Irrational Numbers

Worked example

1. 1

   $\sqrt{7}=2.6457513\dots$ — 7 is not a square number, so no number times itself makes exactly 7. The decimal never ends or repeats: irrational.

2. 2

   $\pi =3.1415926\dots$ — Mathematicians have proved that the digits of $\pi$ never end and never fall into a repeating pattern: irrational.

3. 3

   $\frac{22}{7}=3.142857142857\dots$ — Divide 22 by 7. The block 142857 repeats for ever. It is one integer over another, so it is rational.

4. 4

   $\sqrt{\frac{16}{9}}=\frac{\sqrt{16}}{\sqrt{9}}=\frac{4}{3}$ — 16 and 9 are both square numbers, so the root of the top and of the bottom comes out. A fraction is left: rational.

Worked example

1. 1

   $125^{\frac{1}{3}}=\sqrt[3]{125}$ — A power of $\frac{1}{3}$ means cube root. Turn it back into a root so you can work it out.

2. 2

   $\sqrt[3]{125}=5$ — 5 × 5 × 5 = 125, so the root comes out as a whole number: rational.

3. 3

   $3^0=1$ — Any number to the power 0 is 1, a whole number: rational.

4. 4

   $4^{-1}=\frac{1}{4}$ — A negative power means 1 over the number, so this is a fraction: rational.

Worked example

1. 1

   $66\frac{2}{3}\%=\frac{2}{3}$ — A percentage means out of 100, so it can always be written as a fraction: rational.

2. 2

   $4.2\times 10^{-5}=0.000042$ — Standard form is just a decimal that stops: rational.

3. 3

   $0.4=\frac{4}{10}=\frac{2}{5}$ — To test the root of a decimal, first write the decimal as a fraction in its simplest form.

4. 4

   $\sqrt{\frac{2}{5}}=0.6324555\dots$ — 2 and 5 are not square numbers, so the root does not come out. The decimal never ends: irrational.

Worked example

1. 1

   $5^2=25,6^2=36$ — Square both ends. Squaring keeps the order, so a root between 5 and 6 must come from a number between 25 and 36.

2. 2

   $25<30<36$ — Pick a whole number strictly between 25 and 36. 30 is not a square number, so its root will not come out.

3. 3

   $\sqrt{30}=5.477\dots$ — Write $\sqrt{30}$ as your answer. It is between 5 and 6, and its decimal never ends or repeats, so it is irrational.

Worked example

1. 1

   $6.5^2=42.25,7.5^2=56.25$ — Square both ends, exactly as before. They are decimals, so work carefully: 6.5 × 6.5 = 42.25.

2. 2

   $42.25<50<56.25$ — Pick a whole number between them that is not square. Avoid 49, because 49 = 7 × 7. 50 is fine.

3. 3

   $\sqrt{50}=7.071\dots$ — It is between 6.5 and 7.5, and 50 is not square, so $\sqrt{50}$ is irrational.

Worked example

1. 1

   $\sqrt{5}\times \sqrt{5}=5$ — A square root times itself gives back the number under it. 5 is a whole number: rational.

2. 2

   $\sqrt{2}\times \sqrt{8}=\sqrt{16}$ — Multiply the numbers under the roots: 2 × 8 = 16, under one root sign.

3. 3

   $\sqrt{16}=4$ — 16 is a square number, so the root comes out: rational.

4. 4

   $\frac{6\pi }{2\pi }=3$ — $\pi$ is on the top and the bottom, so it cancels, leaving 6 ÷ 2 = 3: rational.

5. 5

   $(4+\sqrt{3})+(4-\sqrt{3})=8$ — The $+\sqrt{3}$ and the −√3 cancel each other, leaving 4 + 4 = 8: rational.

6. 6

   $2+\sqrt{3}=3.732\dots$ — Here nothing cancels the $\sqrt{3}$. Adding 2 only moves it along, so the decimal still never ends: irrational.