What is a surd — irrational roots explained — Mathematics

Before this

Squares and square roots · coming soon
Converting decimals to fractions · coming soon
Pythagoras' theorem · coming soon

This leads to

Simplifying surds
Proof by contradiction · coming soon

A square root that will not come out

Start with something you already know. The square root of 9 is 3, because $3 \times 3 = 9$ . The square root of 16 is 4, because $4 \times 4 = 16$ . The square root of 100 is 10. Each answer is a whole number, and you can check each one by multiplying.

Now ask the same question about 2. Which number, multiplied by itself, gives 2?

It is not 1, because $1 \times 1 = 1$ . It is not 2, because $2 \times 2 = 4$ . So the answer lies somewhere between 1 and 2. Type √2 into a calculator, and the screen shows something like this:

2 \approx 1.414213562

That looks like an answer. So check it the way you checked the others. Multiply it by itself:

1.414213562 \times 1.414213562 = 1.999999998944 \dots

That is very close to 2. It is not 2. The calculator did not give you the square root of 2. It gave you a decimal that stops after nine places, because the screen has no more room. A computer with more room shows more digits. They never stop, and no block of digits ever starts to repeat.

So here is the question this lecture answers: what kind of number is √2? It sounds like a small question. The answer surprised the best mathematicians of the ancient world, and it is still one of the most important facts about numbers.

Two kinds of number

To answer it, we need to sort numbers into two kinds. First, one word. The whole numbers 0, 1, 2, 3, … together with the negative whole numbers −1, −2, −3, … are called integers. From now on, we use that word.

A rational number is a number that can be written as a fraction $\frac{p}{q}$ , where $p$ and $q$ are integers and $q$ is not zero. The name comes from ratio. A rational number is a ratio of two integers.

Many numbers are rational even when they do not look like fractions:

$5$ is rational, because $5 = \frac{5}{1}$ .
$- 3$ is rational, because $- 3 = \frac{- 3}{1}$ .
$0.75$ is rational, because $0.75 = \frac{3}{4}$ .
$0. \dot{3}$ is rational, because $0. \dot{3} = \frac{1}{3}$ .

An irrational number is a number that cannot be written as a fraction of two integers. Not with large integers, and not with clever ones. Never.

What a decimal can tell you

There is a second way to tell the two kinds apart. Look at the decimal.

When you change a fraction into a decimal, only two things can happen. The decimal stops, like $\frac{3}{4} = 0.75$ . Or a block of digits repeats forever, like $\frac{1}{3} = 0.333 \dots$ or $\frac{1}{7} = 0.142857142857 \dots$

Why only these two? Look inside the division. Here is $1 \div 7$ , worked out by long division. At each step, you divide by 7 and keep a remainder.

Step	Divide	Digit	Remainder
1	10 ÷ 7	1	3
2	30 ÷ 7	4	2
3	20 ÷ 7	2	6
4	60 ÷ 7	8	4
5	40 ÷ 7	5	5
6	50 ÷ 7	7	1
7	10 ÷ 7	1	3

A remainder after dividing by 7 can only be 0, 1, 2, 3, 4, 5 or 6. If the remainder is ever 0, the division stops. If it is never 0, there are only six possible remainders. So within six steps, a remainder must come back. Here, step 7 starts exactly like step 1. From that moment, the division repeats everything it did before, and the digits 142857 repeat with it.

The same thing happens for every fraction. The bottom number limits how many remainders are possible, so the division must either stop or repeat.

This gives us a test. If a number is rational, its decimal stops or repeats. Turn that around. If a decimal never stops and never repeats, the number cannot be rational. It is irrational.

Hunting for a fraction equal to √2

Here is the difficulty. We cannot use the decimal test on √2 directly. We can never see all of its digits. Perhaps they do repeat, but only after a block of a million digits. Looking at digits cannot settle the question. We need a different kind of argument.

