A Rational Number is a number that can be written in the form of p/q where p, q are integers, and q ≠ 0. You can learn about the General Properties of Rational Numbers like Closure, Commutative, Associative, Distributive, Identity, Inverse, etc. here. Not just the regular properties we have all listed all the properties that we know regarding Rational Numbers.
- Closure Property
- Commutative Property
- Associative Property
- Distributive Property
- Identity and Inverse Properties of Rational Numbers
For two rational numbers x, y the addition, subtraction, multiplication results always yield a rational number. The Closure Property isn’t applicable for the division as division by zero isn’t defined. In other words, we can say that closure property is applicable for division too other than zero.
4/7 + 2/3 =26/21
4/3 – 2/4 = 6/12
3/5. 2/3 = 6/15
Considering two rational numbers x, y the addition and multiplication are always commutative. Subtraction doesn’t obey commutative property. You can get a clear idea of this property by having a look at the solved examples.
Commutative Law of Addition: x+y = y+x Ex: 1/3+2/3 = 3/3
Commutative Law of Multiplication: x.y = y.x Ex: 1/2.2/3 =2/3.1/2 =2/6
Subtraction x-y≠y-x Ex: 4/3-1/3 = 3/3 whereas 1/3-4/3=-3/3
Division isn’t commutative x/y ≠y/x Ex: 3/9÷1/2=6/9 whereas 1/2 ÷3/9 =9/6
Rational Numbers obey the Associative Property for Addition and Multiplication. Let us assume x, y, z to be three rational numbers then for Addition, x+(y+z)=(x+y)+z
whereas for Multiplication x(yz)=(xy)z
Ex: 1/3 + (1/4 + 3/3) = (1/3+ 1/4) + 3/3
Let us consider three rational numbers x, y, z then x . (y+z) = (x . y) + (x . z). We will prove the property by considering an example.
Ex: 1/3.(1/4+2/5) =(1/3.1/4)+(1/3.2/5)
Thus, L.H.S = R.H.S
Identity Property: We know 0 is called Additive Identity and 1 is called Multiplicative Identity of Rational Numbers.
Ex: 1/4+0 = 1/4(Additive Identity)
5/3.1 = 5/3(Multiplicative Identity)
Inverse Property: For a Rational Number x/y additive inverse is -x/y and multiplicative inverse is y/x.
Ex: Additive Inverse of 2/3 is -2/3
Multiplicative Inverse of 4/5 is 5/4
There are few other properties that you need to be aware of Rational Numbers and they are explained below.
If a/b is a rational number and m is a non-zero integer then a/b =(a*m)/(b*m).
In other words, we can say that the rational number remains unaltered if we multiply both the numerator and denominator with the same integer.
Ex: 2/3 = 2*2/3*2 = 4/6, 2*3/3*3 = 6/9, 2*4/3*4 = 8/12….
If a/b is a rational number and m is a common divisor then a/b = (a÷m)/(b÷m)
On dividing the numerator and denominator of a rational number with a common divisor the rational number remains unchanged.
Ex: 36/42 =36÷6/42÷6 = 6/7
Consider a/b, c/d to be two rational numbers.
Then a/b = c/d ⇒ a*d = b*c
Ex: 2/4 =4/8 ⇒ 2.8=4.4
For each and every Rational Number n, any of the following conditions hold true.
(i) n>0, (ii) n=0, (iii) n<0
Ex: 3/4 is greater than 0.
0/5 is equal to 0.
-3/4 is less than 0.
For any two rational numbers a, b any one condition is true
(i) a>b, (ii) a=b, (iii) a<b
Ex: 2/3 and 2/5 are two rational numbers and 2/3 is greater than 2/5
If 4/8 and 8/16 are two rational numbers then 4/8 = 8/16
If -4/7 and 3/4 are two rational numbers then -4/7 is less than 3/4
In the case of three rational numbers a > b, b > c then a>c
If 4/5, 16/30, -8/15 are three rational numbers then 4/5 >16/30 and 16/30 is greater than -8/15 then 4/5 is also greater than -8/15.