + __b a x + _c a = 0 Step 2 Add − _c a to both sides in preparation for
completing the square on the left side. x2 + __b ax = −_c a Lesson 13-4 Muhammad bin Musa al-
Khwarizmi SMP08ALG_SE2_C13_L04.indd 797 2/23/07 5:42:00 PM 798 Using Algebra to Prove Step 3 To
complete the square add the square of half the coeffi cient of x to both sides. (See Lesson 12-2.) x2 + _b
a x + ( __b 2a) 2 = − _c a + ( __b 2a) 2 Step 4 The left side is now the square of a binomial. (x + __b 2a) 2 =
− _c a + ( __b 2a) 2 Step 5 Take the power of the fraction to eliminate parentheses on the right side. (x +
__b 2a) 2 = − _c a + b2 ___ 4a2 Step 6 To add the fractions on the right side, fi nd a common
denominator. (x + __b 2a) 2 = − ___ 4ac 4a2 + b2 ___ 4a2 Step 7 Add the fractions. (x + __b 2a) 2 = b
_______ 2 - 4ac 4a2 Step 8 Now the equation has the form t 2 = k, with t = x + __b 2a and k = b _______
2 - 4ac 4a2 . This is where the discriminant b2 - 4ac becomes important. If b2 - 4ac ≥ 0, then there are
real solutions. They are found by taking the square roots of both sides. x + __b 2a = ± b _______ 2 - 4ac
4a2 Step 9 The square root of a quotient is the quotient of the square roots. x + __b 2a = ± √ b ________
2 - 4ac 2a Step 10 This is beginning to look like the formula. Add − __b 2a to each side. x = − __b 2a ± √ b
________ 2 - 4ac 2a Step 11 Adding the fractions results in the Quadratic Formula. x = −b ± √ b2 - 4ac
____________ 2a What if b2 - 4ac < 0? Then the quadratic equation has no real number solutions. The
formula still works, but you have to take square roots of negative numbers to get solutions. You will
study these nonreal solutions in a later course. Chapter 13 SMP08ALG_SE2_C13_L04.indd 798 2/21/07
8:21:36 AM A History and Proof of the Quadratic Formula 799 Questions COVERING THE IDEAS 1.
Multiple Choice The earliest known problems that led to the solving of quadratic equations were studied
about how many years ago? A 1,175 B 1,700 C 2,500 D 3,700 2. In what civilization do quadratic
equations fi rst seem to have been considered and solved? 3. What is the signifi cance of the work of al-
Khwarizmi in the history of the Quadratic Formula? In 4 and 5, suppose two numbers sum to 53 and
have a product of 612. Show your work in fi nding the numbers. 4. Use the Quadratic Formula. 5. Use
the Babylonian Method. 6. Suppose a rectangular room has a fl oor area of 54 square yards. Find two
different lengths and widths that this fl oor might have. In 7 and 8, suppose a rectangular room has a fl
oor area of 144 square yards and that the perimeter of its fl oor is 50 yards. 7. Find its length and width
by solving a quadratic equation using the Quadratic Formula or factoring. 8. Find its length and width
using a more ancient method. 9. Find two numbers whose sum is 15 and whose product is 10. 10. In the
proof of the Quadratic Formula, each of Steps 1–11 tells what was done but does not name the property
of real numbers. For each step, name the property (or properties) from the following list. i. Addition
Property of Equality ii. Multiplication Property of Equality iii. Distributive Property of Multiplication over
Addition iv. Equal Fractions Property v. Power of a Quotient Property vi. Quotient of Square Roots
Property vii. Defi nition of square root Lesson 13-4 SMP08ALG_NA_SE2_C13_L04.indd 799 6/5/07
9:35:06 AM 800 Using Algebra to Prove APPLYING THE MATHEMATICS 11. Solve the equation 7x2 - 6x - 1
= 0 by following the steps in the derivation of the Quadratic Formula. 12. Explain why there are no real
numbers x and y whose sum is 10 and whose product is 60. 13. In a Chinese text that is thousands of
years old, the following problem is given: The height of a door is 6.8 more than its width. The distance
between its corners is 10. Find the height and width of the door. 14. Here is an alternate proof of the
Quadratic Formula. Tell what was done to get each step. ax2 + bx + c = 0 a. 4a2x2 + 4abx + 4ac = 0 b.
4a2x2 + 4abx + 4ac + b2 = b2 c. 4a2x2 + 4abx + b2 = b2 - 4ac d. (2ax + b)2 = b2 - 4ac e. 2ax + b = ± b2 -
4ac f. 2ax = −b ± b2 - 4ac g. x = −b ± b2 - 4ac _____________ 2a REVIEW 15. Consider the following
statement. (Lessons 13-2, 13-1) A number that is divisible by 8 is also divisible by 4. a. Write the
statement in if-then form. b. Decide whether the statement you wrote in Part a is true or false. If it is
false, fi nd a counterexample. c. Write the converse of the statement you wrote in Part a. d. Decide
whether the statement you wrote in Part c is true or false. If it is false, fi nd a counterexample. 16. Solve
x2 + 5x = 30. (Lesson 12-6) In 17–19, an open soup can has volume V = πr2h and surface area S = πr2 +
2πrh, where r is the radius and h is the height of the can. 17. Use common monomial factoring to
rewrite the formula for S. (Lesson 11-4) 18. Find each of the following. (Lesson 11-2) a. the degree of V
b. the degree of S Chapter 13 r h SMP08ALG_NA_SE2_C13_L04.indd 800 6/5/07 9:35:16 AM A History
and Proof of the Quadratic Formula 801 19. If the can has a diameter of 8 cm and a height of 12 cm,
about how many milliliters of soup can it hold? (1 L = 1,000 cm3) (Lesson 5-4) 20. Solve this system by
graphing. ⎧ ⎨ ⎩ y = ⎪x⎥ y = __1 4 x2 (Lessons 10-1, 4-9) 21. Skill Sequence Simplify each expression.
(Lessons 8-7, 8-6) a. √8 + √5 b. √8 · √5 c. _______ √8 · √5 √2 EXPLORATION 22. In a book or on the
Internet, research al-Khwarizmi and fi nd another contribution he made to mathematics or other
sciences. Write a paragraph about your fi ndings.

