Worked examples
The question
Question: is −5 rational or irrational?
−5
Rational means: it can be written as a fraction, with whole numbers on top and at the bottom.
Then its decimal stops, or repeats for ever.
Irrational means: no fraction works.
Its decimal never stops and never repeats.
rational = a fraction of whole numbers
decimal stops or repeats
1 Write it as a fraction
Any whole number can go over 1.
−5 is −5/1.
= −5 = −51
So −5 is rational.
Every whole number is rational, negative ones too.
= −5 is rational
2 Check
Let's check: −5 divided by 1 is −5.
The fraction is the same number.
So the answer is right!
−5 ÷ 1 = −5
Let's look back.
First: write it as a fraction.
Last: check.
▶ Watch it solved
The question
Question: is √20 rational or irrational?
√20
Rational means: it can be written as a fraction, with whole numbers on top and at the bottom.
Then its decimal stops, or repeats for ever.
Irrational means: no fraction works.
Its decimal never stops and never repeats.
rational = a fraction of whole numbers
decimal stops or repeats
1 Look for a square number
Square numbers are a number times itself.
3 times 3 is 9.
4 times 4 is 16.
5 times 5 is 25.
Squares: 3² = 9, 4² = 16, 5² = 25
20 is between 16 and 25.
So it is not a square number.
= 16 < 20 < 25
So √20 is not a whole number.
A calculator gives 4.4721, and so on.
The digits never stop and never repeat.
The square root of a whole number is either whole, or irrational.
= √20 = 4.4721360…
Careful!
4.47 is only the rounded decimal.
It is close, but not exact.
Careful: not √20 = 4.47 ✗
So √20 is irrational.
Its decimal never stops and never repeats.
= √20 is irrational
2 Check
Let's check: 4.47 times 4.47 is 19.9809, close to 20 but not 20.
More digits only get closer.
They never stop.
So the answer is right!
4.47 × 4.47 = 19.9809 ≠ 20
Let's look back.
First: look for a square number.
Last: check.
Answer = √20 is irrational
▶ Watch it solved
The question
Question: is 0.121212… rational or irrational?
0.121212...
Rational means: it can be written as a fraction, with whole numbers on top and at the bottom.
Then its decimal stops, or repeats for ever.
Irrational means: no fraction works.
Its decimal never stops and never repeats.
rational = a fraction of whole numbers
decimal stops or repeats
1 Look at the decimal
The block 1 2 repeats for ever.
A decimal that repeats is rational.
12 repeats for ever
Careful!
It goes on for ever, but it repeats.
Irrational decimals never repeat.
Careful: not 0.121212… is irrational ✗
2 Write it as a fraction
A repeating block of 2 digits goes over 99.
a block of 2 over 99
So 0.121212 repeating is 12/99, which makes simpler: 4/33.
= 0.121212… = 1299 = 433
So 0.121212… is rational.
It is a fraction of whole numbers.
= 0.121212... is rational
3 Check
Let's check: 4 divided by 33.
We get 0.121212, and so on.
The same decimal.
So the answer is right!
4 ÷ 33 = 0.121212…
Let's look back.
First: look at the decimal.
Then: write it as a fraction.
Last: check.
Answer = 0.121212... is rational
▶ Watch it solved
