Integers :
All the natural numbers, 0 and negatives of counting numbers i.e.,{....-3, -2, -1, 0, 1, 2, 3,....} together form the set of Integers. Integers can be divided into following three types. They are,
(i). Positive Integers : (1, 2, 3, 4...) is the set of all positive integers.
(ii).Negative Integers : (-1, -2, -3, -4,...) is the set of all negative integers.
(iii).Non-Positive and Non-Negative Integers : 0 is neither positive nor negative. So, {0, 1, 2, 3,...} represents the set of non-negative integers, while {0, -1, -2, -3 ,...}represents the set of non-positive integers.
Properties of Integers:
1.Identity Property:
(i). 0 + c = c + 0 = c
Example:
0 + (+5) = (+5) + 0 = +5
0 + (-2) = (-2) + 0 = -2 and
(ii). 1 x c = c x 1 = c
Example:
1 x (+5) = (+5) x 1= +5
1 x (-2) = (-2) x 1 = -2
2.Zero property:
0 x c = c x 0 = 0
Example:
0 x (+5) = (+5) x 0= 0
0 x (-2) = (-2) x 0 = 0
3.Commutative property:
(i). a + b = b+ a
Example:
(-7) + (+5) = (+5) +(-7)
(-2) = (-2) and
(ii). a x b = b x a
Example:
(-7) x (+5) = (+5) x (-7)
(-35) = (-35)
4.Associative property:
(i). (a + b) + c = a + (b + c)
Example:
( (-7) + (+5)) + (+3) = (-7) + ( (+5) + (+3))
(+1) = (+1) and
(ii). (i). (a x b) x c = a x (b x c)
Example:
( (-2) x (+2)) x (+3) = (-2) x ( (+2) x (+3))
(-12) = (-12)
5.Distributive property:
(a + b) x c = (a x c) x (b x c)
Example:
( (-2) + (+3)) x (+2) = ( (-2) x (+2) ) + ( (+3) x (+2))
(+1) x (+2) = (-4) + (6)
(+2) = (+2)
6.Inverse property:
(+c) + (-c) = 0
Example:
(-5) + (+5) = 0
The number of the form $\frac{p}{q}$ is called a rational number. where p & q are integers and
q < 0. The rational number comes from the word ratio.
Basic properties of rational numbers:
1. Closure property:
For rational numbers$\frac{a}{b}$ and $\frac{c}{d}$ , $\frac{a}{b}$ x $\frac{c}{d}$ is a unique rational number.
2. Identity property:
A unique rational number, 1, exists such that
1 x $\frac{a}{b}$ = $\frac{a}{b}$ x 1 = $\frac{a}{b}$ ; 1 is the multiplicative identity element.
3. Zero property:
For each rational number $\frac{a}{b}$, 0 x $\frac{a}{b}$ = $\frac{a}{b}$ x 0 = 0.
4. Commutative property:
For rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ , $\frac{a}{b}$ x $\frac{c}{d}$ = $\frac{c}{d}$ x $\frac{a}{b}$.
5. Associative property:
For rational numbers $\frac{a}{b}$ , $\frac{c}{d}$ and $\frac{e}{f}$ ,
($\frac{a}{b}$ x $\frac{c}{d}$) x $\frac{e}{f}$ = $\frac{a}{b}$ x ($\frac{c}{d}$ x $\frac{e}{f}$)
6. Distributive property:
For rational numbers $\frac{a}{b}$ , $\frac{c}{d}$ and $\frac{e}{f}$ ,
$\frac{a}{b}$ x ($\frac{c}{d}$ + $\frac{e}{f}$) = ($\frac{a}{b}$ x $\frac{c}{d}$) + ($\frac{a}{b}$ x $\frac{e}{f}$ ).
7. Multiplicative inverse:
For every non zero rational number $\frac{a}{b}$ ,a unique rational number, $\frac{b}{a}$,exists such that,
$\frac{a}{b}$ x $\frac{b}{a}$ = $\frac{b}{a}$ x $\frac{a}{b}$ = 1.
Example Problems
1.Prove the Multiplicative property of rational numbers with example values.
Solution:
By Multiplicative property:
$\frac{a}{b}$ x $\frac{b}{a}$ = $\frac{b}{a}$ x $\frac{a}{b}$ = 1.
Let us assume ,
a = 2, b= 3
Apply the values of a and b in the basic equation.
We get,
$\frac{a}{b}$ x $\frac{b}{a}$ = $\frac{b}{a}$ x $\frac{a}{b}$
$\frac{2}{3}$ x $\frac{3}{2}$ = $\frac{3}{2}$ x $\frac{2}{3}$
1 = 1.
2.Find a +($\frac{b}{c}$) ,if the values are a=2 ,b=4 and c=3.
Solution: Let a=2,b=4 and c=3
a +($\frac{b}{c}$) = 2 + ($\frac{4}{3}$)
= ( $\frac{2x3}{3}$ ) + ($\frac{4}{3}$)
= $\frac{6+4}{3}$
= $\frac{10}{3}$
So , a +($\frac{b}{c}$) = $\frac{10}{3}$
Related Tags
Integers ,Rational Numbers , Integers And Rational Numbers Examples