Before the argument, try the hunt yourself. Suppose √2 is a fraction $\frac{p}{q}$ . Then $\frac{p}{q} \times \frac{p}{q} = 2$ . Multiply both sides by $q \times q$ . The fraction disappears, and you get a statement about integers only:

p^{2} = 2 q^{2}

So the hunt is this. Find two integers where the square of one is exactly twice the square of the other. Choose a bottom number $q$ below. The page finds the best top number $p$ for it, and compares $p^{2}$ with $2 q^{2}$ .

Explore — hunting for a fraction equal to √2

Move the slider to choose the bottom number q. Watch how close the fraction gets. Then look at the last line.

Bottom number q = 5

For each bottom number $q$ , the best top number $p$ is the integer nearest to $q \times 1.41421 \dots$

$q$	Best $p$	$\frac{p}{q}$	$(\frac{p}{q})^{2}$	$p^{2}$	$2 q^{2}$
5	7	1.4	1.96	49	50
12	17	1.416667	2.006944	289	288
29	41	1.413793	1.998811	1681	1682
70	99	1.414286	2.000204	9801	9800
408	577	1.414216	2.000006	332 929	332 928

The fractions get closer and closer to √2, but $p^{2}$ and $2 q^{2}$ always miss each other, here by exactly 1.

The fractions get closer and closer to √2. With $q = 408$ , the fraction $\frac{577}{408}$ squares to about 2.000006. That is close enough for any engineer. But look at the last line every time. $p^{2}$ and $2 q^{2}$ are never equal. For the best fractions, they miss by exactly 1. Try every $q$ up to 1000, and you will never find a match.

Trying is not proving, though. Perhaps the right fraction has a bottom number of a billion. To be certain, we need an argument that covers every possible fraction at once.

Why no fraction can ever work

The argument uses a method called proof by contradiction. It works like a detective who suspects a story is false. The detective says: "Suppose the story is true. What else would have to be true?" Then the detective follows the consequences, one careful step at a time. If the story leads to something impossible, the story was false.

Here, the story is "√2 is a fraction". We follow it until it breaks. This is the hard part of the lecture. Spend time here. Each step below gives one line and the reason it is allowed. Read the reason before you go on.

Step through — the proof that √2 is irrational

Press Next to add one step. Read the reason before you press Next again.

Each step gives one line of the argument and the reason it is allowed.

Suppose √2 = $\frac{p}{q}$ , where $p$ and $q$ are integers and the fraction is in its lowest terms. Why: every fraction can be cancelled down until the top and bottom have no common factor.
Square both sides: $2 = \frac{p ^{2}}{q ^{2}}$ . Why: if two numbers are equal, their squares are equal.
Multiply both sides by $q^{2}$ : $p^{2} = 2 q^{2}$ . Why: this removes the fraction, so everything is an integer.
So $p^{2}$ is even. Why: $2 q^{2}$ is 2 times an integer.
So $p$ is even. Why: an odd number squared is always odd, because $(2 k + 1)^{2} = 4 k^{2} + 4 k + 1$ is one more than an even number. If $p$ were odd, $p^{2}$ would be odd.
Write $p = 2 k$ . Then $4 k^{2} = 2 q^{2}$ , so $q^{2} = 2 k^{2}$ . Why: substitute into step 3, then divide both sides by 2.
So $q^{2}$ is even, and so $q$ is even. Why: the same reasoning as steps 4 and 5.
Now $p$ and $q$ are both even, so they share the factor 2. But step 1 said they share no common factor. Why: both cannot be true. Every step from 2 to 7 was correct, so the assumption in step 1 is what fails.

Conclusion. No fraction is equal to √2. So √2 is irrational.

Look at what just happened. We did not test fractions one by one. We showed that any fraction equal to √2 would have to be in lowest terms and not in lowest terms at the same time. That is impossible. So no such fraction exists, however large its top and bottom numbers are.

