Basic operations with whole numbers

Addition

When you add numbers, you combine them to find a total. This total is called the sum.

Example:

$13.5+2.7$. Set it up like this:

$$\begin{array}{r}13.5\\ +2.7\\ 16.2\end{array}$$

Because the ones place, decimal point, and tenths place are aligned, you can add as you normally would.

Addition works the same way when the signs differ. For example, $7+(-3)=4$ - adding a negative number is the same as subtracting its positive value.

Subtraction

You can think of subtraction in two ways:

• Removing a number
• Adding the negative of that number

This can make subtraction easier when working with negative numbers. The result of subtraction is called the difference.

Example:

$$\begin{array}{r}5-2=3\\ 5+(-2)=3\end{array}$$

Both expressions give the same result. In the second line, the $2$ becomes negative, and the problem is rewritten as addition.

Example:

$14.05-3.08$. Line up the decimal points and subtract column by column, borrowing when a digit is too small:

$$\begin{array}{r}14.05\\ -3.08\\ 10.97\end{array}$$

The hundredths digit ($5$) is smaller than $8$, so you need to borrow, but the tenths digit is $0$. Borrow $1$ from the $4$ in the ones place (it becomes $3$), which makes the tenths $10$. Then borrow $1$ from the tenths (it becomes $9$), which makes the hundredths $15$. Now subtract column by column: $15-8=7$, $9-0=9$, $3-3=0$, and $1-0=1$, giving $10.97$. Borrowing works the same way across the decimal point as it does between whole-number places.

Multiplication

Example:

$4\times 3$ means adding three groups of $4$:

$$\begin{array}{rl}4\times 3& =4+4+4\\ & =12\end{array}$$

The product is $12$.

Division

Division is the reverse of multiplication. It breaks a number into equal-sized groups. The result of division is called the quotient. The quotient can tell you how many equal groups you have, or how large each group is, depending on how the problem is set up.

Example:

$20÷5$ asks: how many $5$’s add up to $20$?

$$5+5+5+5=20$$

So $20÷5=4$, because $20$ contains $4$ groups of $5$.

Order of operations

When a problem has more than one operation, work through it in a set order: parentheses first, then exponents, then multiplication and division (left to right), and finally addition and subtraction (left to right). This order is often remembered with the acronym PEMDAS.

Example:

$3+4\times 2$. Multiply before you add:

$$\begin{array}{rl}3+4\times 2& =3+8\\ & =11\end{array}$$

Multiplication and division share a step, so work them left to right: $12÷3\times 2=4\times 2=8$, not $12÷6=2$.

Factors

Integers include $-3,-2,-1,0,1,2,3,$ and so on (no decimals or fractions).

Example:

The factors of $12$ are $1,2,3,4,6,12$. It often helps to list them in pairs:

$$1,12$$

$$2,6$$

$$3,4$$