From Wikipedia, the free encyclopedia

The quadratic formula for the roots of the general quadratic

equation
In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation
that can be rearranged in standard form as

where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0.

If a = 0, then the equation is linear, not quadratic, as there is no   term. The numbers a, b,
and c are the coefficients of the equation and may be distinguished by calling them,
respectively, the quadratic coefficient, the linear coefficient and the constant or free term. [1]

The values of x that satisfy the equation are called solutions of the equation,
and roots or zeros of the expression on its left-hand side. A quadratic equation has at most
two solutions. If there is no real solution, there are two complex solutions. If there is only
one solution, one says that it is a double root. A quadratic equation always has two roots, if
complex roots are included and a double root is counted for two. A quadratic equation can
be factored into an equivalent equation

where r and s are the solutions for x. Completing the square on a quadratic equation in

standard form results in the quadratic formula, which expresses the solutions in terms of a, b,
and c. Solutions to problems that can be expressed in terms of quadratic equations were
known as early as 2000 BC.
Because the quadratic equation involves only one unknown, it is called "univariate". The
quadratic equation only contains powers of x that are non-negative integers, and therefore it
is a polynomial equation. In particular, it is a second-degree polynomial equation, since the
greatest power is two.

Solving the quadratic equation

Figure 1. Plots of quadratic function y = ax  + bx + c, varying each coefficient separately while the other
2

coefficients are fixed (at values a = 1, b = 0, c = 0)

A quadratic equation with real or complex coefficients has two solutions, called roots. These

two solutions may or may not be distinct, and they may or may not be real.
Factoring by inspection
It may be possible to express a quadratic equation ax  + bx + c = 0 as a product (px + q)
2

(rx + s) = 0. In some cases, it is possible, by simple inspection, to determine values

of p, q, r, and s that make the two forms equivalent to one another. If the quadratic equation
is written in the second form, then the "Zero Factor Property" states that the quadratic
equation is satisfied if px + q = 0 or rx + s = 0. Solving these two linear equations provides
the roots of the quadratic.
For most students, factoring by inspection is the first method of solving quadratic equations
to which they are exposed.  If one is given a quadratic equation in the form x  + bx + c = 0,
[2]:202–207 2

the sought factorization has the form (x + q)(x + s), and one has to find two
numbers q and s that add up to b and whose product is c (this is sometimes called "Vieta's
rule"  and is related to Vieta's formulas). As an example, x  + 5x + 6 factors as (x + 3)(x + 2).
[3] 2

The more general case where a does not equal 1 can require a considerable effort in trial and
error guess-and-check, assuming that it can be factored at all by inspection.
Except for special cases such as where b = 0 or c = 0, factoring by inspection only works for
quadratic equations that have rational roots. This means that the great majority of quadratic
equations that arise in practical applications cannot be solved by factoring by inspection. [2]:207

Completing the square

Main article: Completing the square
Figure 2. For the quadratic function y = x  − x − 2, the points where the graph crosses the x-axis, x = −1 and x =
2

2, are the solutions of the quadratic equation x  − x − 2 = 0.

2

The process of completing the square makes use of the algebraic identity

which represents a well-defined algorithm that can be used to solve any quadratic equation.

 Starting with a quadratic equation in standard form, ax  + bx + c = 0
[2]:207 2

1. Divide each side by a, the coefficient of the squared term.

2. Subtract the constant term c/a from both sides.
3. Add the square of one-half of b/a, the coefficient of x, to both sides. This "completes
the square", converting the left side into a perfect square.
4. Write the left side as a square and simplify the right side if necessary.
5. Produce two linear equations by equating the square root of the left side with the
positive and negative square roots of the right side.
6. Solve each of the two linear equations.
We illustrate use of this algorithm by solving 2x  + 4x − 4 = 0
2

The plus-minus symbol "±" indicates that both x = −1 + √3 and x = −1 − √3 are solutions of

the quadratic equation. [4]
Quadratic formula and its derivation
Main article: Quadratic formula
Completing the square can be used to derive a general formula for solving quadratic
equations, called the quadratic formula.  The mathematical proof will now be briefly
[5]

summarized.  It can easily be seen, by polynomial expansion, that the following equation is
[6]

equivalent to the quadratic equation:

Taking the square root of both sides, and isolating x, gives:

Some sources, particularly older ones, use alternative parameterizations of the quadratic
equation such as ax  + 2bx + c = 0 or ax  − 2bx + c = 0 ,  where b has a magnitude one half of
2 2 [7]

the more common one, possibly with opposite sign. These result in slightly different forms
for the solution, but are otherwise equivalent.
A number of alternative derivations can be found in the literature. These proofs are simpler
than the standard completing the square method, represent interesting applications of other
frequently used techniques in algebra, or offer insight into other areas of mathematics.
A lesser known quadratic formula, as used in Muller's method provides the same roots via
the equation

This can be deduced from the standard quadratic formula by Vieta's formulas, which assert
that the product of the roots is c/a.
One property of this form is that it yields one valid root when a = 0, while the other root
contains division by zero, because when a = 0, the quadratic equation becomes a linear
equation, which has one root. By contrast, in this case, the more common formula has a
division by zero for one root and an indeterminate form 0/0 for the other root. On the other
hand, when c = 0, the more common formula yields two correct roots whereas this form
yields the zero root and an indeterminate form 0/0.
Reduced quadratic equation
It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is
one. This is done by dividing both sides by a, which is always possible since a is non-zero.
This produces the reduced quadratic equation: [8]

where p = b/a and q = c/a. This monic equation has the same solutions as the original.