The proof needs only one fact: an odd number squared is odd. Check it with numbers if you like: $3^{2} = 9$ , $7^{2} = 49$ and $1 5^{2} = 225$ . All are odd. The algebra in step 5 shows why this must always happen.

That is it. That is the whole idea. √2 is irrational. You do not have to accept it because a book says so. You followed the reasoning yourself, and the reasoning leaves no other possibility.

The same kind of argument works for √3, √5, √6, and the square root of every positive integer that is not a perfect square. The general proof uses prime factors, and you will meet it later. Here is the result it gives.

So what is a surd?

Surds

A surd is a root that cannot be simplified to a whole number or a fraction. Its value is irrational.

For a positive integer $n$ : if $n$ is a perfect square, $n$ is a whole number. If $n$ is not a perfect square, $n$ is a surd.

So for a square root of an integer, the test is one question: is the number under the root a perfect square? The perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, … Learn to recognise them quickly. They do almost all the work.

Two more points complete the definition.

First, the same idea works for other roots. $38 = 2$ , because $2 \times 2 \times 2 = 8$ . So $38$ is not a surd. But $310$ is a surd, because 10 is not a perfect cube.

Second, the symbol √ always means the positive square root. The number 9 has two square roots, 3 and −3, because $(- 3) \times (- 3) = 9$ as well. But $9$ means 3 only. This agreement keeps every answer single and clear.

Stop and think

Which of these are surds: $49$ , $0.49$ , $50$ ?

Show the answer

Only $50$ . $49 = 7$ . Also $0.49 = 0.7$ , because $0.7 \times 0.7 = 0.49$ . A root sign does not make a surd, and neither does a decimal under the root. The only question is whether some whole number or fraction, multiplied by itself, gives the number. For 50, none does: $7^{2} = 49$ and $8^{2} = 64$ , and 50 is not a perfect square.

√2 is an exact point on the number line

An irrational number can sound vague, like a number that never quite arrives. It is not vague at all. √2 has an exact position on the number line, and you can find it with a ruler and a compass.

Draw a square with sides of length 1. By Pythagoras, its diagonal $d$ satisfies $d^{2} = 1^{2} + 1^{2} = 2$ . So the diagonal has length exactly √2. Put the compass point at 0, open it to the far end of the diagonal, and swing the diagonal down onto the number line.

The diagonal of a square with side 1 has length √2. Swung down with a compass, it lands on an exact point of the number line.

The point where the arc lands is √2. Not approximately. Exactly. The decimal 1.414… is only a description of that point, and every decimal you write down falls slightly short. The surd √2 is the exact name of the point.

This is why mathematicians keep surds as surds. If a square has an area of 2 cm², its side is √2 cm. If you write 1.41 cm instead, you lose some accuracy at once. Use that rounded side in more calculations, and the small error grows. Writing √2 cm loses nothing.

Worked examples

Worked Example 1 — Surd or not?

Which of these are surds: $49$ , $50$ , $16$ , $3$ , $100$ , $20$ , $7$ , $11$ ?

Ask one question of each: is the number under the root a perfect square?

Root	Perfect square?	Result
$49$	Yes, $7^{2}$	$= 7$ , not a surd
$50$	No, between $7^{2}$ and $8^{2}$	surd
$16$	Yes, $4^{2}$	$= 4$ , not a surd
$3$	No	surd
$100$	Yes, $1 0^{2}$	$= 10$ , not a surd
$20$	No, between 16 and 25	surd
$7$	No	surd
$11$	No	surd

The surds are $50$ , $3$ , $20$ , $7$ and $11$ .

Worked Example 2 — Fractions and decimals under the root

Are $\frac{1}{4}$ , $\frac{9}{4}$ and $0.01$ surds?

The perfect-square test was for integers. For fractions and decimals, go back to the meaning: is there a fraction which, multiplied by itself, gives the number?

$\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ . So $\frac{1}{4} = \frac{1}{2}$ . Rational. Not a surd.

