Commutative, Associative and Distributive Laws

TLDR:

Commutative Laws:	a + b  =  b + a a * b  =  b * a
Associative Laws:	(a + b) + c  =  a + (b + c) (a * b) * c  =  a * (b * c)
Distributive Law:	a * (b + c)  =  a * b  +  a * c

Commutative law

The “Commutative Laws” say we can swap numbers over and still get the same answer.

Why “commutative”, because the numbers can travel back and forth like a commuter.

When we add:

a = 10
b = 20
print(a + b == b + a)

When we multiply:

a = 3
b = 21.42
print(a * b == b * a)

When we do percentages:

# 8% of 50 = 50% of 8 = 4
percent = lambda part, whole:float(whole) / 100 * float(part)
a = 8
b = 50
print(percent(a, b) == percent(b, a))

NOTE

The Commutative Law does not work for subtraction or division: $12/3=4$, but $3/12=\frac{1}{4}$

Associative Laws

The “Associative Laws” say that it doesn’t matter how we group the numbers (i.e. which we calculate first).

When we add:

a = 1.1
b = 2.1
c = 3.1
print((a + b) + c == a + (b + c))

This: $(2+4)+5=6+5=11$, has the same answer as this: $2+(4+5)=2+9=11$

When we multiply:

a = 3.18
b = 5
c = 3.1
print((a * b) * c == a * (b * c))

This: $(3\ast 4)\ast 5=12\ast 5=60$, has the same answer as this: $3\ast (4\ast 5)=3\ast 20=60$

Sometimes it is easier to add or multiply in a different order:

What is 19 + 36 + 4? Try to use associative law.
$19+36+4=19+(36+4)=19+40=59$

Or to rearrange a little:

What is 2 * 16 * 5? Try to use associative law.
$2\ast 16\ast 5=(2\ast 5)\ast 16=10\ast 16=160$

NOTE

The Associative Law does not work for subtraction or division: $(9–4)–3=5–3=2$, but 9 – (4 – 3) = 9 – 1 = 8

Distributive Law

The “Distributive Law” is the BEST one of all, but needs careful attention.

3 lots of (2+4) is the same as 3 lots of 2 plus 3 lots of 4 So, the 3 can be “distributed” across the 2+4, into 32 and 34.

And we write it like this:

$a\ast (b+c)=a\ast b+a\ast c$

Try the calculations yourself:

$3\ast (2+4)=3\ast 6=18$

$3\ast 2+3\ast 4=6+12=18$

With distributive law we get the same answer when we:

• multiply a number by a group of numbers added together, or
• do each multiply separately then add them

What is 6 * 204? Try to use distributive law to simplify.
$6\ast 204=6\ast 200+6\ast 4=1,200+24=1,224$

What is 16 * 6 + 16 * 4? Try to use distributive law with combining.
$16\ast 6+16\ast 4=16\ast (6+4)=16\ast 10=160$

We can use it in subtraction too:

What is 263 - 243? Try to use distributive law.
$26\ast 3-24\ast 3=\ast \ast (26-24)\ast \ast \ast 3=2\ast 3=6$

What is 67 + 27 + 37 + 57 + 4*7?

$$6\ast 7+2\ast 7+3\ast 7+5\ast 7+4\ast 7=(6+2+3+5+4)\ast 7=20\ast 7=140$$

NOTE

The Distributive Law does not work for division: $24/(4+8)=24/12=2$, but $24/4+24/8=6+3=9$