Number Theory and the

Real Number System

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 Number Theory Definitions
• The set of Natural Numbers = Set of counting numbers and is represented as N = {1,2,3,4,5...}

• The set of Whole Numbers = Set of natural numbers or counting numbers and zero and is
represented as W = {0,1,2,3,4,5,...}

• The set of Integers = Set of negative and positive natural or counting numbers and zero = Set of
negative and positive whole numbers and is represented as Z = {...-3,-2,-1,0,1,2,3...}

• The set of Rational Numbers = The set of numbers that CAN be written as fractions. Example -⅓, -
⅔, 0, ⅕ ⅖. Or the set of negative and positive fractions including zero. A rational number can
always be written as a decimal, whether terminating or recurring.

• The set of Irrational Numbers = The set of numbers that CANNOT be written as fractions. Example
-√3, √7, √5/4. Irrational numbers do not terminate or recur.

• The set of Real Numbers = The union of the set of rational and irrational numbers.
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 The Set Real Numbers

• The union of the rational numbers and the irrational numbers is the set of
real numbers.
• The sets that make up the real numbers are called subsets of the real
numbers.

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 Example 1: Classifying Real Numbers

Consider the following set of numbers:

 3 
−7, − , 0,0.6, 5,  , 7.3, 81 
 4 
List the numbers in the set that are
a. natural numbers b. whole numbers c. Integers
d. rational numbers e. irrational numbers f. real numbers

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 Example 1: Classifying Real Numbers (continued)

 3 
Solution: −7, − , 0,0.6, 5,  , 7.3, 81 
 4 
a. natural numbers 81 Because 81 = 9

because whole numbers include 0

b. whole numbers 0, 81
and the natural numbers
because integers include whole
c. integers 0, 81, -7 numbers and the negative natural
numbers
because these numbers can be
d. rational numbers 0, 81, -7, -¾, 0.6, & 7.3 expressed as a quotient or as a
terminating or repeating decimal
because neither terminate nor
e. irrational numbers 5 ,π have blocks of repeating digits

because real numbers have all

f. real numbers 0, 81, -7, -¾, 0.6, 5 , 7.3, & π
the above numbers as subsets

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 Properties of the Real Numbers

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 Properties of the Real Numbers

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 Properties of the Real Numbers

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 Example 2: Identifying Properties of Real
Numbers

Name the property illustrated:

a. 3  7 = 7  3 Commutative property of multiplication

b. (4 + 7) + 6 = 4 + (7 + 6) Associative property of addition

(
c. 2 3 + 5 = 6 + 2 5 ) Distributive property of
multiplication over addition

d. 17 + (−17) = 0 Inverse property of addition

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 Computation:

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 The Integers; Order of Operations

12
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 Order of Operations (BODMAS)

1. Perform all operations within Brackets.

2. Evaluate all exponential expressions.
3. Perform all operations that are Of Division.
4. Perform all operations that are Multiplications.
5. Finally, do all Additions and Subtractions in the order in which they
occur, working from left to right.

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 Example 6: Using the Order of Operations

Simplify 62 – 24 ÷ 22 · 3 + 1.

Solution: There are no brackets, thus, we begin by evaluating exponential

expressions.
62 – 24 ÷ 22 · 3 + 1 = 36 – 24 ÷ 4 · 3 + 1
= 36 – 6 · 3 + 1
= 36 – 18 + 1
= 18 + 1
= 19

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 Test yourself

Ed. 4 of the book (available in the library)

Ex. Set 5.2

Exercises 1-23, 31-35, 43-45, 53-66, 67-76, 81-92, 98

15
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 38 1/2 of it + 5 /4 x3 x3 +2 /8 x 25 + 26

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 5 + 25 ÷ 5 (-1 × -5) + 7 (-2) – 1

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 Properties of Exponents

Property Meaning Examples

The Product Rule When multiplying exponential 96 · 912 = 96 + 12
expressions with the same base,
bm · bn = bm + n = 918
add the exponents. Use this sum
as the exponent of the common
base.
The Power Rule When an exponential expression is (34)5 = 34·5 = 320
raised to a power, multiply the
(bm)n = bmn (53)8 = 53·8 = 524
exponents. Place the product of the
exponents on the base and remove
the parentheses.
The Quotient Rule When dividing exponential expressions
512
with the same base, subtract the
4
= 512 − 4 = 58
bm exponent in the denominator from the 5
m−n
= b exponent in the numerator. Use this 9 40
bn difference as the exponent of the 5
= 9 40 −5 = 935
common base. 9