$\frac{3}{2} \times \frac{3}{2} = \frac{9}{4}$ . So $\frac{9}{4} = \frac{3}{2}$ . Rational. Not a surd.

Write the decimal as a fraction first: $0.01 = \frac{1}{100}$ . Then $\frac{1}{10} \times \frac{1}{10} = \frac{1}{100}$ . So $0.01 = 0.1$ . Rational. Not a surd.

Check: $0.1 \times 0.1 = 0.01$ . Correct.

Worked Example 3 — Where does √50 sit?

Find the two integers that $50$ lies between. Then give $50$ to 1 decimal place without a calculator's square root key.

$7^{2} = 49$ and $8^{2} = 64$ . Since 50 lies between 49 and 64, $50$ lies between 7 and 8.

Now narrow it down. $7. 1^{2} = 50.41$ , which is just above 50. So $50$ lies between 7.0 and 7.1.

To round correctly, test the halfway point: $7.0 5^{2} = 49.7025$ . That is below 50, so $50$ is above 7.05. It rounds up.

50 \approx 7.1 (1 decimal place)

Check with a calculator: $50 = 7.0710 \dots$ Correct. Notice the sign ≈. The surd $50$ is exact; 7.1 is only close to it.

Worked Example 4 — A square garden

A square garden has an area of 20 m². Find the exact length of its side. Then give the length to 3 significant figures (3 s.f.).

If the side is $s$ metres, then $s^{2} = 20$ . So $s = 20$ . We take the positive root, because a length cannot be negative.

Is $20$ a surd? $4^{2} = 16$ and $5^{2} = 25$ . 20 is not a perfect square. So yes: the exact side is $20$ m, a surd between 4 and 5.

For 3 s.f.: $4.4 7^{2} = 19.9809$ and $4.4 8^{2} = 20.0704$ . So the side lies between 4.47 and 4.48. The halfway test: $4.47 5^{2} = 20.025625$ , which is above 20. So the side is below 4.475, and it rounds down.

s = 20 m \approx 4.47 m (3 s.f.)

In the next lecture, you will see that $20$ can be written in a simpler exact form, $25$ .

Three classic mistakes

Trap 1 — A root sign does not make a surd

$49$ , $\frac{9}{4}$ and $0.01$ all have root signs, and none of them is a surd. A root is a surd only when its value is irrational. Always ask: does some whole number or fraction square to give this?

Trap 2 — √2 is not equal to 1.414

Write $2 \approx 1.414$ , never $2 = 1.414$ . Square 1.414 and you get 1.999396, not 2. The equals sign means "exactly the same number". A rounded decimal is never exactly a surd.

Trap 3 — A long decimal is not the same as an irrational one

$\frac{1}{7} = 0.142857142857 \dots$ looks long and messy, but it repeats, so it is rational. In the same way, $\frac{22}{7} = 3.142857 \dots$ is rational. It is close to π, which is about 3.14159, but it is not equal to π. Irrational does not mean "long". It means "never repeats and never stops".

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

Which of these are surds: $36$ , $37$ , $\frac{16}{25}$ , $0.04$ , $39$ ?

Hint 1
For each one, ask: does some whole number or fraction, multiplied by itself (or by itself three times, for the cube root), give this number?

Hint 2
Write 0.04 as a fraction with 100 on the bottom. For the cube root, list the cubes: 1, 8, 27.

Full answer
$36 = 6$ , not a surd. $37$ is a surd, because 37 lies between 36 and 49. $\frac{16}{25} = \frac{4}{5}$ , not a surd. $0.04 = \frac{4}{100}$ , so $0.04 = \frac{2}{10} = 0.2$ , not a surd. $39$ is a surd, because 9 lies between the cubes 8 and 27. So the surds are $37$ and $39$ .
Between which two consecutive integers does $75$ lie? Then give $75$ to 1 decimal place without the square root key.

Hint 1
Find the perfect squares on either side of 75.

