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Unit 8: Kinds of Number

Where each one came from.

Unit 8 of 28 in Problem-solving maths for kids. Its 7 lessons are Building the Number Line, The Numbers With No Fraction, Working With Fractions, Decimals and Rounding, Surds Behave Like Letters, Naming the Number and The Whole Family — below is everything each one explains, and a question or two from it to try.

The answers are marked in the app by value — 1/2, 0.5 and 2/4 all count — and every one is re-derived by sympy before it ships.

This unit opens with a fortnight’s trial of everything — no card needed — or with a family plan, bought in the iPhone app. The first two units of every track are free for ever. Try it in the app.

📏 Building the Number Line

Counting came first

The counting numbers are 1, 2, 3, and on. They are as old as sheep.

Add zero and you have the whole numbers — and zero took humans thousands of years to accept, because "how many sheep is none?" did not feel like a number.

Then someone asked what 3 - 5 is, and the counting numbers had no answer.

Negatives fill in the other side

The integers are the whole numbers plus their negatives: … -3, -2, -1, 0, 1, 2, 3 …

Now every subtraction works. 3 - 5 = -2, and the number line stretches both ways forever.

But 7 ÷ 2 still had no answer.

Fractions fill in the gaps

A rational number is any number you can write as one integer over another: 3/4, -2/7, and also 5, which is 5/1.

That covers every division. And rationals are packed unbelievably tightly — between any two of them there is always another.

So surely they fill the line completely? They do not.

Try it yourself

Work it out.

-7 + 12

Work it out. Two minuses meeting.

-8 - (-3)

Answer them in the app

🌊 The Numbers With No Fraction

A number that broke a school

The ancient Greeks believed every number was a fraction. Then someone proved that sqrt(2) is not — and the story goes that it did not end well for them.

The proof is short. Suppose sqrt(2) = a/b in lowest terms. Squaring gives 2b^2 = a^2, so a^2 is even, so a is even. Put a = 2c and you get b^2 = 2c^2, so b is even too.

But they were in lowest terms — they cannot *both* be even. So no such fraction exists.

Irrational means "not a ratio"

Numbers like sqrt(2), sqrt(3) and pi are irrational. Their decimals run forever and never settle into a repeating pattern.

That is the give-away, and it works both ways:

- A decimal that stops or repeats is a fraction.
- One that does neither is irrational.

Turning a repeating decimal into a fraction

Let x = 0.3333…. Then 10x = 3.3333….

Subtract the first from the second and the endless tail cancels exactly: 9x = 3, so x = 1/3.

The same trick works for any repeating block — you just multiply by 10, 100 or 1000 depending on how long the block is.

Try it yourself

Which decimal is definitely a fraction?

  • 0.363636…
  • 3.14159265…
  • 0.101001000100001…
  • sqrt(5)

Use the trick on 0.7777… — write it as a fraction in lowest terms.

Answer them in the app

🍕 Working With Fractions

Adding needs a shared bottom

You cannot add quarters to thirds any more than you can add apples to oranges. First make them the same kind.

1/4 + 1/3 becomes 3/12 + 4/12 = 7/12.

The shared bottom is the lowest common multiple from unit 5. That is why that lesson came first.

Multiplying is the easy one

Multiplying needs no shared bottom at all. Tops together, bottoms together:

2/3 × 4/5 = 8/15

And "of" means multiply, so two thirds of four fifths is exactly that sum. Multiplying two fractions below 1 always makes something smaller — which surprises people who expect multiplication to grow things.

Dividing flips the second one

To divide by a fraction, turn it upside down and multiply.

1/2 ÷ 1/4 becomes 1/2 × 4/1 = 2.

And that answer is right in plain English too: how many quarters fit inside a half? Two. Dividing by something small gives something big.

Try it yourself

Add them. Write your answer as a fraction in lowest terms.

2/5 + 1/3

Subtract them. Write your answer as a fraction in lowest terms.

3/4 - 1/6

Answer them in the app

📍 Decimals and Rounding

The point continues the pattern

Going left, each place is worth ten times more. Going right, ten times less.

So after the units come tenths, hundredths, thousandths — the same rule, never interrupted. The point is not a break; it just marks where the ones are.

0.25 is two tenths and five hundredths, which is 25/100, which is 1/4.

Which fractions stop and which go on forever

A fraction gives a terminating decimal only when its bottom (in lowest terms) is built from 2s and 5s — the factors of ten.

1/8 stops, because 8 is 2^3. 1/3 does not, because 3 is neither.

So you can tell before dividing. Look at the prime factors of the bottom, exactly as in unit 5.

Rounding

To round to one decimal place, look at the next digit along. Five or more rounds up, less than five rounds down.

3.47 becomes 3.5. 3.44 becomes 3.4.

Round at the very end of a calculation, never in the middle — rounding twice quietly moves the answer.

Try it yourself

Write 0.75 as a fraction in lowest terms.

Write 3/8 as a decimal.

Answer them in the app

🌿 Surds Behave Like Letters

Multiplying and dividing roots

Roots multiply straight through: sqrt(3) × sqrt(12) = sqrt(36) = 6.

And they divide the same way: sqrt(50)/sqrt(2) = sqrt(25) = 5.

So two irrational numbers can multiply to give a perfectly ordinary whole one. That is worth noticing before it surprises you.

Adding is where they act like letters

You cannot add sqrt(2) and sqrt(3) into anything shorter, exactly as you cannot add x and y.

But like terms do collect: 3 sqrt(2) + 5 sqrt(2) = 8 sqrt(2).

So simplify each surd first. sqrt(8) is really 2 sqrt(2), and only then can you see what will combine.

Try it yourself

Work it out.

sqrt(3) × sqrt(12)

Work it out.

sqrt(50)/sqrt(2)

Answer them in the app

🏷️ Naming the Number

Deciding which family a number belongs to

Ask three questions in order.

Is it a whole number, possibly negative? Then it is an integer.
Can it be written as one integer over another? Then it is rational.
Otherwise it is irrational — and if it involves i, it is complex.

sqrt(16) looks irrational and is not: it is 4.

Adding two irrationals can give a rational

Here is a surprise: (2 + sqrt(3)) and (2 - sqrt(3)) are both irrational, but they add to 4.

And they multiply to 4 - 3 = 1.

So the irrational numbers are not "closed" — combining two of them can land you back among the ordinary ones. The rationals never do that.

Try it yourself

Which of these is an integer?

  • sqrt(49)
  • sqrt(50)
  • 22/7
  • 0.3

What is sqrt(49)?

Answer them in the app

🌳 The Whole Family

Each one contains the last

Counting numbers sit inside the integers. Integers sit inside the rationals. Rationals and irrationals together make the real numbers — the whole unbroken line.

And the reals sit inside the complex numbers, which is where unit 2 took you.

Every step happened because somebody asked a question the numbers they had could not answer.

Rationalising the bottom

Nobody likes a square root underneath a fraction. 1/sqrt(2) is tidied by multiplying top and bottom by sqrt(2):

1/sqrt(2) = sqrt(2)/2

It is the same number, written in a form you can compare and add. And it is exactly the conjugate trick from unit 2, wearing different clothes.

Try it yourself

Which question forced people to invent negative numbers?

  • What is 3 take away 5?
  • What is 7 divided by 2?
  • What is the square root of 2?
  • What is the square root of -1?

And which one forced complex numbers?

  • What is the square root of -1?
  • What is 3 take away 5?
  • What is 1 divided by 3?
  • What is the square root of 2?

Answer them in the app