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 Example 1: Using the Zero Exponent Rule

Use the zero exponent rule to simplify:

a. 70=1

b.  0 = 1

c. (−5)0 = 1

d. −50 = −1

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 The Negative Exponent Rule

• If b is any real number other than 0 and m is a

natural number, 1
b −m = m
.
b

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 Example 2: Using the Negative Exponent Rule

Use the negative exponent rule to simplify:

1 1 1
a. 8− 2 = = =
8 2
8  8 64

1 −3 1 1
b. 5 = 3 = =
5 5  5  5 125

1 1 −1
c. 7 = 1 =
7 7

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 Powers of Ten

1. A positive exponent tells how many zeros follow

the 1. For example, 109, is a 1 followed by 9 zeros:
1,000,000,000.
2. A negative exponent tells how many places there are
to the right of the decimal point. For example, 10-9 has
nine places to the right of the decimal point.
10^5= 100,000 10 -9 = 0.000000001
10^-6= 0.000001

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 Scientific Notation

A positive number is written in scientific notation when it is expressed in

the form a  10n , where a is a number greater than or equal to 1 and less
than 10 (1 ≤ a < 10), and n is an integer.

*Powers of 10
a= Number between 1 and 10 but <10
{1,2,3,4,5,6,7,8,9} equal to or greater than 1

b= Integer

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 Convert Scientific Notation to Decimal Notation

• If n is positive, move the decimal point in a to the

right n places.
• If n is negative, move the decimal point in a to the
left |n| places.

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 Example 3: Converting from Scientific to
Decimal Notation

Write each number in decimal notation:

a. 2.6  107 b. 1.1  10-4

Solution:

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 Converting From Decimal to Scientific
Notation

To write the number in the form a  10n:

• Determine a, the numerical factor. Move the decimal point in the given
number to obtain a number greater than or equal to 1 and less than 10.
• Determine n, the exponent on 10n. The absolute value of n is the number
of places the decimal point was moved. The exponent n is positive if the
given number is greater than or equal to 10 and negative if the given
number is between 0 and 1.

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 Example 4: Converting from Decimal
Notation to Scientific Notation

Write each number in scientific notation:

a. 4,600,000 b. 0.000023

Solution:

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 Computations with Scientific Notation

• We use the product rule for exponents to multiply

numbers in scientific notation:
(a 10n)  (b  10m) = (a  b)  10n+m
Add the exponents on 10 and multiply the other parts of
the numbers separately.

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 Example 6: Multiplying Numbers in Scientific
Notation

Multiply: (3.4  109)(2  10-5). Write the product in

decimal notation.

Solution: (3.4  109)(2  10-5) = (3.4  2)(109  10-5)

= 6.8  109+(-5)
= 6.8  104
= 68,000

0.34  108

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 Computations with Scientific Notation

• We use the quotient rule for exponents to divide numbers in scientific

notation:

a 10 n  a  n−m
=    10
b 10 m
b

Subtract the exponents on 10 and divide the other parts of the numbers
separately.

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 Example 7: Dividing Numbers In Scientific
Notation
−7
Divide: 8.4  10 . Write the quotient in decimal notation.
4 10 − 4

Solution:

 8.4  10 

−7 −7
8.4 10
−4
=    − 4  Regroup factors.
4 10  4   10 
− 7 − ( −4 )
= 2.110
Subtract the exponents.
= 2.110-3
= 0.0021 Write the quotient in decimal notation.
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 Example 9: The National Debt

As of December 2008, the national debt was $10.8

trillion, or 10.8  1012 dollars.
At that time, the COUNTRY population was
approximately 306,000,000, or
3.06  108.
If the national debt was evenly divided among every
individual in the COUNTRY, how much would each
citizen have to pay?
Solution: The amount each citizen would have to pay is
the total debt, 1.08  1013, divided among the number
of citizens, 3.06  108.
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 Example 9: The National Debt continued

1.08 1013  1.08   1013 

=  8 
3.06 10  3.06   10 
8

13−8
 0.353 10
= 0.353 105
= 3.53 104
= 35,300
Every citizen would have to pay approximately
$35,300 to the federal government to pay off the
national debt.
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