The Laws of Arithmetic
Ever add up a shopping bill in your head by grouping the pounds that make a round number, or work
out 6 \times 13 by doing 6 \times 10 and
6 \times 3 separately? Every one of those handy shortcuts is powered by
three quiet rules — the laws of arithmetic — that promise the answer won't change
when you rearrange the sum.
Adding and
multiplying follow three
simple, dependable laws. They are the rules that say which moves are
always safe — when you can reorder numbers, regroup them, or split one apart without ever
changing the answer. Each law has a friendly name: commutative (swap),
associative (regroup) and distributive (share out). Learn
these three and you can rearrange almost any sum into an easier one — that is the whole secret
of fast mental maths.
1. Commutative: order doesn't matter (swap)
You can swap the two numbers around and the answer stays the same. This is
true for both adding and multiplying:
a + b = b + a
a \times b = b \times a
So 3 + 5 is the same as 5 + 3 (both
8), and 2 \times 4 is the same as
4 \times 2 (both 8). Think of two
amounts being poured into one pile — it makes no difference which you pour first.
Why is this handy? Because you get to choose the easier order. To add
2 + 47, flip it to 47 + 2 and just count
on two: 49. Counting on from the bigger number is far less work.
Three apples and four oranges cost exactly the same as four oranges and three apples — seven
pieces of fruit either way. Order is something we care about; the total doesn't.
+
= 7
+
= 7
2. Associative: grouping doesn't matter (regroup)
When you add (or multiply) three numbers, it doesn't matter which pair you combine first. The
brackets just say "do this bit first", and you can move them freely — that is what
regrouping means:
(a + b) + c = a + (b + c)
(a \times b) \times c = a \times (b \times c)
Try (2 + 3) + 4: that is 5 + 4 = 9. Now
2 + (3 + 4): that is 2 + 7 = 9. Same
total. This lets you pick the friendly pair first — for
7 + 8 + 2 it is easier to do 8 + 2 = 10
first, then 7 + 10 = 17. Hunting for a pair that makes ten is the
commonest mental-maths move there is, and the associative law is what makes it legal.
It works for multiplying too. For 2 \times 9 \times 5, regroup the
2 and the 5:
9 \times (2 \times 5) = 9 \times 10 = 90.
3. Distributive: the star of the show (share out)
This is the most powerful law. Multiplying a number by a sum is the same as
sharing that multiplication out over each part and then adding:
a \times (b + c) = a \times b + a \times c
The array below makes it obvious. Draw a rows of
b + c squares, so there are a \times (b + c)
squares in all. Now slice the array down the line between the b
columns and the c columns: you get two smaller arrays, one
a \times b and one a \times c. No square
moved, so the two pieces must add up to the whole. Step through it, and press
Refresh for a brand-new example.
Why these laws make mental maths easy
The distributive law is the trick that turns a hard multiplication into easy ones. To work out
6 \times 13, share it out by splitting
13 into 10 + 3:
6 \times 13 = 6 \times (10 + 3) = 6 \times 10 + 6 \times 3 = 60 + 18 = 78
Here are two more worked examples — watch how each law does a job:
-
Distributive. 8 \times 12 = 8 \times (10 + 2) = 80 + 16 = 96.
Two easy products instead of one tricky one.
-
Commutative + associative together.
4 \times 7 \times 25 = 7 \times (4 \times 25) = 7 \times 100 = 700 —
swap the order, regroup the friendly pair, done in your head.
The distributive law is exactly the
area model at work,
and it is the engine behind almost every multiplication shortcut you will ever learn.
Three friends each get a party bag holding four cookies and two sweets. How many treats
altogether? You could fill one bag (six treats) and triple it, or count all the cookies and
all the sweets and add. Both give 3 \times (4 + 2) = 3 \times 4 + 3 \times 2 = 18 —
that is the distributive law, sharing the 3 across both kinds of treat.
× 3 bags = 18 treats
When the laws stop working
The three laws are loyal friends of addition and multiplication
— but subtraction and division do not play along. With those
two, order and grouping really do change the answer.
- 5 - 3 = 2, but 3 - 5 = -2 — not the same!
- (8 - 4) - 2 = 2, but 8 - (4 - 2) = 6 — regrouping changed it.
- 8 \div 2 = 4, but 2 \div 8 = \tfrac{1}{4} — order matters here too.
These laws work for + and ×, but
subtraction and division are fussy about order and grouping:
- You may not swap a subtraction or a division:
a - b is usually not b - a, and
a \div b is usually not b \div a.
- You may not move the brackets freely either:
(a - b) - c is usually not a - (b - c).
If you have five cookies and eat three, two are left. But you cannot start with three cookies
and "eat five" — there aren't enough! 5 - 3 and
3 - 5 are simply not the same story, which is exactly why
subtraction is not commutative.
− 3 eaten = 2 left
See it explained
Khan Academy uses the distributive property to break a product into easier pieces: