Lesson 2: Adding and Subtracting Rational Numbers

Property

We can use negative numbers to indicate a decrease or a loss, and positive numbers to indicate an increase or a gain.
An increase or decrease is called a net change.

Examples

• If the temperature was $8^{\circ}$ and changed by $-12^{\circ}$, the new temperature is $-4^{\circ}$.
• A stock was worth 25 dollars per share and is now worth 18 dollars. The net change in value is $-7$ dollars.
• Corinne's bank account had 150 dollars, and she wrote a check for 200 dollars. Her account's value changed by $-200$ dollars, resulting in a balance of $-50$ dollars.

Explanation

Signed numbers are perfect for showing change. A positive number shows a gain or increase, like the temperature rising. A negative number shows a loss or decrease, like a stock price dropping. This is the 'net change.'

Property

To add the integers $p$ and $q$: begin at zero and draw the line segment (arrow) to $p$.
Starting at the endpoint $p$, draw the line segment representing $q$. Where it ends is the sum $p + q$.
An arrow pointing right is positive, and a negative arrow points left.
Each arrow is a quantity with both length (magnitude) and direction (sign).

Examples

• To calculate $3 + 4$, start at 0, move 3 units right, and then move 4 more units right. You land at 7. So, $3 + 4 = 7$.
• To find $-6 + 4$, start at 0, move 6 units left to $-6$, then move 4 units right. You land at $-2$. So, $-6 + 4 = -2$.
• To compute $-3 + (-5)$, start at 0, move 3 units left to $-3$, then move 5 more units left. You land at $-8$. So, $-3 + (-5) = -8$.

Explanation

Think of adding on a number line as taking a journey. Positive numbers are steps to the right, and negative numbers are steps to the left. Your final position is the sum of the integers.

Property

We denote gains by positive numbers and losses by negative numbers.
We can find the net change in a quantity by adding those gains and losses.

Examples

• A football team gains 12 yards and then loses 5 yards. The net change in position is found by the sum $(+12) + (-5) = +7$ yards.

• The temperature drops 8 degrees overnight and then rises 15 degrees during the day. The net change in temperature is $(-8) + (+15) = +7$ degrees.