Hint 2
Try $8. 6^{2}$ and $8. 7^{2}$ . Then test the halfway point, $8.6 5^{2}$ .

Full answer
$8^{2} = 64$ and $9^{2} = 81$ , so $75$ lies between 8 and 9. Next, $8. 6^{2} = 73.96$ and $8. 7^{2} = 75.69$ , so it lies between 8.6 and 8.7. Halfway: $8.6 5^{2} = 74.8225$ , which is below 75. So $75$ is above 8.65, and $75 \approx 8.7$ (1 decimal place). A calculator gives 8.660…, which agrees.
A square tile has an area of 45 cm². Give the exact length of its side, say whether it is a surd, and give the length to 3 significant figures.

Hint 1
The side $s$ satisfies $s^{2} = 45$ . Is 45 a perfect square?

Hint 2
Try $6.7 0^{2}$ and $6.7 1^{2}$ , then the halfway point $6.70 5^{2}$ .

Full answer
The exact side is $45$ cm. Since $6^{2} = 36$ and $7^{2} = 49$ , 45 is not a perfect square, so $45$ is a surd. Next, $6.7 0^{2} = 44.89$ and $6.7 1^{2} = 45.0241$ . Halfway: $6.70 5^{2} = 44.957025$ , which is below 45, so the side is above 6.705. The side is $45$ cm $\approx 6.71$ cm (3 s.f.).
Ama says: " $0.04 = 0.2$ , so $0.4$ must also be 0.2." Is she right? Is $0.4$ a surd? Explain how you know.

Hint 1
Test Ama's claim the way you test any square root: square 0.2.

Hint 2
Write $0.4 = \frac{40}{100}$ . Square $\frac{40}{10}$ and see what you get. Then ask: is $40$ a surd? What happens if you divide a surd by 10?

Full answer

Ama is wrong. $0. 2^{2} = 0.04$ , not 0.4. In fact $0. 6^{2} = 0.36$ and $0. 7^{2} = 0.49$ , so $0.4$ lies between 0.6 and 0.7. (It is about 0.632.)

To decide whether it is a surd, look at $\frac{40}{10}$ . Its square is $\frac{40}{100} = 0.4$ , and it is positive. So $0.4 = \frac{40}{10}$ .

40 is not a perfect square, so $40$ is irrational. Now suppose $\frac{40}{10}$ were a fraction $\frac{a}{b}$ . Then $40 = \frac{10 a}{b}$ , which is also a fraction. That is impossible. So $0.4$ is irrational. It is a surd.

Notice the method: assume the opposite, and follow it to something impossible. It is the same idea as the proof for √2.

The bigger idea

Take a sheet of A4 paper, the size used in most of the world. It measures 210 mm by 297 mm, and $297 \div 210 \approx 1.414$ . That is √2, and it is not an accident. Fold the sheet in half and the half has exactly the same shape as the whole. A ratio of long side to short side $r$ does this only if $r = \frac{2}{r}$ , which means $r^{2} = 2$ .

The ancient Pythagoreans believed that every quantity in the world was a ratio of whole numbers. The proof you just read broke that belief. In fact, there are far more irrational numbers than rational ones. Pick a point on the number line completely at random, and it is almost certain to be irrational. The fractions we use every day are the rare ones.

Key concepts in this lesson

A rational number can be written as $\frac{p}{q}$ with integers $p$ and $q$ , where $q \neq = 0$ , and its decimal either stops or repeats.
An irrational number cannot be written as a fraction of two integers, and its decimal never stops and never repeats.
A surd is a root that cannot be simplified to a whole number or a fraction, such as √2, √7 or $310$ .
For a positive integer $n$ , $n$ is a surd exactly when $n$ is not a perfect square.
√2 is irrational because assuming $2 = \frac{p}{q}$ in lowest terms forces $p$ and $q$ to both be even, which is a contradiction.
A surd is an exact value, and a decimal such as 1.414 is only an approximation, so we write $2 \approx 1.414$ .