Real numbers are all the numbers that include both rational numbers and irrational numbers in the number system. They can be positive, negative, fractions, or decimals and are represented by R.
When arithmetic operations such as addition, subtraction, multiplication, and division are performed on real numbers, the result also follows certain rules depending on the types of numbers involved.
Rule 1: Operations on Two Rational Numbers
When arithmetic operations such as addition, subtraction, multiplication, or division are performed on two rational numbers, the result is always a rational number.
Examples:
- Addition: 0.25 + 0.25 = 0.50 = 50/100 (rational)
- Subtraction: 0.20 − 0.10 = 0.10 = 10/100 (rational)
- Multiplication: 0.4 × 184 = 73.6 = 736/10 (rational)
- Division: 0.252 ÷ 0.4 = 0.63 = 63/100 (rational)
Rule 2: Operations on Two Irrational Numbers
When arithmetic operations are performed on two irrational numbers, the result may be rational or irrational.
Examples:
- Addition: √2 + √3 (irrational)
- Subtraction: √3 − √3 = 0 (rational)
- Multiplication: √5 × √5 = 5 (rational), √3 × √5 = √15 (irrational)
- Division: √8 ÷ √8 = 1 (rational), √5 ÷ √3 (irrational)
Rule 3: Operations on a Rational Number and an Irrational Number
When arithmetic operations are performed between a rational number and an irrational number, the result is usually irrational. This happens because irrational numbers cannot be expressed as a fraction of integers, so combining them with rational numbers generally keeps the result irrational.
- Addition: 3 + √5 = 3 + √5 (irrational)
- Subtraction: 5√6 − 3 (irrational)
- Multiplication: 3 × √5 = 3√5 (irrational), except when the rational number is 0
- Division: 4 ÷ √2 = 2√2 (irrational)
Solved Examples
Example 1: Simplify (2√3 + √7) + (3√3 – 4√7)
= (2√3 + √7) + (3√3 – 4√7)
= 2√3 + √7 + 3√3 – 4√7
= 2√3 + 3√3 + √7 – 4√7
= 5√3 - 3√7
Example 2: Simplify (-√3) × (-4√3)
= (-√3) × (- 4√3)
= 4(√3)(√3)
= 4 × 3
= 12
Example 3: Simplify (9√5 / 3√5)
= (9√5 / 3√5)
= 9√5 / 3√5
= 3
Example 4: Simplify 34(√ 3) - √3(3 + √3)
= 34(√3) - √3(3 + √3)
= 34(√3) - 3√3 - 3
= 31√3 - 3
Example 5: Show that 7√7 is an irrational number.
Let us assume, to the contrary, that 7√7 is rational.
That is, we can find coprime a and b (b ≠ 0) such that 7√7 = ab
Rearranging, we get √7 = ab/7
Since 7, a and b are integers, ab/7 is rational, and so √7 is rational.
But this contradicts the fact that √7 is irrational.
So, we conclude that 7√7 is irrational.
Example 6: Explain why (17 × 5 × 13 × 3 × 7 + 7 × 13) is a composite number.
17 × 5 × 13 × 3 × 7 + 7 × 13 …(i)
= 7 × 13 (17 × 5 × 3 + 1)
= 7 × 13 (255 + 1)
= 7 × 13 × 256
Since the number has factors 7, 13, and 256, it has more than two factors.
Therefore, the number is composite.
Example 7: Prove that 3 + 2√3 is an irrational number.
Let us assume to the contrary, that 3 + 2√3 is rational.
So that we can find integers a and b (b ≠ 0).
Such that 3 + 2√3 = ab, where a and b are coprime.
Rearranging the equations, we get since a and b are integers, we get a2b−32 is rational and so √3 is rational.
But this contradicts the fact that √3 is irrational.
So we conclude that 3 + 2√3 is irrational.