The quadratic formula for the solutions of the reduced quadratic equation, written in terms of
its coefficients, is:

or equivalently:
Discriminant

Figure 3. Discriminant signs

In the quadratic formula, the expression underneath the square root sign is called
the discriminant of the quadratic equation, and is often represented using an upper case D or
an upper case Greek delta: [9]

A quadratic equation with real coefficients can have either one or two distinct real roots, or
two distinct complex roots. In this case the discriminant determines the number and nature of
the roots. There are three cases:

 If the discriminant is positive, then there are two distinct roots

both of which are real numbers. For quadratic equations with rational coefficients, if

the discriminant is a square number, then the roots are rational—in other cases they
may be quadratic irrationals.
 If the discriminant is zero, then there is exactly one real root

sometimes called a repeated or double root.

 If the discriminant is negative, then there are no real roots. Rather, there are two distinct
(non-real) complex roots [10]

which are complex conjugates of each other. In these expressions i is the imaginary

unit.
Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if
and only if the discriminant is non-negative.

Geometric interpretation
Graph of y = ax  + bx + c, where a and the discriminant b  − 4ac are positive, with
2 2

 Roots and y-intercept in red

 Vertex and axis of symmetry in blue
 Focus and directrix in pink

Visualisation of the complex roots of y = ax  + bx + c: the parabola is rotated 180° about its vertex (orange).
2

Its x-intercepts are rotated 90° around their mid-point, and the Cartesian plane is interpreted as the complex
plane (green).[11]

The function f(x) = ax  + bx + c is a quadratic function.  The graph of any quadratic function
2 [12]

has the same general shape, which is called a parabola. The location and size of the parabola,
and how it opens, depend on the values of a, b, and c. As shown in Figure 1, if a > 0, the
parabola has a minimum point and opens upward. If a < 0, the parabola has a maximum
point and opens downward. The extreme point of the parabola, whether minimum or

maximum, corresponds to its vertex. The x-coordinate of the vertex will be located at  ,

and the y-coordinate of the vertex may be found by substituting this x-value into the
function. The y-intercept is located at the point (0, c).
The solutions of the quadratic equation ax  + bx + c = 0 correspond to the roots of the
2

function f(x) = ax  + bx + c, since they are the values of x for which f(x) = 0. As shown in
2

Figure 2, if a, b, and c are real numbers and the domain of f is the set of real numbers, then
the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis.
As shown in Figure 3, if the discriminant is positive, the graph touches the x-axis at two
points; if zero, the graph touches at one point; and if negative, the graph does not touch the x-
axis.
Quadratic factorization
The term

is a factor of the polynomial

if and only if r is a root of the quadratic equation

It follows from the quadratic formula that

In the special case b  = 4ac where the quadratic has only one distinct root (i.e. the
2

discriminant is zero), the quadratic polynomial can be factored as

Graphical solution

Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation 2x  + 4x − 4 = 0.
2

Although the display shows only five significant figures of accuracy, the retrieved value of xc is
0.732050807569, accurate to twelve significant figures.
A quadratic function without real root: y = (x − 5)  + 9. The "3" is the imaginary part of the x-intercept. The
2

real part is the x-coordinate of the vertex. Thus the roots are 5 ± 3i.

The solutions of the quadratic equation

may be deduced from the graph of the quadratic function

which is a parabola.
If the parabola intersects the x-axis in two points, there are two real roots, which are the x-
coordinates of these two points (also called x-intercept).
If the parabola is tangent to the x-axis, there is a double root, which is the x-coordinate of the
contact point between the graph and parabola.
If the parabola does not intersect the x-axis, there are two complex conjugate roots. Although
these roots cannot be visualized on the graph, their real and imaginary parts can be. [13]

Let h and k be respectively the x-coordinate and the y-coordinate of the vertex of the

parabola (that is the point with maximal or minimal y-coordinate. The quadratic function
may be rewritten

Let d be the distance between the point of y-coordinate 2k on the axis of the parabola, and a
point on the parabola with the same y-coordinate (see the figure; there are two such points,
which give the same distance, because of the symmetry of the parabola). Then the real part of
the roots is h, and their imaginary part are ±d. That is, the roots are

or in the case of the example of the figure

Avoiding loss of significance

Although the quadratic formula provides an exact solution, the result is not exact if real
numbers are approximated during the computation, as usual in numerical analysis, where real
numbers are approximated by floating point numbers (called "reals" in many programming
languages). In this context, the quadratic formula is not completely stable.
This occurs when the roots have different order of magnitude, or, equivalently,
when b  and b  − 4ac are close in magnitude. In this case, the subtraction of two nearly equal
2 2

numbers will cause loss of significance or catastrophic cancellation in the smaller root. To

avoid this, the root that is smaller in magnitude, r, can be computed as   where R is the
root that is bigger in magnitude.
A second form of cancellation can occur between the terms b  and 4ac of the discriminant,
2

that is when the two roots are very close. This can lead to loss of up to half of correct
significant figures in the roots. [7][14]

Examples and applications

The trajectory of the cliff jumper is parabolic because horizontal displacement is a linear function of

time  , while vertical displacement is a quadratic function of time  . As a result, the path follows

quadratic equation  , where   and   are horizontal and vertical components of the original
velocity, a is gravitational acceleration and h is original height. The a value should be considered negative
here, as its direction (downwards) is opposite to the height measurement (upwards).

