Irrational & Real Numbers
โAn irrational number never ends and never repeats โ and no fraction can hold it.โ
The hook
Write โ2 as a decimal โ 1.41421356โฆ โ no end, no pattern. A number you can never write as a fraction: the arrival of the irrationals.
In plain words
Numbers you can write as a fraction (integer/integer) are rational; those you can't are irrational. Together they form the reals.
The intuition
Spread a number out as a decimal: if it stops (0.5) or repeats the same block (0.333โฆ), it's rational. If it never stops and never repeats, it's irrational โ like โ2, ฯ, the ratio of a circle's circumference.
How it's built
The reals split into two kinds โ rational (writable as a fraction) and irrational (not). Every point on the number line matches one real number, and the irrationals fill the gaps between the rationals.
Example
Is โ2 irrational? Since 1.4ยฒ=1.96 and 1.5ยฒ=2.25, โ2 sits between 1.4 and 1.5. No matter how many digits you add it never lands exactly, and it's been proven it can't be written as a fraction.
Common mistake
A โ sign doesn't automatically mean irrational. โ9=3 and โ16=4 are perfect squares, so they're rational. Irrationals come from roots that don't land, like โ2 and โ3.
Where it's used
A circle's circumference and area (ฯ), a diagonal's length (โ2), natural growth (e) โ measuring real shapes and nature exactly requires irrational numbers.
Where it came from
The Pythagoreans believed 'every number is a fraction', but Hippasus showed โ2 was not, shaking the school; legend says he was cast out for it.
Prerequisites
Quick check
Which of these is irrational?
- 0.5
- 1/3
- โ2โ
- โ4
Practice
Choose the irrational number.
Answer: 2
- 1.5 = 3/2 and 2/7 are fractions โ rational
- โ4 = 2 โ rational
- โ3 = 1.732โฆ never lands โ irrational
Takeaway: A non-perfect-square root is what makes an irrational.
Choose the rational number.
Answer: 2
- โ2, โ5, ฯ never stop or repeat โ irrational
- โ9 = 3 โ rational
Takeaway: When the inside is a perfect square, the root tidies up to a rational.
Is 0.777โฆ rational? Write why.
Answer: undefined
- The digit 7 repeats in a regular pattern (a repeating decimal)
- 0.777โฆ = 7/9, which is a fraction
- โ rational
Takeaway: A repeating decimal can be written as a fraction, so it's rational.
In one line, explain why โ2 can't be written as a fraction (why it's irrational).
Answer: undefined
- โ2 = 1.41421356โฆ โ the decimal never ends
- and no block of digits ever repeats
- so it can't be written as an integer/integer fraction
Takeaway: A decimal that never ends and never repeats = irrational.