Algebra 1
Negative numbers
Negatives turn up in almost every answer from here on. Getting comfortable with them now means a later problem adds only one new wrinkle at a time, instead of two at once. The friendliest way in is to think of a negative as a debt: −5 means you owe $5.
Start with which negative is bigger, because that's where the eye gets fooled. Would you rather owe $5 or owe $2? Owing less leaves you better off, so −2 is the bigger number and −5 is the smaller one, even though 5 is bigger than 2. On the number line it's the same story: the bigger the debt, the farther left you sit, and farther left means smaller 1.5.f1.
There are really two separate facts about −5 here, and keeping them apart is what makes negatives stop feeling backwards. One is its value (−5, which can be negative). The other is its distance from 0: how far it sits from 0, ignoring direction, which is 5, and a distance is never negative.
So −5 is farther from 0 than −2 (distance 5 beats distance 2), and at the same time −5 is the smaller value. That distance has a name, absolute value, written |−5| = 5.
Adding is a walk along the line. Adding a positive steps you right; adding a negative steps you left. To work out −3 + (−5), stand on −3 and take 5 steps left, landing on −8. The debt view agrees: owe $3, then owe $5 more, and you owe $8.
Subtracting has one move that catches almost everyone, so slow down for it. The plain cases first: subtracting is the same as adding the opposite, so 5 − 8 is 5 + (−8) = −3 (stand on 5, walk 8 left).
Now the surprising one. Subtracting a negative looks like it should make a number smaller, so 3 − (−4) pulls you toward −1 or so. But taking away a debt makes you richer: wipe out a $4 debt and you're $4 better off, so 3 − (−4) climbs to 3 + 4 = 7. Subtracting a negative adds.
Read the two signs separately as you go: the first is the operation (subtract), the second is the number's sign (a negative four), and the move stops being slippery.
Multiplying and dividing come down to counting signs. Same signs give a positive; different signs give a negative. So (−6)(3) = −18 and (−6)(−3) = 18; −20 ÷ 4 = −5 and −20 ÷ (−5) = 4. For a longer chain, just count the negative factors: an even count comes out positive, an odd count negative.
New words
- 1.5.d1 Opposite (additive inverse): the same distance from 0 on the other side; the opposite of 5 is -5. Subtracting a number is the same as adding its opposite.
- 1.5.d2 Absolute value |a|: how far a is from 0 on the number line, a distance, so it is never negative. |5| = 5 and |-5| = 5 (both sit 5 units from 0); |0| = 0. Opposite and absolute value pair up: opposites share the same absolute value (|5| = |-5|) but have opposite signs.
- 1.5.d3 Sign rules (×, ÷): same signs → positive; different signs → negative. In a product, an even number of negative factors → positive; an odd number → negative.
Worked example
- 1.5.w1 −3 + (−5) = −8. Two debts pile up; or stand on −3 and step 5 to the left.
- 1.5.w2 4 + (−9) = −5. Adding a negative steps left: start at 4, walk 9 left.
- 1.5.w3 5 − 8 = 5 + (−8) = −3. Subtracting is adding the opposite, so this is a step 8 to the left of 5.
- 1.5.w4 3 − (−4) = 3 + 4 = 7. Subtracting a negative adds: wiping out a $4 debt leaves you $4 richer. This is the one to slow down on.
- 1.5.w5 (−6)(3) = −18. Different signs, so the product is negative.
- 1.5.w6 (−6)(−3) = 18. Same signs, so the product is positive.
- 1.5.w7 −20 ÷ 4 = −5. Different signs, so the quotient is negative.
- 1.5.w8 −20 ÷ (−5) = 4. Same signs, so the quotient is positive.
- 1.5.w9 |5| = 5, |−5| = 5, |0| = 0. Each is a distance from 0, so none can be negative.
- 1.5.w10 Which is farther from 0, −8 or 3? Compare distances: |−8| = 8 and |3| = 3, so −8 is farther, even though −8 is the smaller value. Distance and value are two different questions.
Before the set mixes things up, reset on a clean one: −10 + 10. Start on −10 and step 10 to the right, landing exactly on 0. A debt of $10 paid off by $10 leaves you even.
There's a tidy-sounding rule that "two negatives make a positive." It's true for multiplying, but not for adding. Owing $3 and then owing $5 more leaves you owing $8, so −3 + (−5) = −8, not +8. When you see two negatives, check which operation you're actually doing before you reach for that rule.
Check yourself
- 1.5.c1 Which is greater, −7 or −3? Explain with the number line or with debt. (−3, because owing $3 is better off than owing $7, and −3 sits to the right of −7 on the line.)
- 1.5.c2 Rewrite 6 − (−2) as an addition, then evaluate it. (6 − (−2) = 6 + 2 = 8, because subtracting a negative adds.)
- 1.5.c3 (−2)(−2)(−2): is the result positive or negative before you multiply, and how do you know? (Three negative factors, an odd count, so it's negative, which the arithmetic confirms: −8.)
You can now order negatives by thinking in debt, add and subtract by walking the line, turn subtracting-a-negative into adding, multiply and divide by counting signs, and read absolute value as a distance that's never negative.
When a set switches between adding, subtracting, multiplying, and dividing signs, you have to pick the right move each time instead of repeating one, which is harder now and is exactly what carries the skill into next week. Each answer is at the end of the lesson, and the worked examples above cover every move in the set, including the subtract-a-negative one.
Add / subtract:
Reveal answerHide to problem 1
-11Reveal answerHide to problem 2
-4Reveal answerHide to problem 3
5Reveal answerHide to problem 4
-4Reveal answerHide to problem 5
-8Reveal answerHide to problem 6
7Reveal answerHide to problem 7
-5Reveal answerHide to problem 8
0Multiply / divide:
Reveal answerHide to problem 9
-20Reveal answerHide to problem 10
12Reveal answerHide to problem 11
-21Reveal answerHide to problem 12
-4Reveal answerHide to problem 13
6Reveal answerHide to problem 14
-3Reveal answerHide to problem 15
-8Reveal answerHide to problem 16
1Absolute value (distance from 0):
Reveal answerHide to problem 17
7Reveal answerHide to problem 18
3Reveal answerHide to problem 19
0Reveal answerHide to problem 20
10(In 6 and 7 you're subtracting a negative; 15 and 16 are chains of negatives, so track each sign; for 17 to 20, read absolute value as distance from 0, which is always ≥ 0.)