NUMBERS
What are Natural Numbers?
All whole numbers, with the exception of zero, are referred to as
natural numbers. In both our daily words and actions, these numerals are
frequently employed. Numbers are used in a variety of ways around the
world, including to count things, symbolize or exchange money, gauge
temperature, keep time, and more. They are referred to as "natural
numbers" since they are used to count items.
All positive numbers from 1 to infinity are considered natural
numbers and are therefore a component of the number system. Counting
numbers are another name for natural numbers.
Set of Natural Numbers
A set is a group of objects (numbers in this context). In mathematics, the
set of natural numbers is represented by the symbols {1,2,3,...}. The letter
N stands for the collection of natural numbers. N = {1,2,3,4,5,...∞}
Difference Between Natural Numbers and Whole Numbers
Natural numbers, such as 1, 2, 3, 4, and so on, are all positive numbers.
They are the typical numbers that you count, and they continue to infinity.
While all natural numbers, including 0 (for example, 1, 2, 3, 4, etc.), are
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considered whole numbers. All whole numbers and their opposites are
included in integers. For instance, -4, -3, -2, -1, 0, 1, 2, 3, 4, and so forth.
The distinction between a whole number and a natural number is
displayed in the table below.
Natural Number Whole Number
The set of natural numbers is N= The set of whole numbers is
{1,2,3,...∞} W={0,1,2,3,...}
The smallest whole number
The smallest natural number is 1.
is 0.
All natural numbers are whole Each whole number is a
numbers, but all whole numbers are natural number, except
not natural numbers. zero.
OPERATIONS with REAL NUMBERS
If a, b,c belong to the set of ℝ of real numbers, then
1.)a + b and ab belong to R Closure law
2.)a+b=b+a Commutative law of Addition
3.)a + (b+c) = (a+b) + c Associative law of Addition
4.)ab = ba Commutative law of Multiplication
5.)a(bc) = (ab)c Associative law of Multiplication
6.)a ( b + c) = ab + ac Distributive law
7.)a + 0 = 0 + a = a, 1(a) = (a)1 = a
Zero is called the identity with respect to addition, 1 is called the
identity with respect to multiplication.
8.) For any a there is a number x in ℝ such that x +a = 0
x is called the inverse of a with respect to addition and denoted by
-a
9.) For any a ≠ 0 there is a number x in ℝ such that ax = 1
x is called the inverse of a with respect to multiplication and is
denoted by a-1 or 1/a
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In general any set, such as ℝ , whose members satisfy the above is called
These enable us to operate according to the usual rules of algebra.
field.
INEQUALITIES
If a – b is a non – negative number a is greater than or equal to b or
b is less than or equal to a and write respectively a ≥ b or b ≤ a. If there is
no possibility that a = b, we write a > b or b < a. Geometrically, a > b if
the point on the real axis corresponding to a lies to the right of the point
corresponding to b.
Examples: 3 < 5 or 5 > 3 ; -2 < -1 or -1 > -2 ; x ≤ 3 means that x is a real
number which may be 3 or less than 3.
If a , b and c are any given real numbers, then:
1.) Either a > b, a = b or a < b Law of Trichotomy
2.) If a > b and b > c , then a > c Law of transitivity
3.) If a > b, then a + c > b + c
4.) If a > b and c > 0 , then ac > bc
5.) If a > b and c < 0, then ac < bc
ABSOLUTE VALUE OF REAL NUMBERS
The absolute value of a real number a, denoted by │a│, is defined
as a if a > 0, -a if a < 0, and 0 if a = 0.
Examples: │-5│= 5, │+2│= 2, │-1/2│= ½
1.) │ab│= │a││b│ or │abc…m│= │a││b││c│ … │m│
2.) │a + b│≤ │a│ + │b│ or │a + b + c + … + m│≤ │a│+ │b│+ │c│+ …
│m│
3.) │a – b│ ≥ │a│ - │b│
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The distance between any two points ( real numbers) a and b on
the real axis is │a - b│= │b - a│.
