Rules that numbers always follow

Your school is holding a function in the hall, and your class has to set out 24 chairs. Riya puts them in 4 rows of 6. Arjun says, "It would look better as 6 rows of 4." They argue for five minutes about which way needs more chairs.

You already know the answer, don't you? It is 24 chairs either way. Turning the whole arrangement sideways does not add or remove a single chair: 4 × 6 = 24 and 6 × 4 = 24.

That tiny fact is an example of a property of numbers: a rule that is true not just for 4 and 6, but for every pair of numbers you could ever pick. This lesson is about those rules. They are not extra things to memorise. They are the reasons your mental maths already works, and once you can see them, hard sums start to look easy.

Long before anyone wrote numbers down, people counted: one goat, two goats, three goats. The numbers we count with, 1, 2, 3, 4, 5, …, are called natural numbers (or counting numbers). The dots mean they never stop. However big a number you think of, you can always add 1 and get a bigger one.

Now think about an empty biscuit tin. How many biscuits are in it? None. We need a number for "nothing", and that number is 0. When we put 0 together with all the counting numbers, we get the whole numbers: 0, 1, 2, 3, 4, …

So every natural number is a whole number, but one whole number, 0, is not a natural number.

Picture the whole numbers as marks on a long straight road, equally spaced, starting at 0 and running off to the right for ever. This is the number line.

• Moving right means getting bigger. Adding 3 means jump 3 steps right.
• Moving left means getting smaller. Subtracting 3 means jump 3 steps left.
• Multiplying 4 × 3 means make 4 jumps of 3 steps each, starting at 0: you land on 12.

The number just after a number is its successor (the successor of 99 is 100). The number just before is its predecessor (the predecessor of 100 is 99). Every whole number has a successor, but 0 has no whole-number predecessor: there is nothing to its left on our road. (In the Extend layer you will see what lives to the left of 0.)

Amma puts 5 laddoos on one plate and 3 laddoos on another. How many laddoos altogether? 5 + 3 = 8. Now start with the second plate: 3 + 5 = 8. Of course. The laddoos are the same laddoos; you only counted them in a different order.

Mathematicians call this the commutative property of addition. "Commute" means to travel back and forth, like a daily commute: the numbers can swap places and the answer stays the same. Teachers often call these turn-around facts.

It works for big numbers too: 348 + 57 = 405 and 57 + 348 = 405. And it gives you a free trick. To add 2 + 69, turn it around and count on from the bigger number: 69, then 70, 71. Much faster than counting on 69 steps from 2!

Look at the chairs again, but this time read down the columns instead of along the rows. There are 6 columns, and each column has 4 chairs. So the same picture shows 6 × 4 as well as 4 × 6. One picture, two multiplications, the same 24 chairs.

An arrangement of objects in neat rows and columns is called an array. Arrays are everywhere: eggs in a tray (5 rows of 6 = 30), windows on a building, seats in a cinema, stamps on a sheet. Every array is a picture of the commutative property of multiplication.

This is why you only need to learn half of the times tables. If you know 7 × 8 = 56, you already know 8 × 7 = 56.

At the kirana shop you buy rice for ₹38, dal for ₹45 and oil for ₹55. The shopkeeper adds them up in his head in a flash. How?

Doing it in the written order, you would do 38 + 45 = 83, then 83 + 55 = 138. That works, but the numbers are awkward. The shopkeeper spots that 45 and 55 make 100, so he does that pair first: 45 + 55 = 100, then 100 + 38 = 138. Same answer, far less effort.

When you add three numbers, you can choose which two to add first. We show the pair we do first with brackets: (38 + 45) + 55 and 38 + (45 + 55) both give 138. This is the associative property of addition. To "associate" means to keep company with: you choose which numbers keep company first.

The same trick works for multiplication. Try 2 × 7 × 5. In the written order, 2 × 7 = 14, then 14 × 5 = 70. But if you first multiply 2 × 5 = 10, then 10 × 7 = 70, it is almost instant.

A cricket example: at a practice net there are 3 bowlers, each bowls 5 overs, and each over is 6 balls. Total balls = 3 × 5 × 6. Group (3 × 5) × 6 = 15 × 6 = 90, or 3 × (5 × 6) = 3 × 30 = 90. Both give 90 balls.

A school garden has 6 rows of plants with 14 plants in each row. How many plants?

14 is a bit big to multiply in your head, so break it apart into 10 and 4. Think of the garden as two smaller gardens side by side: a part with 6 rows of 10, and a part with 6 rows of 4.

• 6 rows of 10 = 60 plants.
• 6 rows of 4 = 24 plants.
• Together: 60 + 24 = 84 plants.