The golden ratio is found as the positive solution of the quadratic equation 

The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas
—are quadratic equations in two variables.
Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as
large involves solving a quadratic equation.
The process of simplifying expressions involving the square root of an expression involving
the square root of another expression involves finding the two solutions of a quadratic
equation.
Descartes' theorem states that for every four kissing (mutually tangent) circles,
their radii satisfy a particular quadratic equation.
The equation given by Fuss' theorem, giving the relation among the radius of a bicentric
quadrilateral's inscribed circle, the radius of its circumscribed circle, and the distance
between the centers of those circles, can be expressed as a quadratic equation for which the
distance between the two circles' centers in terms of their radii is one of the solutions. The
other solution of the same equation in terms of the relevant radii gives the distance between
the circumscribed circle's center and the center of the excircle of an ex-tangential
quadrilateral.

History
Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets)
could solve problems relating the areas and sides of rectangles. There is evidence dating this
algorithm as far back as the Third Dynasty of Ur.  In modern notation, the problems
[15]

typically involved solving a pair of simultaneous equations of the form:

which is equivalent to the statement that x and y are the roots of the equation: [16]:86

The steps given by Babylonian scribes for solving the above rectangle problem, in terms
of x and y, were as follows:
1. Compute half of p.
2. Square the result.
3. Subtract q.
4. Find the (positive) square root using a table of squares.
5. Add together the results of steps (1) and (4) to give x.

In modern notation this means calculating  , which is equivalent to the modern

day quadratic formula for the larger real root (if any)   with a = 1, b = −p, and c = q.
Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece,
China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050
BC to 1650 BC), contains the solution to a two-term quadratic equation.  Babylonian
[17]

mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC

used geometric methods of dissection to solve quadratic equations with positive roots. [18]

 Rules for quadratic equations were given in The Nine Chapters on the Mathematical Art, a
[19]

Chinese treatise on mathematics.  These early geometric methods do not appear to have
[19][20]

had a general formula. Euclid, the Greek mathematician, produced a more abstract

geometrical method around 300 BC. With a purely geometric approach Pythagoras and
Euclid created a general procedure to find solutions of the quadratic equation. In his
work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but
giving only one root, even when both roots were positive. [21]

In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not
completely general) solution of the quadratic equation ax  + bx = c as follows: "To the
2

absolute number multiplied by four times the [coefficient of the] square, add the square of
the [coefficient of the] middle term; the square root of the same, less the [coefficient of the]
middle term, being divided by twice the [coefficient of the] square is the value."
(Brahmasphutasiddhanta, Colebrook translation, 1817, page 346)  This is equivalent to:
[16]:87
The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic
formula for solving quadratic equations, as well as quadratic indeterminate
equations (originally of type ax/c = y ). Muhammad ibn Musa al-
[clarification needed : this is linear, not quadratic]

Khwarizmi (Persia, 9th century), inspired by Brahmagupta,  developed a set of [original research?]

formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full
solution to the general quadratic equation, accepting one or two numerical answers for every
quadratic equation, while providing geometric proofs in the process.  He also described the [22]

method of completing the square and recognized that the discriminant must be positive, [22]

 which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th
[23]:230

century) who gave geometric figures to prove that if the discriminant is negative, a quadratic
equation has no solution.  While al-Khwarizmi himself did not accept negative solutions,
[23]:234

later Islamic mathematicians that succeeded him accepted negative solutions,  as well [22]:191

as irrational numbers as solutions.  Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in
[24]

particular was the first to accept irrational numbers (often in the form of a square root, cube
root or fourth root) as solutions to quadratic equations or as coefficients in an equation.  The [25]

9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.
[26]

The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the
first European book to include the full solution to the general quadratic equation.  His [27]

solution was largely based on Al-Khwarizmi's work.  The writing of the Chinese [22]

mathematician Yang Hui (1238–1298 AD) is the first known one in which quadratic

equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu
Yi.  By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The
[28]

quadratic formula covering all cases was first obtained by Simon Stevin in 1594.  In [29]

1637 René Descartes published La Géométrie containing the quadratic formula in the form

we know today.

Advanced topics
Alternative methods of root calculation
Vieta's formulas
Main article: Vieta's formulas

Figure 5. Graph of the difference between Vieta's approximation for the smaller of the two roots of the
quadratic equation x  + bx + c = 0 compared with the value calculated using the quadratic formula. Vieta's
2

approximation is inaccurate for small b but is accurate for large b. The direct evaluation using the quadratic
formula is accurate for small b with roots of comparable value but experiences loss of significance errors for
large b and widely spaced roots. The difference between Vieta's approximation versus the direct computation
reaches a minimum at the large dots, and rounding causes squiggles in the curves beyond this minimum.

Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients.

