Classify the following numbers as rational or irrational:
Find the explicit fractions in case they are rational.
- (i) √81
- (ii) √12
- (iii) 0.33333 …
- (iv) 0.123451234512345 …
- (v) 1.01001000100001 … (Notice the pattern: Is it repeating a single block?)
- (vi) 23.560185612239874790120
Step-by-step solution
Idea: A number is rational if it is a fraction of integers; its decimal then terminates or repeats. A decimal that never ends and never repeats a fixed block is irrational. Square roots of non-perfect squares are irrational.
(i) √81
- √81 = 9 = 91: rational.½ mark
(ii) √12
- √12 = √(4 × 3) = 2√3. If 2√3 = r were rational, √3 = r2 would be rational, but √3 is irrational. So √12 is irrational.½ mark
(iii) 0.33333 …
- 0.333… = 0.3: let x = 0.3, 10x = 3.3, 9x = 3, x = 13. Rational.½ mark
(iv) 0.123451234512345 …
- The block 12345 repeats: 0.12345. Let x = 0.12345; 100000x = 12345.12345; so 99999x = 12345.½ mark
- x = 1234599999 = 411533333 (divide by 3). Rational.½ mark
(v) 1.01001000100001 … (Notice the pattern: Is it repeating a single block?)
- The 1s are separated by one 0, then two 0s, then three, then four, …. The gaps keep growing, so no fixed block repeats, and the decimal never ends. Non-terminating and non-repeating: irrational.½ mark
(vi) 23.560185612239874790120
- The decimal stops, so it is terminating and therefore rational. Dropping the final 0, it has 20 decimal places: 23.56018561223987479012 = 23560185612239874790121020.½ mark
- Numerator and denominator are both divisible by 4 (and the numerator is not divisible by 5): 58900464030599686975325000000000000000000 in lowest terms.½ mark
Check: 4115 × 3 = 12345 and 33333 × 3 = 99999 ✓. 589004640305996869753 × 4 = 2356018561223987479012 and 25 × 1018 × 4 = 1020 ✓.
Answer to write in the exam
(i)
√81 = 9 = 91
∴ Rational
(ii)
√12 = 2√3
√3 irrational ⇒ 2√3 irrational
∴ Irrational
(iii)
x = 0.3, 10x = 3.3
9x = 3
∴ x = 13, rational
(iv)
x = 0.12345, 100000x = 12345.12345
99999x = 12345
∴ x = 1234599999 = 411533333, rational
(v)
Number of 0s between 1s: 1, 2, 3, 4, … (growing)
Non-terminating, no repeating block
∴ Irrational
(vi)
Terminating decimal ⇒ rational
= 23560185612239874790121020
∴ = 58900464030599686975325000000000000000000, rational
Common mistakes that cost marks
- Calling (vi) irrational because it is long and looks random. It stops, so it is rational.
- Calling (v) rational because it ‘has a pattern’. A pattern is not enough; it must repeat the same block for ever.
- Calling √12 rational because 12 is a whole number; 12 is not a perfect square.
How this can come in the exam
Which of the following is irrational?
- √0.09
- 0.21
- √50
- √18√2
Show answer
(C) √50
√0.09 = 0.3, 0.21 = 733, √18√2 = √9 = 3, but √50 = 5√2 is irrational.
Assertion (A): 2.101001000100001… is irrational.
Reason (R): A decimal that is non-terminating and non-repeating represents an irrational number.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
The zeros between the 1s keep increasing, so the decimal never repeats a block; R is the rule that makes A true.
Try one yourself
Classify √49, √20 and 0.272727… and give fractions where possible.
Show answer
√49 = 7 = 71 rational; √20 = 2√5 irrational; 0.27 = 2799 = 311 rational.
More questions like this
- The number 0.9 (which means 0.99999 … ) is a rational number. Using algebra (let x = 0.9, multiply by 10, and subtract), explain why 0.9 is exactly equal to 1.
- We have seen that the repeating block of 17 is a cyclic number. Try to find more numbers (n) whose reciprocals (1n) produce decimals with repeating blocks that are cyclic.
- Consider this puzzle: What is the square root of −1? We know that 1 × 1 = 1. We also know that (−1) × (−1) = 1. There is no Real Number that, when multiplied by itself, results in a negative number. Thus, √−1 cannot exist on number line.
- Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division:
- Prove that √5 is an irrational number.