Real Numbers

Worked example

1. 1

   $7=\frac{7}{1}$ — Any integer can be written over 1. That makes it one integer over another, so 7 is rational as well as an integer.

2. 2

   $\sqrt{2}=1.41421356\dots$ — No fraction times itself makes exactly 2. The decimal never ends and never repeats, so $\sqrt{2}$ is irrational.

3. 3

   $\text{real numbers}=\text{rationals and irrationals together}$ — Every number is either rational or irrational, never both. Together the two groups fill every point on the number line.

Worked example

1. 1

   $6,11,15,24,27,35,49$ — The list. Test every number against one word at a time.

2. 2

   $\text{even: }6,24$ — They end in 6 and 4, so 2 divides them exactly. The rest end in 1, 5, 7 or 9, so they are odd.

3. 3

   $\text{multiple of }7:35=7\times 5,49=7\times 7$ — Both are in the 7 times table. The others leave a remainder when you divide by 7.

4. 4

   $\text{multiple of }3:6,15,24,27$ — Add the digits: 1 + 5 = 6, 2 + 4 = 6, 2 + 7 = 9. Each total is in the 3 times table, so the number is too.

5. 5

   $\text{multiple of }9:27$ — 2 + 7 = 9, which is in the 9 times table. 15 and 24 only reach 6, so they are not.

6. 6

   $\text{factor of }30:6,15$ — 30 ÷ 6 = 5 and 30 ÷ 15 = 2, with nothing left. 35 and 49 are bigger than 30, so they cannot be factors.

Worked example

1. 1

   $49=7\times 7$ — 7 times itself makes 49, so 49 is a square number. No other number in the list is a whole number times itself.

2. 2

   $27=3\times 3\times 3$ — 3 times itself, then times 3 again, makes 27, so 27 is a cube number.

3. 3

   $11=1\times 11$ — The only way to make 11 is 1 × 11. It has exactly two factors, so it is prime.

4. 4

   $15=3\times 5,35=5\times 7$ — They are odd, but they split into smaller numbers. That gives more than two factors, so they are not prime.

Worked example

1. 1

   $0.6=\frac{6}{10}=\frac{3}{5}$ — 0.6 means 6 tenths. The decimal stops, so it is a fraction: rational.

2. 2

   $0.333\dots =\frac{1}{3}$ — 1 ÷ 3 = 0.333… The 3 repeats for ever, and a repeating decimal can always be written as a fraction: rational.

3. 3

   $\sqrt{9}=3$ — 9 = 3 × 3 is a square number, so the root comes out as a whole number. $3=\frac{3}{1}$: rational.

4. 4

   $\sqrt{10}=3.16227\dots$ — 10 is not a square number, so the root does not come out. The decimal never ends or repeats: irrational.

Worked example

1. 1

   $\sqrt{121},3.14,-0.8,\sqrt{50},\frac{2}{9},64$ — The list. Before you test anything, find the true value of each number.

2. 2

   $\sqrt{121}=11$ — 11 × 11 = 121, so this is just 11. It is a natural number, and also a prime.

3. 3

   $\sqrt{121}\text{is not a square number}$ — It looks like one, but its value is 11, and 11 is not a whole number times itself. The square number here is 64 = 8 × 8.

4. 4

   $3.14=\frac{314}{100}$ — A decimal that stops is a fraction, so 3.14 is rational. It is only close to $\pi =3.14159\dots$, which never ends.

5. 5

   $-0.8=-\frac{4}{5}$ — The minus sign does not change the test. It is a fraction, so it is rational, but it is not an integer.

6. 6

   $\frac{2}{9}=0.222\dots$ — It is already a fraction of two integers, so it is rational. The decimal repeats for ever, but that does not make it irrational.

7. 7

   $\sqrt{50}=7.0710678\dots$ — 50 is not a square number, so the root does not come out. The decimal never ends or repeats: this is the irrational one.