Whole numbers and the properties of operations

Counting numbers and whole numbers

The numbers you first learned, \(1, 2, 3, 4, \dots\), are the counting numbers (also called natural numbers). Add zero to the list and you get the whole numbers:

There's no largest whole number: whatever number you name, adding \(1\) gives a bigger one.

Each digit in a whole number has a place value. In \(47{,}305\) the \(4\) means \(4\) ten-thousands, the \(7\) means \(7\) thousands, the \(3\) means \(3\) hundreds, the \(0\) means no tens and the \(5\) means \(5\) ones:

On a number line, whole numbers are evenly spaced starting from \(0\), and numbers get larger as you move right. We compare them with \(<\) ("is less than") and \(>\) ("is greater than"): \(19 < 91\) and \(500 > 50\).

Properties of addition and multiplication

Adding or multiplying two whole numbers always gives another whole number. These two operations also follow a few rules that let you rearrange a calculation to make it easier.

There's one more fact about multiplication worth remembering: anything times zero is zero.

Subtraction and division

Subtraction undoes addition, and division undoes multiplication. Every subtraction can be checked with an addition, and every division with a multiplication:

• \(63 - 28 = 35\) because \(35 + 28 = 63\)
• \(72 \div 8 = 9\) because \(9 \cdot 8 = 72\)

Neither operation is commutative: \(10 - 4 = 6\) but \(4 - 10\) isn't even a whole number, and \(12 \div 3 = 4\) but \(3 \div 12\) isn't a whole number either. We'll need negative numbers and fractions for those.

When a division doesn't come out evenly, there's a remainder. Dividing \(29\) cookies among \(4\) friends gives each one \(7\), with \(1\) left over:

The remainder is always smaller than the divisor. If it weren't, you could hand out one more round.

Practice