The roots   of the quadratic polynomial   satisfy

These results follow immediately from the relation:

which can be compared term by term with

The first formula above yields a convenient expression when graphing a quadratic function.
Since the graph is symmetric with respect to a vertical line through the vertex, when there are
two real roots the vertex's x-coordinate is located at the average of the roots (or intercepts).
Thus the x-coordinate of the vertex is given by the expression

The y-coordinate can be obtained by substituting the above result into the given quadratic
equation, giving

As a practical matter, Vieta's formulas provide a useful method for finding the roots of a
quadratic in the case where one root is much smaller than the other. If | x 2| << | x 1|, then x 
1 + x 2 ≈ x 1, and we have the estimate:

The second Vieta's formula then provides:

These formulas are much easier to evaluate than the quadratic formula under the condition of
one large and one small root, because the quadratic formula evaluates the small root as the
difference of two very nearly equal numbers (the case of large b), which causes round-off
error in a numerical evaluation. Figure 5 shows the difference between (i) a direct evaluation
using the quadratic formula (accurate when the roots are near each other in value) and (ii) an
evaluation based upon the above approximation of Vieta's formulas (accurate when the roots
are widely spaced). As the linear coefficient b increases, initially the quadratic formula is
accurate, and the approximate formula improves in accuracy, leading to a smaller difference
between the methods as b increases. However, at some point the quadratic formula begins to
lose accuracy because of round off error, while the approximate method continues to
improve. Consequently, the difference between the methods begins to increase as the
quadratic formula becomes worse and worse.
This situation arises commonly in amplifier design, where widely separated roots are desired
to ensure a stable operation (see step response).
Trigonometric solution
In the days before calculators, people would use mathematical tables—lists of numbers
showing the results of calculation with varying arguments—to simplify and speed up
computation. Tables of logarithms and trigonometric functions were common in math and
science textbooks. Specialized tables were published for applications such as astronomy,
celestial navigation and statistics. Methods of numerical approximation existed,
called prosthaphaeresis, that offered shortcuts around time-consuming operations such as
multiplication and taking powers and roots.  Astronomers, especially, were concerned with
[30]

methods that could speed up the long series of computations involved in celestial
mechanics calculations.
It is within this context that we may understand the development of means of solving
quadratic equations by the aid of trigonometric substitution. Consider the following alternate
form of the quadratic equation,

[1]   
where the sign of the ± symbol is chosen so that a and c may both be positive. By
substituting

[2]   
and then multiplying through by cos θ, we obtain
2

[3]   
Introducing functions of 2θ and rearranging, we obtain

[4]   

[5]   
where the subscripts n and p correspond, respectively, to the use of a negative or positive
sign in equation [1]. Substituting the two values of θn or θp found from
equations [4] or [5] into [2] gives the required roots of [1]. Complex roots occur in the
solution based on equation [5] if the absolute value of sin 2θp exceeds unity. The amount of
effort involved in solving quadratic equations using this mixed trigonometric and logarithmic
table look-up strategy was two-thirds the effort using logarithmic tables alone.  Calculating
[31]

complex roots would require using a different trigonometric form. [32]

To illustrate, let us assume we had available seven-place logarithm and trigonometric

tables, and wished to solve the following to six-significant-figure accuracy:

1. A seven-place lookup table might have only 100,000 entries, and computing
intermediate results to seven places would generally require interpolation between
adjacent entries.

2.
 3.

4.

5.

6.

7.  (rounded to six significant figures)

Solution for complex roots in polar coordinates

If the quadratic equation   with real coefficients has two complex roots—the case

where   requiring a and c to have the same sign as each other—then the solutions for the
roots can be expressed in polar form as [33]

where   and 
Geometric solution

Figure 6. Geometric solution of ax  + bx + c = 0 using Lill's method. Solutions are −AX1/SA, −AX2/SA
2

The quadratic equation may be solved geometrically in a number of ways. One way is
via Lill's method. The three coefficients a, b, c are drawn with right angles between them as
in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a
diameter. If this cuts the middle line AB of the three then the equation has a solution, and the
solutions are given by negative of the distance along this line from A divided by the first
coefficient a or SA. If a is 1 the coefficients may be read off directly. Thus the solutions in
the diagram are −AX1/SA and −AX2/SA. [34]
Carlyle circle of the quadratic equation x  − sx + p = 0.
2

The Carlyle circle, named after Thomas Carlyle, has the property that the solutions of the
quadratic equation are the horizontal coordinates of the intersections of the circle with
the horizontal axis.  Carlyle circles have been used to develop ruler-and-compass
[35]

constructions of regular polygons.
Generalization of quadratic equation
The formula and its derivation remain correct if the coefficients a, b and c are complex
numbers, or more generally members of any field whose characteristic is not 2. (In a field of
characteristic 2, the element 2a is zero and it is impossible to divide by it.)
The symbol

in the formula should be understood as "either of the two elements whose square is b  − 4ac,
2

if such elements exist". In some fields, some elements have no square roots and some have
two; only zero has just one square root, except in fields of characteristic 2. Even if a field
does not contain a square root of some number, there is always a quadratic extension
field which does, so the quadratic formula will always make sense as a formula in that
extension field.
Characteristic 2
In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not
hold. Consider the monic quadratic polynomial

over a field of characteristic 2. If b = 0, then the solution reduces to extracting a square root,
so the solution is

and there is only one root since

In summary,
See quadratic residue for more information about extracting square roots in finite fields.
In the case that b ≠ 0, there are two distinct roots, but if the polynomial is irreducible, they
cannot be expressed in terms of square roots of numbers in the coefficient field. Instead,
define the 2-root R(c) of c to be a root of the polynomial x  + x + c, an element of
2

the splitting field of that polynomial. One verifies that R(c) + 1 is also a root. In terms of the
2-root operation, the two roots of the (non-monic) quadratic ax  + bx + c are
2

and

For example, let a denote a multiplicative generator of the group of units of F4, the Galois
field of order four (thus a and a + 1 are roots of x  + x + 1 over F4. Because (a + 1)  = a, a +
2 2

