Signed numbers

Step 1 of 5Warm-up

One quick question from the last lesson. Answer from memory first. Pulling it out of your head is what makes it stick.

What placement score do you need to be allowed into MATH 1050?

A score of 400 places you into MATH 1050 (College Algebra). A 300 places you into MATH 1010.

Step 2 of 5The idea

A signed number is a number with a + or − sign. Think of a number line: positive numbers are to the right of 0, negative numbers are to the left.

Adding. Same signs: add the sizes, keep the sign. Different signs: subtract the smaller size from the bigger one and keep the sign of the bigger one.

Subtracting. Subtracting a number is the same as adding its opposite: $a - b = a + (-b)$.

Multiplying and dividing. Same signs give a positive answer. Different signs give a negative answer.

The "size" of a number is its absolute value, its distance from 0. For example, $|-7| = 7$.

Why this shows up on the placement test: almost every algebra problem has negative numbers inside it. One sign slip turns a right method into a wrong answer.

Step 3 of 5Worked examples

Example 1: $-8 + 3$

The signs are different, so subtract the sizes: $8 - 3 = 5$.
The bigger size is 8, and it was negative, so the answer is negative.
$-8 + 3 = -5$. Check with money: you owe $8 and pay back $3. You still owe $5.

Example 2: $-4 - (-9)$, then $(-6)(-2) \div (-3)$

Change subtracting into adding the opposite: $-4 - (-9) = -4 + 9$.
Different signs: $9 - 4 = 5$, and 9 is the bigger size (positive), so the answer is $5$.
For the second one, go left to right. $(-6)(-2) = 12$ (same signs → positive).
$12 \div (-3) = -4$ (different signs → negative).

Step 4 of 5Practice

Type your answer and press Check. If you miss twice, open the worked solution. Then mark how it went.

$-4 + (-7)$

Same signs: add the sizes, $4 + 7 = 11$, and keep the negative sign. Answer: $-11$.

$-9 + 15$

Different signs: $15 - 9 = 6$. The bigger size, 15, is positive, so the answer is $6$.

$3 - 13$

Rewrite as $3 + (-13)$. Different signs: $13 - 3 = 10$. The bigger size is negative, so the answer is $-10$.

$-6 - (-8)$

Subtracting $-8$ means adding $8$: $-6 + 8 = 2$.

$(-7)(6)$

$7 \times 6 = 42$. Different signs, so the answer is negative: $-42$.

$\dfrac{-36}{-4}$

$36 \div 4 = 9$. Same signs, so positive: $9$.

Which expression has a positive value?

$(-2)(-3)(-1)$
$-5 + 2 - 4$
$(-8) \div (-2)$
$-3 - (-1)$

(a) $(-2)(-3) = 6$, then $6 \cdot (-1) = -6$. Negative.

(b) $-5 + 2 = -3$, then $-3 - 4 = -7$. Negative.

(c) Same signs, so positive: $(-8) \div (-2) = 4$. This one.

(d) $-3 + 1 = -2$. Negative.

The temperature is $-8^\circ$ at 6 a.m. and rises $5^\circ$ by noon. What is the temperature at noon?

Rising means adding: $-8 + 5$. Different signs: $8 - 5 = 3$, and the bigger size is negative. Answer: $-3^\circ$.

Step 5 of 5Wrap-up

You can now work with negative numbers using three rules: add the opposite to subtract; same signs give a positive product; different signs give a negative product.

Read or watch next: OpenStax Elementary Algebra 2e, 1.3 Add and Subtract Integers and 1.4 Multiply and Divide Integers (free textbook, with more practice at the end of each section).

Also good: Khan Academy: Negative numbers (short videos). Quick rules: Number skills cheat sheet.

Something unclear? Ask your teacher bot a follow-up question, for example: "Why does minus a negative become plus?" It knows this course and can explain things another way.