So 6 × 14 = 6 × 10 + 6 × 4 = 84. This is the distributive property. "Distribute" means to hand out, like distributing sweets: the 6 is handed out to both parts, the 10 and the 4.

You can break a number apart in any way that helps. For 7 × 12, split 12 into 10 + 2: 7 × 10 + 7 × 2 = 70 + 14 = 84. For 8 × 25, split 25 into 20 + 5: 8 × 20 + 8 × 5 = 160 + 40 = 200.

It even works with subtraction. 9 × 19 is awkward, but 19 is just 20 − 1. Nine twenties are 180, and then take away one 9: 180 − 9 = 171. So 9 × 19 = 9 × 20 − 9 × 1 = 171. Shopkeepers use this all the time with prices like ₹99 or ₹199.

Zero behaves in two very different ways, depending on what you do with it.

Adding or subtracting 0 changes nothing. If you have 7 marbles and your friend gives you 0 more, you still have 7: 7 + 0 = 7. If you give away 0 marbles, you still have 7: 7 − 0 = 7. Because adding 0 leaves every number exactly as it was, 0 is called the additive identity. "Identity" means the number keeps its identity: it stays itself.

Multiplying by 0 wipes everything out. Think of 5 boxes with 0 pencils in each. How many pencils? None: 5 × 0 = 0. Turn it around: 0 boxes of 5 pencils is also no pencils: 0 × 5 = 0. Whatever the number, times zero is zero. Even 1,000,000 × 0 = 0.

That last row is the famous one. You cannot divide by zero.

Here is a way to see it. Division undoes multiplication. 12 ÷ 3 = 4 because 4 × 3 = 12. So for 8 ÷ 0 we need a number that, multiplied by 0, gives 8. But anything times 0 is 0, never 8. No such number exists. So 8 ÷ 0 has no answer, and mathematicians say it is undefined.

Try it on a calculator or phone: type 8 ÷ 0 and it will show Error or Cannot divide by zero. The calculator is not broken. It is telling you the question has no answer.

The number 1 is to multiplication what 0 is to addition.

• Multiplying by 1 changes nothing. One box with 9 mangoes holds 9 mangoes: 9 × 1 = 9, and 1 × 9 = 9. So 1 is the multiplicative identity.
• Dividing by 1 changes nothing. Share 9 mangoes with 1 person (just yourself!) and you get all 9: 9 ÷ 1 = 9.
• A number divided by itself is 1 (as long as it is not 0). Share 9 mangoes among 9 people and each gets 1: 9 ÷ 9 = 1.

But be careful: adding 1 does change a number. 9 + 1 = 10. Adding 1 gives the next number on the number line, the successor. So 0 is special for adding, and 1 is special for multiplying. Mixing them up is one of the most common slips in maths.

Take some socks and try to put them into pairs. With 8 socks you make 4 pairs and nothing is left over, so 8 is even. With 7 socks you make 3 pairs and one sock is left alone, so 7 is odd.

• Even numbers: 0, 2, 4, 6, 8, 10, 12, … (they end in 0, 2, 4, 6 or 8).
• Odd numbers: 1, 3, 5, 7, 9, 11, 13, … (they end in 1, 3, 5, 7 or 9).

Zero is even: zero socks make zero pairs with nothing left over.

Now the magic: you can predict whether an answer will be even or odd without working it out. Just think about the lonely leftover socks.

The surprising row is odd + odd = even. Each odd number has one lonely sock. Put two odd numbers together and the two lonely socks become a pair! So 7 + 9 = 16 is even, and so is 99 + 101 = 200.

Multiplying has its own rules: a number times an even number is always even, because you get groups of pairs. Only odd × odd is odd: 3 × 5 = 15, 7 × 7 = 49.

Here is a question that sounds odd at first: if you add two whole numbers, is the answer always a whole number?

Try some: 3 + 5 = 8, 0 + 12 = 12, 4,567 + 8,999 = 13,566. Yes, every time. Adding whole numbers never takes you outside the family of whole numbers. We say the whole numbers are closed under addition: the family is like a closed room, and adding never lets you escape.

Multiplication is the same: 7 × 6 = 42, 0 × 9 = 0. Always a whole number. So whole numbers are closed under multiplication too.

But subtraction can let you escape! 5 − 8 is not a whole number. (On a thermometer you would say −3, a number less than zero, which is not in our family.) And division can escape too: 7 ÷ 2 = 3½, a fraction. So whole numbers are not closed under subtraction or division.

Every clever shortcut you have ever learned is one of these properties in disguise. Here are five favourites. Try each one on paper before looking at the answers.