Adding and Subtracting Positive and Negative Numbers

Positive and negative signs

Numbers can be either positive, negative, or zero. Positive numbers represent values greater than zero, while negative numbers represent values less than zero.

If a number has no sign in front of it, it is a positive number.

When writing numbers, a positive sign is often assumed. For example, $5$ means exactly the same as $+5$. However, a negative sign must always be written, such as $-5$.

Zero is neither positive nor negative; it is the neutral center point of the number line.

Adding signed numbers

Adding two numbers means combining their values. The direction of movement on a number line depends on the sign of the number being added.

Before adding integers or decimals with mixed signs, remember that adding a positive number moves the value to the right. Adding a negative number moves the value to the left.

When you add a negative number, it is mathematically identical to subtracting a positive number. For example, $6 + (-4)$ produces the same result as $6 - 4$. Both operations move the value $4$ units to the left.

• Like signs simplified: $+(+)$ becomes $+$.
• Unlike signs simplified: $+(-)$ becomes $-$.

Subtracting signed numbers

Subtracting a number means taking its value away. The rules for subtracting integers or signed decimals reverse the direction of addition.

Subtracting a positive number moves the value to the left, which decreases the total. Subtracting a negative number moves the value to the right, which increases the total.

When you subtract a negative number, it is mathematically identical to adding a positive number. For example, $3 - (-5)$ produces the same result as $3 + 5$. Both operations move the value $5$ units to the right.

• Unlike signs simplified: $-(+)$ becomes $-$.
• Like signs simplified: $-(-)$ becomes $+$.

When two like signs are next to each other, they simplify to a positive sign. When two unlike signs are next to each other, they simplify to a negative sign.

Number-line model

Drawing number lines provides a reliable way to visualize any addition or subtraction problem involving signed numbers.

Start at the first number in the expression. Then, use the simplified sign to determine which way to move. Move right for a positive simplified sign, and move left for a negative simplified sign.

Why subtracting a negative adds

Subtracting a negative number often feels counterintuitive, but it can be modeled mathematically using zero pairs. A zero pair consists of one positive unit and one negative unit, which together equal zero.

Suppose you want to calculate $4 - (-2)$. You start with $4$ positive units, but you cannot take away $2$ negative units because you do not currently have any.

To solve this, add $2$ zero pairs to your starting group. Your total value is still exactly $4$, but you now have $2$ negative units available to remove. When you take away those $2$ negative units, the $2$ positive units from the zero pairs are left behind, leaving a final total of $6$.

Worked examples

Applying the sign rules systematically makes calculations reliable. These operations are essential when solving integer word problems.

Example 1: Adding a negative number

Question: What is $-5 + (-3)$?

Method:

1. Identify the adjacent signs. Here, a positive and a negative sign are next to each other, forming $+(-)$.
2. Simplify the unlike signs into a single negative sign. The expression becomes $-5 - 3$.
3. Start at $-5$ on a number line and move $3$ units to the left.

Answer: $-8$.

Check: Adding a negative value means the starting number should decrease. Since $-8$ is less than $-5$, the result makes sense.

Example 2: Subtracting a negative number

Question: Calculate $12 - (-7)$.

Method:

1. Identify the adjacent signs. Here, two negative signs are next to each other, forming $-(-)$.
2. Simplify the like signs into a positive sign. The expression becomes $12 + 7$.
3. Add the numbers normally.

Answer: $19$.

Check: Subtracting a negative is the same as adding its opposite. The opposite of $-7$ is $7$, and $12 + 7 = 19$.

Example 3: Multiple signed operations

Question: Evaluate $-4 - (+6) - (-9)$.

Method:

1. Simplify the first pair of adjacent signs, $-(+)$, which become a negative sign, leaving $-4 - 6$.
2. Simplify the second pair of adjacent signs, $-(-)$, which become a positive sign, leaving $+ 9$.
3. Rewrite the complete simplified expression as $-4 - 6 + 9$.
4. Calculate from left to right. First, find $-4 - 6 = -10$.
5. Finally, calculate $-10 + 9$.

Answer: $-1$.

Check: Starting at $-4$ and moving left $6$ units reaches $-10$. Moving right $9$ units from $-10$ reaches $-1$.

Common mistakes

Avoiding simple sign errors is crucial for accurate calculations.

• Treating the starting sign as an operation: The sign of the first number is just a starting position. Do not apply the "two negatives make a positive" rule to $-3 - 4$. There are no adjacent signs to simplify here. The calculation simply starts at $-3$ and moves $4$ units left, resulting in $-7$.
• Ignoring the minus sign during substitution: When substituting a negative number into an expression with a minus sign, you must keep both signs. For example, if $x = -2$, then $5 - x$ becomes $5 - (-2)$, which simplifies to $5 + 2$.
• Moving the wrong direction on the number line: Always double-check whether the final simplified operation is addition (move right) or subtraction (move left).

Frequently asked questions

Understanding why signs interact the way they do is foundational for all operations with negative numbers.

Do these sign rules apply to fractions and decimals?

Yes. The rules for adjacent signs are universal and apply to fractions, decimals, algebraic terms, and all real numbers.

Why do two negatives make a positive when adding and subtracting?

When a negative sign means "the opposite of," then a double negative means "the opposite of the opposite." The opposite of moving left is moving right, which is the positive direction on a number line.

What happens if there are three signs together?

Simplify them in pairs from the inside out. For example, $5 - (-(-2))$ simplifies step by step. The inner $-(-)$ becomes a positive, turning the expression into $5 - (+2)$, which further simplifies to $5 - 2$.