The Laws of Arithmetic in Algebra

You have used this law since primary school without ever naming it. Look at three rows of five dots and five rows of three dots. It is the same collection turned on its side, so 3 times 5 and 5 times 3 both give 15. Adding behaves the same way: 4 plus 7 and 7 plus 4 are both 11. This is the commutative law, and commute just means to swap places. Written with letters it says a plus b equals b plus a, and ab equals ba. Because letters simply stand for numbers, anything true for every pair of numbers stays true for the letters. That is why you are allowed to tidy n times 7 into 7n.

Now for the limits of that law, because it does not cover everything. Take 9 minus 4, which is 5, and swap it to 4 minus 9, which is negative 5. Those are different numbers, so subtraction cannot be swapped. Division is just as fussy: 12 shared among 3 people gives 4 each, but 3 shared among 12 people gives only a quarter each. That single counter-example is enough. A law in maths has to hold for every number without exception, so finding one pair where it fails proves the rule is not a law. Remember it this way: swapping is safe for plus and times, and never safe for minus and divide.

The second law is about grouping rather than order. Add 2, 8 and 5 by first grouping the 2 and 8: you get 10 plus 5, which is 15. Group the 8 and 5 instead and you get 2 plus 13, which is also 15. Multiplying behaves the same way, so 2 times 5 times 7 gives 70 whichever pair you handle first. This is the associative law, and with letters it says bracket a plus b close bracket plus c equals a plus bracket b plus c. It is genuinely useful: to work out 37 plus 19 plus 1, group the 19 and 1 to make 20. And once again, subtraction refuses to play along.

The third law is the most powerful, and a picture makes it obvious. Draw a rectangle 4 tall and 13 wide, then cut the 13 into 10 and 3. The left piece has area 40 and the right piece has area 12, so the whole thing is 52, which is exactly 4 times 13. Now put a letter on one part of the side. A rectangle 3 tall, with a side split into n and 2, has pieces of area 3n and 6, so 3 lots of the bracket n plus 2 comes to 3n plus 6. That is the distributive law: the multiplier out the front reaches every part inside the bracket. Check it at n equal to 5 and both sides give 21.

So what are these laws actually for? Two things. First, tidying up. To simplify 5 times n times 4, the commutative law lets you swap the n and the 4, and the associative law lets you group the two numbers together, giving 20n. Second, they make you faster at arithmetic. To work out 7 times 23 in your head, split the 23 into 20 plus 3 and distribute the 7: 140 plus 21 is 161. That is the distributive law doing the work. Keep the three names straight — commutative means order can swap, associative means grouping can move, and distributive means the multiplier reaches everything inside a bracket.