1 is the unique solution of the quadratic equation x  + a = 0. On the other hand, the
2

polynomial x  + ax + 1 is irreducible over F4, but it splits over F16, where it has the two
2

roots ab and ab + a, where b is a root of x  + x + a in F16

2

Pythagorean theorem, the well-known geometric theorem that the

sum of the squares on the legs of a right triangle is equal to
the square on the hypotenuse (the side opposite the right angle)—or,
in familiar algebraic notation, a  + b  = c . Although the theorem has
2 2 2

long been associated with Greek mathematician-

philosopher Pythagoras (c. 570–500/490 BCE), it is actually far older.
Four Babylonian tablets from circa 1900–1600 BCE indicate some
knowledge of the theorem, with a very accurate calculation of
the square root of 2 (the length of the hypotenuse of a right triangle
with the length of both legs equal to 1) and lists of
special integers known as Pythagorean triples that satisfy it (e.g., 3, 4,
and 5; 3  + 4  = 5 , 9 + 16 = 25). The theorem is mentioned in the
2 2 2

Baudhayana Sulba-sutra of India, which was written between 800 and

400 BCE. Nevertheless, the theorem came to be credited to Pythagoras.
It is also proposition number 47 from Book I of Euclid’s Elements.
According to the Syrian historian Iamblichus (c. 250–330 CE),
Pythagoras was introduced to mathematics by Thales of Miletus and
his pupil Anaximander. In any case, it is known that Pythagoras
traveled to Egypt about 535 BCE to further his study, was captured
during an invasion in 525 BCE by Cambyses II of Persia and taken to
Babylon, and may possibly have visited India before returning to the
Mediterranean. Pythagoras soon settled in Croton (now Crotone, Italy)
and set up a school, or in modern terms a monastery
(see Pythagoreanism), where all members took strict vows of secrecy,
and all new mathematical results for several centuries were attributed
to his name. Thus, not only is the first proof of the theorem not
known, there is also some doubt that Pythagoras himself actually
proved the theorem that bears his name. Some scholars suggest that
the first proof was the one shown in the figure. It was probably
independently discovered in several different cultures.

Pythagorean theorem
Visual demonstration of the Pythagorean theorem. This may be the original proof of the
ancient theorem, which states that the sum of the squares on the sides of a right
triangle equals the square on the hypotenuse (a2 + b2 = c2). In the box on the left, the
green-shaded a2 and b2 represent the squares on the sides of any one of the identical
right triangles. On the right, the four triangles are rearranged, leaving c2, the square on
the hypotenuse, whose area by simple arithmetic equals the sum of a2 and b2. For the
proof to work, one must only see that c2 is indeed a square. This is done by
demonstrating that each of its angles must be 90 degrees, since all the angles of a
triangle must add up to 180 degrees.
Encyclopædia Britannica, Inc.
Book I of the Elements ends with Euclid’s famous “windmill” proof of
the Pythagorean theorem. (See Sidebar: Euclid’s Windmill.) Later in
Book VI of the Elements, Euclid delivers an even easier demonstration
using the proposition that the areas of similar triangles are
proportionate to the squares of their corresponding sides. Apparently,
Euclid invented the windmill proof so that he could place the
Pythagorean theorem as the capstone to Book I. He had not yet
demonstrated (as he would in Book V) that line lengths can be
manipulated in proportions as if they were commensurable numbers
(integers or ratios of integers). The problem he faced is explained in
the Sidebar: Incommensurables.

A great many different proofs and extensions of the Pythagorean

theorem have been invented. Taking extensions first, Euclid himself
showed in a theorem praised in antiquity that any symmetrical regular
figures drawn on the sides of a right triangle satisfy the Pythagorean
relationship: the figure drawn on the hypotenuse has an area equal to
the sum of the areas of the figures drawn on the legs. The semicircles
that define Hippocrates of Chios’s lunes are examples of such an
extension. (See Sidebar: Quadrature of the Lune.)

In the Nine Chapters on the Mathematical Procedures (or Nine

Chapters), compiled in the 1st century CE in China, several problems
are given, along with their solutions, that involve finding the length of
one of the sides of a right triangle when given the other two sides. In
the Commentary of Liu Hui, from the 3rd century, Liu Hui offered a
proof of the Pythagorean theorem that called for cutting up the
squares on the legs of the right triangle and rearranging them
(“tangram style”) to correspond to the square on the hypotenuse.
Although his original drawing does not survive, the next figure shows
a possible reconstruction.
 “tangram” proof of the Pythagorean theorem by Liu Hui
This is a reconstruction of the Chinese mathematician's proof (based on his written
instructions) that the sum of the squares on the sides of a right triangle equals the
square on the hypotenuse. One begins with a2 and b2, the squares on the sides of the
right triangle, and then cuts them into various shapes that can be rearranged to form c 2,
the square on the hypotenuse.
Encyclopædia Britannica, Inc.

he Pythagorean theorem has fascinated people for nearly 4,000 years;

there are now more than 300 different proofs, including ones by the
Greek mathematician Pappus of Alexandria (flourished c. 320 CE), the
Arab mathematician-physician Thābit ibn Qurrah (c. 836–901), the
Italian artist-inventor Leonardo da Vinci (1452–1519), and even U.S.
Pres. James Garfield (1831–81).

The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Erik
Gregersen, Senior Editor.
LEARN MORE in these related Britannica articles:

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 Euclidean geometry: Pythagorean theorem

For a triangle △ABC the Pythagorean theorem has two parts: (1) if ∠ACB is a right

angle, then a2 + b2 = c2; (2) if a2 + b2 = c2, then ∠AC…

history of Mesopotamia: The achievements of ancient Mesopotamia

…a highly commendable achievement that Pythagoras’ law (that the sum of the
squares on the two shorter sides of a right-angled triangle equals the square on the
longest side), even though it was never formulated, was being applied as early as the
18th century bce. Technical accomplishments were perfected in…


 mathematics: Geometric and algebraic problems

…(now commonly known as the Pythagorean theorem) more than a thousand years

before the Greeks used it.…

Euclid, Greek Eukleides, (flourished c. 300 BCE, Alexandria, Egypt),

the most prominent mathematician of Greco-Roman antiquity, best
known for his treatise on geometry, the Elements.

