Order of Operations with Rational Numbers

Try two different orders on $-6 + 4 \times \left(-\tfrac{1}{2}\right)$.

• Multiply first: $4 \times \left(-\tfrac{1}{2}\right) = -2$, and then $-6 + (-2) = -8$.
• Add first: $-6 + 4 = -2$, and then $-2 \times \left(-\tfrac{1}{2}\right) = 1$.

Two orders give two different results, $-8$ and $1$. An expression should have the same value for everyone who reads it, so everyone has to use the same order. That shared order is called the order of operations.

Which order makes sense? Look at what the symbols mean. The product $4 \times \left(-\tfrac{1}{2}\right)$ means $4$ copies of $-\tfrac{1}{2}$. That is one quantity, $-2$. A multiplication or a division builds one quantity. Addition and subtraction then combine those quantities. So the shared order builds first and combines second.

In the expression above, the parentheses hold a single number, $-\tfrac{1}{2}$, so there is nothing to work out inside them. You multiply first and then add, so the value of the expression is $-8$.

Each step in the order is one operation on two rational numbers, so you need all four operations ready. Below, $p$ and $q$ stand for any rational numbers.

Adding. Think of numbers as points on a number line. The absolute value of $q$, written $|q|$, is the distance from $0$ to $q$. The sum $p + q$ is the number $|q|$ units from $p$ in the direction of $q$: the positive direction when $q$ is positive, and the negative direction when $q$ is negative. When $q = 0$ you do not move, so the sum is $p$. For example, $-3.5 + 2$ is $2$ units from $-3.5$ in the positive direction, so $-3.5 + 2 = -1.5$.

Subtracting. The additive inverse of $q$, also called its opposite, is the number $-q$ for which $q + (-q) = 0$. Opposites always sum to $0$. Subtracting $q$ undoes adding $q$, and adding $-q$ undoes it too. So subtracting $q$ means adding its additive inverse: $p - q = p + (-q)$. For example, $2 - (-1.5) = 2 + 1.5 = 3.5$.

Multiplying and dividing follow rules you already know, so here is only a short reminder of the words. In a product, the numbers you multiply are the factors. In a quotient $p \div q$, the number $q$ that you divide by is the divisor. The reciprocal of a nonzero number $q$ is $\tfrac{1}{q}$, the number whose product with $q$ is $1$. For example, the reciprocal of $-\tfrac{3}{4}$ is $-\tfrac{4}{3}$, because $-\tfrac{3}{4} \times \left(-\tfrac{4}{3}\right) = \tfrac{12}{12} = 1$. Zero has no reciprocal, because $0$ times any number is $0$, never $1$. Dividing by a nonzero number means multiplying by its reciprocal.

The table collects all four operations with their conditions.

Takeaway. For an expression built from grouping symbols and the four operations, everyone gets the same value because everyone works in the same order: grouping symbols first, then multiplication and division from left to right, then addition and subtraction from left to right. Each step is one operation on rational numbers. To add a number, move by its absolute value in its direction. To subtract a number, add its additive inverse. For a product or quotient of nonzero numbers, an odd count of negatives gives a negative result and an even count gives a positive one. Before you divide, check that the divisor is not $0$.

Next, you will use these skills to solve multi-step word problems and to choose the operation that fits a real-world situation.