## Postulates for the natural numbers

Every natural number a has a successor a + 1. (For example, the successor of 5 if 6, the successor of 10 is 11, and so on.)

Every natural number a (except 1) has a predecessor a – 1. (The predecessor of 4 is 3, the predecessor of 9 is 8, and so on.)

The natural numbers can be put in order. In other words, if a and b are two natural numbers, then either a is greater than b (written a > b), a is less than b (a < b), or a is equal to b (a = b.)

The order doesn’t make a difference (commutative) property of addition: a + b = b + a (for any two given numbers a and b).

The order doesn’t make a difference (commutative) property of multiplication: a x b = b x a (for any two given numbers a and b).

The where you put the parentheses doesn’t make a difference property of addition: (a + b) + c = a + (b + c). For any three numbers a, b, and c. (This property is known as the associative property of addition, because it says that it doesn’t matter which numbers associate with each other.) For example,

(5 + 4) + 3 = 5 + (4 + 3)
(9) + 3 = 5 + (7)
12 = 12

The where you put the parentheses doesn’t make a difference (associative) property of multiplication: (a x b) x c = a x (b x c). For any three numbers, a b, and c. For example,

(2 x 6) x 4 = 2 x (6 x 4)
(12) x 4 = 2 + (24)
48 = 48

Precedence rules for arithmetic expressions

When an expression contains more than one type of operation, perform the operations in the following order:

Top priority: an operation inside parentheses. For example, in the expression 8 x (7 + 6), perform the addition before the multiplication, 8 x (7 + 6) = 8 x 13 = 104.

Second priority: exponentiations. For example, in the expression 4 x 32, perform the exponentiation before the multiplication. 4 x 32 = 4 x 9 = 36

Third priority: multiplications and divisions. For example, in the expression 8 + 7 x 6, perform the multiplication before the addition. 8 + 7 x 6 = 8 + 42 = 50.

When an expression contains more than one operation with the same priority, perform the operations in order from left to right. For example, here is how we work out this expressions:

7 + 3 x 6 – 4 x 2 + 9

Calculate 3 x 6 first:

7 + 18 – 4 x 2 + 9

Next, calculate 4 x 2:

7 + 18 – 8 + 9

Next, calculate 7 + 18:

25 – 8 + 9

Next, calculate 25 – 8:

17 + 9

The final result is 17 + 9 = 26.