Rational and Irrational Numbers

Rational numbers

A rational number is a number that can be written as one whole number divided by another: a fraction such as $\frac{3}{4}$3/4 or $\frac{1}{3}$1/3. Whole numbers are rational too, because 5 can be written as $\frac{5}{1}$5/1.

Every fraction has a decimal, and you find it by dividing. $\frac{3}{4}=3÷4=0.75$3/4 = 3 ÷ 4 = 0.75, and the division stops. $\frac{1}{3}=1÷3=0.333\dots$1/3 = 1 ÷ 3 = 0.333…, and the division never stops: the digit 3 repeats forever.

Divide 3 by 4 one decimal place at a time. The remainders are 3, then 2, then 0, and at 0 the division stops: $\frac{3}{4}=0.75$3/4 = 0.75.

Divide 1 by 3 and the remainder is 1 at every step, so the digit 3 comes out at every step: $\frac{1}{3}=0.333\dots$1/3 = 0.333…

Why a fraction’s decimal stops or repeats

Dividing by 7 can only leave a remainder of 0, 1, 2, 3, 4, 5 or 6. If the remainder is ever 0, the decimal stops. If it never is, there are only six other remainders, so within six steps one of them must come round again. From that moment the same digits come out in the same order, forever.

So the decimal of every fraction either stops or repeats. $\frac{1}{7}=0.142857142857\dots$1/7 = 0.142857142857…, and the block 142857 repeats without end.

The remainders of 1 ÷ 7 are 1, 3, 2, 6, 4 and 5, and then 1 again, so the digits 142857 repeat.

From the decimal back to the fraction

The journey also runs backwards. A decimal that stops is a number of tenths, hundredths or thousandths: $0.75=\frac{75}{100}$0.75 = 75/100, which simplifies to $\frac{3}{4}$3/4.

A decimal that repeats can be turned back into a fraction too. Call the number x, so x = 0.333… with the 3 repeating forever. Multiply by 10 to get 10x = 3.333…, with the same 3s after the point. Now subtract x from 10x: every repeating 3 after the point cancels, and 10x − x = 3, so 9x = 3 and $x=\frac{3}{9}=\frac{1}{3}$x = 3/9 = 1/3.

So a number is rational exactly when its decimal stops or repeats.

Irrational numbers

Some numbers cannot be written as a fraction at all, and they are called irrational. The best-known is $\sqrt{2}$√2, the positive number that multiplies by itself to make 2.

Fractions can get close to $\sqrt{2}$√2. ${\left(\frac{7}{5}\right)}^2=\frac{49}{25}=1.96$(7/5)² = 49/25 = 1.96, which is just under 2, and ${\left(\frac{3}{2}\right)}^2=\frac{9}{4}=2.25$(3/2)² = 9/4 = 2.25, which is over 2, so $\sqrt{2}$√2 lies between $\frac{7}{5}$7/5 and $\frac{3}{2}$3/2. Closer fractions squeeze nearer to it from both sides, but a proof shows that no fraction squares to exactly 2.

Its decimal begins 1.41421356… and never stops or repeats. It cannot, because a decimal that stopped or repeated would turn back into a fraction.

$\sqrt{2}$√2 lies between $\frac{7}{5}=1.4$7/5 = 1.4 and $\frac{3}{2}=1.5$3/2 = 1.5, much nearer to 1.4. Its decimal begins 1.41421…

$\pi$π is irrational too

$\pi$π is the circumference of a circle divided by its diameter. It is irrational: its decimal begins 3.14159… and never stops or repeats.

The fraction $\frac{22}{7}$22/7 is a good approximation to $\pi$π, but it is not equal to $\pi$π. $\frac{22}{7}=3.142857\dots$22/7 = 3.142857…, which already differs from $\pi$π in the third decimal place. $\frac{22}{7}$22/7 is a fraction, so it is rational, and $\pi$π is not.

Roll a wheel whose diameter is 1 through one full turn. The distance it rolls is its circumference, which is $\pi =3.14159\dots$π = 3.14159…, a little more than 3.

Between 3.14 and 3.145, $\pi$π and $\frac{22}{7}$22/7 are close together, but they are two different points on the line.

The usual mistakes

A root sign alone does not make a number irrational. $\sqrt{49}=7$√49 = 7 is a whole number, and the square root of $\frac{9}{4}$9/4 is $\frac{3}{2}$3/2, a fraction. Both are rational.

A long decimal on a calculator screen does not prove that a number is irrational either, because the screen stops after a few digits. $\frac{1}{7}$1/7 fills the screen with digits and is still a fraction. What decides it is whether the number can be written exactly as one whole number divided by another.