Life
Of Euclid’s life nothing is known except what the Greek
philosopher Proclus (c. 410–485 CE) reports in his “summary” of
famous Greek mathematicians. According to him, Euclid taught
at Alexandria in the time of Ptolemy I Soter, who reigned over Egypt
from 323 to 285 BCE. Medieval translators and editors often confused
him with the philosopher Eukleides of Megara, a contemporary
of Plato about a century before, and therefore called him Megarensis.
Proclus supported his date for Euclid by writing “Ptolemy once asked
Euclid if there was not a shorter road to geometry than through
the Elements, and Euclid replied that there was no royal road to
geometry.” Today few historians challenge the consensus that Euclid
was older than Archimedes (c. 290–212/211 BCE).

Sources And Contents Of The Elements

Euclid compiled his Elements from a number of works of earlier men.
Among these are Hippocrates of Chios (flourished c. 440 BCE), not to
be confused with the physician Hippocrates of Cos (c. 460–375 BCE).
The latest compiler before Euclid was Theudius, whose textbook was
used in the Academy and was probably the one used
by Aristotle (384–322 BCE). The older elements were at once
superseded by Euclid’s and then forgotten. For his subject matter
Euclid doubtless drew upon all his predecessors, but it is clear that the
whole design of his work was his own, culminating in the construction
of the five regular solids, now known as the Platonic solids.

A brief survey of the Elements belies a common belief that it concerns

only geometry. This misconception may be caused by reading no
further than Books I through IV, which cover elementary plane
geometry. Euclid understood that building a logical and rigorous
geometry (and mathematics) depends on the foundation—a
foundation that Euclid began in Book I with 23 definitions (such as “a
point is that which has no part” and “a line is a length without
breadth”), five unproved assumptions that Euclid called postulates
(now known as axioms), and five further unproved assumptions that
he called common notions. (See the table of Euclid’s 10 initial
assumptions.) Book I then proves elementary theorems
about triangles and parallelograms and ends with the Pythagorean
theorem. (For Euclid’s proof of the theorem, see Sidebar: Euclid’s
Windmill Proof.)

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Euclid's axioms

1 Given two points there is one straight line that joins them.

2 A straight line segment can be prolonged indefinitely.

3 A circle can be constructed when a point for its centre and a distance for its radius are given.
 Euclid's axioms

4 All right angles are equal.

If a straight line falling on two straight lines makes the interior angles on the same side less than two right
5 angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the
two right angles.

Euclid's common notions

6 Things equal to the same thing are equal.

7 If equals are added to equals, the wholes are equal.

8 If equals are subtracted from equals, the remainders are equal.

9 Things that coincide with one another are equal.

1
The whole is greater than a part.
0

The subject of Book II has been called geometric algebra because it

states algebraic identities as theorems about equivalent geometric
figures. Book II contains a construction of “the section,” the division of
a line into two parts such that the ratio of the larger to the smaller
segment is equal to the ratio of the original line to the larger segment.
(This division was renamed the golden section in the Renaissance after
artists and architects rediscovered its pleasing proportions.) Book II
also generalizes the Pythagorean theorem to arbitrary triangles, a
result that is equivalent to the law of cosines (see plane trigonometry).
Book III deals with properties of circles and Book IV with the
construction of regular polygons, in particular the pentagon.

Book V shifts from plane geometry to expound a general theory of

ratios and proportions that is attributed by Proclus (along with Book
XII) to Eudoxus of Cnidus (c. 395/390–342/337 BCE). While Book V
can be read independently of the rest of the Elements, its solution to
the problem of incommensurables (irrational numbers) is essential to
later books. In addition, it formed the foundation for a geometric
theory of numbers until an analytic theory developed in the late 19th
century. Book VI applies this theory of ratios to plane geometry,
mainly triangles and parallelograms, culminating in the “application
of areas,” a procedure for solving quadratic problems by geometric
means.

Books VII–IX contain elements of number theory, where number

(arithmos) means positive integers greater than 1. Beginning with 22
new definitions—such as unity, even, odd, and prime—these books
develop various properties of the positive integers. For instance, Book
VII describes a method, antanaresis (now known as the Euclidean
algorithm), for finding the greatest common divisor of two or more
numbers; Book VIII examines numbers in continued proportions, now
known as geometric sequences (such as ax, ax2, ax3, ax4…); and Book
IX proves that there are an infinite number of primes.

According to Proclus, Books X and XIII incorporate the work of the

Pythagorean Theaetetus (c. 417–369 BCE). Book X,
which comprises roughly one-fourth of the Elements, seems
disproportionate to the importance of its classification of
incommensurable lines and areas (although study of this book would
inspire Johannes Kepler [1571–1630] in his search for a cosmological
model).

Books XI–XIII examine three-dimensional figures, in

Greek stereometria. Book XI concerns the intersections of planes,
lines, and parallelepipeds (solids with parallel parallelograms as
opposite faces). Book XII applies Eudoxus’s method of exhaustion to
prove that the areas of circles are to one another as the squares of their
diameters and that the volumes of spheres are to one another as the
cubes of their diameters. Book XIII culminates with the construction
of the five regular Platonic solids (pyramid, cube, octahedron,
dodecahedron, icosahedron) in a given sphere, as displayed in
the animation.
 Platonic solids
These are the only geometric solids whose faces are composed of regular, identical
polygons. Placing the cursor on each figure will show it in animation.
Encyclopædia Britannica, Inc.

