Adding and Subtracting Signed Decimals and Fractions

Here is what adding means. Let $p$ be the number you start at and $q$ be the number you add. The sum $p + q$ is the number that is $|q|$ away from $p$: in the positive direction when $q$ is positive, and in the negative direction when $q$ is negative. If $q = 0$, you don't move, so $p + 0 = p$. If $p = 0$, you start at $0$ and land on $q$, so $0 + q = q$.

Now think about where you land when neither number is $0$:

• Same sign. You move away from $0$ on the side where you started, so the two distances from $0$ add.
• Different signs. You move back toward $0$, so the two distances cancel at least partly. If you move farther than your starting distance from $0$, you pass $0$ and land on the other side.
• Opposites. The opposite of a number, also called its additive inverse, is the number at the same distance from $0$ on the other side, like $2.5$ and $-2.5$. ($0$ is its own opposite.) Here the distances cancel completely: $2.5 + (-2.5) = 0$.

The table turns each case into a calculation.

Subtraction fits the same thinking. Subtracting a number undoes adding it: add $q$, then subtract $q$, and you are back where you started. Adding $q$ moves you $|q|$ in $q$'s direction, so undoing it moves you the same distance the other way. That is exactly what adding the opposite, $-q$, does. So for any numbers $p$ and $q$, subtracting $q$ gives the same result as adding its opposite:

• $5.2 - 1.5 = 5.2 + (-1.5) = 3.7$
• $5.2 - (-1.5) = 5.2 + 1.5 = 6.7$

The first line matches the subtraction you already know. In the second line, the opposite of $-1.5$ is $1.5$, because $-1.5 + 1.5 = 0$. The opposite of any negative number is positive, so subtracting a negative number always moves you in the positive direction.

Start with decimals. The signs tell you whether to add or subtract the absolute values and which sign the result takes. To do that arithmetic, line up the decimal points so that each digit sits with digits of the same place value: ones with ones, tenths with tenths, hundredths with hundredths. If one number has fewer digits after the decimal point, write zeros to the right of those digits. That does not change the number: $6.4 = 6.40$, because $4$ tenths is the same amount as $40$ hundredths.

Fractions use the same sign reasoning. The one new step is making the pieces the same size. The denominator (the bottom number, never $0$) names the size of the pieces, and the numerator (the top number) counts them. You can combine the counts only when the pieces match, so first rewrite the fractions with a common denominator, a denominator that both fractions share. Multiplying the numerator and the denominator by the same number, as long as it is not $0$, changes how a fraction looks but not its value.

A negative fraction such as $-\frac{7}{8}$ is $7$ eighths in the negative direction, so you can write its sign on the numerator: $\frac{-7}{8}$. Then the numerators carry the signs, and you add them just as you add any positive and negative numbers.

Take $\frac{1}{2} + \left(-\frac{7}{8}\right)$ as an example. Since $1 \times 4 = 4$ and $2 \times 4 = 8$, rewrite $\frac{1}{2}$ as $\frac{4}{8}$ so that both fractions are in eighths. The numerators $4$ and $-7$ have different signs, $7 - 4 = 3$, and $-7$ has the larger absolute value, so $4 + (-7) = -3$. The sum is $-3$ eighths. Here is the whole calculation:

The takeaway: Every sum is a move you can calculate. The sign of the number you add gives the direction, and its absolute value gives the distance. When two numbers have the same sign, their distances from $0$ add and the sum keeps that sign. When they have different signs, the distances cancel at least partly, and the sum takes the sign of the number with the larger absolute value; opposites cancel to exactly $0$. To subtract, add the opposite. For decimals, line up the place values. For fractions, use a common denominator and combine the numerators with their signs. Next, you'll apply properties of operations to add and subtract numbers like these.