Properties beyond the whole numbers

The properties you have studied were stated for whole numbers. But mathematicians never stop at one family. What happens on the left of zero, with negative numbers? With fractions? On a clock face, where 9 + 5 = 2? Inside a computer, where 0.1 + 0.2 is not quite 0.3? And in everyday actions such as putting on socks and shoes, where order matters a great deal?

This layer takes the properties on a journey. You will see how the distributive law forces (−3) × (−4) to be +12, why some families gain closure while others lose it, where commutativity breaks in the real world, and how the properties power olympiad puzzles, computer science and cryptography. It ends with projects to try and questions nobody has answered yet.

Extend the number line to the left of 0 and mark −1, −2, −3, … The integers are … −3, −2, −1, 0, 1, 2, 3, … You already meet them: a temperature of −5 °C in Leh in winter, a lift going to basement level −2, a bank balance that is overdrawn, a cricket team's net run rate, a height below sea level.

With integers, subtraction is finally closed: 3 − 5 = −2 is an integer. In fact subtraction becomes addition of the opposite: 3 − 5 = 3 + (−5). Every integer a has an additive inverse −a with a + (−a) = 0. The integers keep closure, commutativity and associativity for + and ×, keep the identities 0 and 1, and keep the distributive law. They are still not closed under division (1 ÷ 2 is not an integer).

The table tells a story. Each new family of numbers was invented partly to repair closure: integers repair subtraction, fractions (rational numbers) repair division. Division by zero is the one hole that no family of ordinary numbers repairs, for the reasons you saw in Deepen.

Integers also give subtraction some new behaviour worth knowing. It is still not commutative (5 − 3 = 2 but 3 − 5 = −2; they are opposites) and still not associative: (10 − 4) − 3 = 3 but 10 − (4 − 3) = 9. But rewritten as addition, 10 + (−4) + (−3) can be regrouped freely, because addition of integers is associative. That is the trick behind adding a column of profits and losses in any order.

The rational numbers (fractions such as ½, −¾, 7/3) keep every property of + and × you know, and add one more: every number except 0 has a reciprocal (multiplicative inverse). ¾ × 4/3 = 1.

The distributive law is especially handy with fractions: ½ × (18 + 6) = 9 + 3 = 12. And the shortcuts still work: ¼ × 36 × 4 = (¼ × 4) × 36 = 1 × 36 = 36. Here the associative and commutative laws let us bring the reciprocals together to make the identity 1.

One important change: with fractions, multiplying does not always make bigger, and dividing does not always make smaller. 12 × ½ = 6, and 12 ÷ ½ = 24. The claim "a × b > a" that was sometimes true for whole numbers is now false for every b between 0 and 1.

It is 9 o'clock. What time will it be in 5 hours? Not 14 o'clock on a 12-hour clock face, but 2 o'clock. On a clock, numbers wrap around. This is called modular arithmetic, here "arithmetic modulo 12".

Does clock addition keep the properties?

• Closed: adding any two clock times gives a clock time. ✓
• Commutative: 9 + 5 and 5 + 9 both give 2. ✓
• Associative: (9 + 5) + 7 = 2 + 7 = 9, and 9 + (5 + 7) = 9 + 12 = 9. ✓
• Identity: adding 12 hours brings you back to the same time, so 12 behaves like 0. ✓
• Inverses: from 9, adding 3 hours gets you to 12, the identity. So 3 is the inverse of 9. Every time has one. ✓

The days of the week are arithmetic modulo 7. If today is Monday, what day is it in 100 days? 100 = 14 × 7 + 2, so 14 full weeks bring you back to Monday and 2 more days give Wednesday.

Outside arithmetic, most ways of combining things are not commutative. That is why the commutative property is worth naming at all. Some examples to think about:

• Words. Join "rain" and "bow": rainbow. Join "bow" and "rain": bowrain. Joining text (called concatenation in computing) is associative but not commutative.
• Turning and flipping. Take a book lying face up. Turn it a quarter turn clockwise, then flip it over left to right. Now start again: flip first, then turn. The book ends up in a different position. Rotations and reflections do not commute.
• Cooking. Boil the rice, then add salt; or add salt, then boil. Similar. Grind the spices, then fry them; or fry them, then grind. Very different.
• Rubik's cube. Twisting the right face then the top face is not the same as top then right. Solving the cube is all about clever orders of moves.

A computer stores most decimal numbers in a fixed number of binary digits, rounding them slightly. Because of this rounding, computer addition of decimals is not always associative. In the Python programming language:

• (0.1 + 0.2) + 0.3 gives 0.6000000000000001
• 0.1 + (0.2 + 0.3) gives 0.6

And with very large numbers: (10,000,000,000,000,000 + 1) − 10,000,000,000,000,000 gives 0, because the computer cannot store the "+ 1" at that size, while (10,000,000,000,000,000 − 10,000,000,000,000,000) + 1 gives 1.

This matters in real life. Weather forecasts, satellite orbits and bank software add millions of numbers. Programmers choose the order of additions carefully, and banks usually store money as whole numbers of paise rather than decimals of rupees, so that whole-number addition, which is associative, does the work.

Multiplying 142,857 by 1 to 6 just rotates its digits. Then 142,857 × 7 = 999,999. The number comes from the repeating decimal of 1/7 = 0.142857142857…, and the pattern is explained by clock arithmetic modulo 7. It is a lovely example of a pattern hiding a property.

Now try these quick-fire challenges. Each is a property in disguise; the answers are in the quiz and summary.

1. 999 × 999 = 998,001. Can you see why, using (1,000 − 1)²?
2. What is 25 × 32 × 125? (Hint: 32 = 4 × 8.)
3. 2,468 × 1,357 is even or odd? What about 2,469 + 1,357?
4. Is 7,777,777 a perfect square? (Look at the last two digits and divide by 4.)

Even and odd numbers look simple, yet they hide questions no one has answered.

Goldbach's conjecture (1742). Every even number from 4 upward seems to be the sum of two primes: 4 = 2 + 2, 10 = 3 + 7 = 5 + 5, 100 = 3 + 97. Computers have checked every even number up to 4 × 10¹⁸ (four billion billion) without finding a counterexample — a distributed search completed in 2013. But nobody has a proof, so it is still a conjecture. This is exactly the difference between "true for many examples" and "always true" that you met in Investigate, at the highest level.

Odd perfect numbers. A perfect number equals the sum of its factors other than itself: 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14. Every perfect number found so far is even. Is there an odd one? No one knows, after more than 2,000 years of searching.