The unevenness of the several books and the varied mathematical

levels may give the impression that Euclid was but an editor
of treatises written by other mathematicians. To some extent this is
certainly true, although it is probably impossible to figure out which
parts are his own and which were adaptations from his predecessors.
Euclid’s contemporaries considered his work final and authoritative; if
more was to be said, it had to be as commentaries to the Elements.

Renditions Of The Elements
In ancient times, commentaries were written by Heron of
Alexandria (flourished 62 CE), Pappus of Alexandria (flourished c.
320 CE), Proclus, and Simplicius of Cilicia (flourished c. 530 CE). The
father of Hypatia, Theon of Alexandria (c. 335–405 CE), edited
the Elements with textual changes and some additions; his version
quickly drove other editions out of existence, and it remained the
Greek source for all subsequent Arabic and Latin translations until
1808, when an earlier edition was discovered in the Vatican.

The immense impact of the Elements on Islamic mathematics is

visible through the many translations into Arabic from the 9th century
forward, three of which must be mentioned: two by al-Ḥajjāj ibn Yūsuf
ibn Maṭar, first for the ʿAbbāsid caliph Hārūn al-Rashīd (ruled 786–
809) and again for the caliph al-Maʾmūn (ruled 813–833); and a third
by Isḥāq ibn Ḥunayn (died 910), son of Ḥunayn ibn Isḥāq (808–873),
which was revised by Thābit ibn Qurrah (c. 836–901) and again
by Naṣīr al-Dīn al-Ṭūsī (1201–74). Euclid first became known in
Europe through Latin translations of these versions.

The first extant Latin translation of the Elements was made about 1120

by Adelard of Bath, who obtained a copy of an Arabic version in Spain,
where he traveled while disguised as a Muslim student. Adelard also
composed an abridged version and an edition with commentary, thus
starting a Euclidean tradition of the greatest importance until
the Renaissance unearthed Greek manuscripts. Incontestably the best
Latin translation from Arabic was made by Gerard of Cremona (c.
1114–87) from the Isḥāq-Thābit versions.

The first direct translation from the Greek without an Arabic

intermediary was made by Bartolomeo Zamberti and published
in Vienna in Latin in 1505, and the editio princeps of the Greek text
was published in Basel in 1533 by Simon Grynaeus. The first English
translation of the Elements was by Sir Henry Billingsley in 1570. The
impact of this activity on European mathematics cannot be
exaggerated; the ideas and methods of Kepler, Pierre de
Fermat (1601–65), René Descartes (1596–1650), and Isaac
Newton (1642 [Old Style]–1727) were deeply rooted in, and
inconceivable without, Euclid’s Elements.

Other Writings
The Euclidean corpus falls into two groups: elementary geometry and
general mathematics. Although many of Euclid’s writings were
translated into Arabic in medieval times, works from both groups have
vanished. Extant in the first group is the Data (from the first Greek
word in the book, dedomena [“given”]), a disparate collection of 94
advanced geometric propositions that all take the following form:
given some item or property, then other items or properties are also
“given”—that is, they can be determined. Some of the propositions can
be viewed as geometry exercises to determine if a figure is
constructible by Euclidean means. On Divisions (of figures)—restored
and edited in 1915 from extant Arabic and Latin versions—deals with
problems of dividing a given figure by one or more straight lines into
various ratios to one another or to other given areas.

Four lost works in geometry are described in Greek sources and

attributed to Euclid. The purpose of the Pseudaria (“Fallacies”), says
Proclus, was to distinguish and to warn beginners against different
types of fallacies to which they might be susceptible in geometrical
reasoning. According to Pappus, the Porisms (“Corollaries”), in three
books, contained 171 propositions. Michel Chasles (1793–1880)
conjectured that the work contained propositions belonging to the
modern theory of transversals and to projective geometry. Like the
fate of earlier “Elements,” Euclid’s Conics, in four books, was
supplanted by a more thorough book on the conic sections with the
same title written by Apollonius of Perga (c. 262–190 BCE). Pappus
also mentioned the Surface-loci (in two books), whose subject can
only be inferred from the title.

Among Euclid’s extant works are the Optics, the first Greek treatise on

perspective, and the Phaenomena, an introduction to mathematical
astronomy. Those works are part of a corpus known as “the Little
Astronomy” that also includes the Moving Sphere by Autolycus of
Pitane.

Two treatises on music, the “Division of the Scale” (a

basically Pythagorean theory of music) and the “Introduction to
Harmony,” were once mistakenly thought to be from The Elements of
Music, a lost work attributed by Proclus to Euclid.

Legacy
Almost from the time of its writing, the Elements exerted a continuous
and major influence on human affairs. It was the primary source of
geometric reasoning, theorems, and methods at least until the advent
of non-Euclidean geometry in the 19th century. It is sometimes said
that, other than the Bible, the Elements is the most translated,
published, and studied of all the books produced in the Western world.
Euclid may not have been a first-class mathematician, but he set a
standard for deductive reasoning and geometric instruction that
persisted, practically unchanged, for more than 2,000 years.

Bartel Leendert van der WaerdenChristian Marinus Taisbak

Method of exhaustion, in mathematics, technique invented by the

classical Greeks to prove propositions regarding the areas and
volumes of geometric figures. Although it was a forerunner of
the integral calculus, the method of exhaustion used neither limits nor
arguments about infinitesimal quantities. It was instead a strictly
logical procedure, based upon the axiom that a given quantity can be
made smaller than another given quantity by successively halving it (a
finite number of times). From this axiom it can be shown, for example,
that the area of a circle is proportional to the square of its radius. The
term method of exhaustion was coined in Europe after the
Renaissance and applied to the rigorous Greek procedures as well as to
contemporary “proofs” of area formulas by “exhausting” the area of
figures with successive polygonal approximations.