Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter goals
By the end of this chapter, you will be able to:
• add and subtract signed numbers without guessing,
• multiply and divide signs correctly,
• use quick checks to catch impossible sign errors,
• stop letting minus signs ruin otherwise good work.
Prerequisites (if this is hard, go back to Unit 0)
If adding and multiplying basic whole numbers still feels unstable, go back to Unit 0 first.
Why this matters
Key idea
A huge amount of “bad algebra” is really bad sign control. The math idea may be correct, but one dropped minus sign destroys the whole solution.
Real life: what it’s for
Signed numbers model direction and change.
• A temperature of −5∘ is below zero.
• A bank balance of −40 means debt.
• A vertical velocity can be positive on the way up and negative on the way down.
If signs feel random, real models feel random too.
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The number-line view
Key idea
Positive means right. Negative means left.
Adding a positive number moves you right. Adding a negative number moves you left.
Worked example (no skipped steps)
Compute:
7 + (−3)
Start at 7. Adding −3 means move 3 units left. You land at 4.
Answer: 4.
Subtraction means “add the opposite”
Recipe (steps that always work)
When you see subtraction, rewrite it as addition of the opposite: 𝑎 − 𝑏 = 𝑎 + (− 𝑏)
Then simplify.
Worked example (no skipped steps)
Compute:
6 − 11
Rewrite subtraction as addition of the opposite:
6 − 11 = 6 + (−11)
Start at 6 and move 11 units left:
−5
Answer: −5.
Worked example (no skipped steps)
Compute:
−8 − (−3)
Subtracting a negative means adding the opposite:
−8 − (−3) = −8 + 3
Now move 3 units right from −8:
−5
Answer: −5.
Chapter 2. Signed numbers without confusion
Multiplication and division signs
Key idea
The sign rules are:
(+)(+) = + ,
(−)(−) = + ,
(+)(−) = − ,
(−)(+) = − .
Same signs give positive. Different signs give negative.
Worked example (no skipped steps)
Compute:
(−4)(7)
Different signs give negative. Multiply the absolute values: 4 · 7 = 28
So:
(−4)(7) = −28
Worked example (no skipped steps)
Compute:
−18
−3
Two negatives divided give a positive:
−18 = 6
−3
Quick mental checks
Key idea
Before moving on, ask whether the sign makes sense.
• −5 + 2 should stay negative because the negative part is larger.
• (−100)(3) cannot be positive.
• (−12) /(−4) cannot be negative.
These tiny checks catch many errors early.
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Bridge (why this matters later)
Sign control matters in all later books. Slope can be positive or negative, trig values change sign by quadrant, derivatives change sign, and deviations in statistics can be above or below the mean.
Common mistakes (and how to avoid them)
• reading −8 − (−3) as “two minuses make plus” without understanding why,
• forgetting that subtraction and multiplication use different sign logic,
• dropping a negative sign between lines,
• treating every pair of minus signs the same way without asking whether the operation is subtraction, multiplication, or division.
Recap: what you’re taking away
Minus signs are not decoration. They carry meaning.
If signed numbers feel stable, later algebra becomes much safer because you stop losing correct solutions to one sign error.
Problems
2.1. Compute: 7 − 12.
2.2. Compute: −9 + 4.
2.3. Compute: −6 − 5.
2.4. Compute: −8 − (−3).
2.5. Compute: (−5)(−6).
2.6. Compute: (−4)(9).
2.7. Compute: −21 .
7
2.8. Compute: −24 .
−6
2.9. Compute: −7 + 15.
2.10. Compute: −11 − (−4).
2.11. Compute: (−8)(−7).
2.12. Compute: 36 .
−9
2.13. Compute: −4 + (−9).
2.14. Compute: 13 − (−6).
2.15. Compute: (−8)(−5).
Chapter 2. Signed numbers without confusion
2.16. Compute: 42 ÷ (−7).
2.17. Compute: −18 ÷ (−3).
2.18. Compute: −9 + 14.
2.19. Compute: −3 − 11.
2.20. Compute: (−6)(4).
Quick checklist (if you miss it, reread and try again)
• I can read subtraction as “add the opposite.”
• I know the sign rules for multiplication and division.
• I can catch sign errors before moving on.
Fractions without panic