For any number a, the product of a and 1 is a.

8 * 1 = 8
1 * 8 = 8

For any number a, the product of a and 0 is 0.

a * 0 = 0
0 * a = 0

4 * 0 = 0
0 * 4 = 0

For every number \frac{a}{b}, where a, b \neq 0, there is exactly one number \frac{b}{a} such that the product of \frac{a}{b} and \frac{b}{a} is 1.

\frac{a}{b} * \frac{b}{a} = 1

\frac{b}{a} * \frac{a}{b} = 1

\frac{2}{3} * \frac{3}{2} = \frac{6}{6} = 1

\frac{1}{4} * \frac{4}{1} = \frac{4}{4} = 1

Any quantity is equal to itself.

For any number a, \quad a = a

7 = 7

2 + 5 = 2 + 5

If one quantity equals a second quantity, then the second quantity equals the first.

For any numbers a and b

If a = b

Then b = a.

If 9 = 5 + 4

Then 5 + 4 = 9

If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity.

For any numbers a, b, and c,
if a = b and b = c

then a = c

If 5+7 = 8+4 and 8+4 = 12

then 5+7 = 12

A quantity may be substituted for its equal in any expression.

If a = b , then a may be replaced by b in any expression.

If n = 15 ,

then 3n = 3*15.

For any numbers a, b, and c.

a(b+c) = ab + ac \quad and
(b+c)a = ba + ca \quad and
a(b-c) = ab - ac \quad and
(b-c)a = ba - ca \quad \quad

3(2+5) = 3*2 + 3*5
3(7) = 6 + 15
21 = 21

The order in which you add or multiply numbers does not change their sum or product.

For any numbers a and b

a + b = b+a \quad and

a*b = b*a

7 + 9 = 9 + 7 \quad and

5*4 = 4*5

The way you group three or more numbers when adding or multiplying does not change their sum or product.

For any numbers a, b, and c

(a + b) + c = a + (b + c)

and

(a*b)*c = a*(b*c) .

(2 + 4) + 6 = 2 + (4 + 6)

and

(3*5)*4 = 3*(5*4).