Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter goals
By the end of this unit, you should be able to:
• follow order of operations without inventing your own rule,
• evaluate simple exponents as repeated multiplication,
• read the equal sign as a balance relationship,
• handle positive and negative numbers without guessing,
• understand that a fraction is a division,
• tell when a fraction is less than 1, equal to 1, or greater than 1,
• rewrite fractions in equivalent forms,
• add, subtract, multiply, and divide basic fractions,
• move between fractions, decimals, and percentages in basic cases.
Why this matters (no fluff)
Key idea
A surprising number of “algebra problems” are not really algebra problems. They are fraction problems, sign problems, order-of-operations problems, or percent problems wearing an algebra costume.
If those basic moves feel shaky, every later chapter feels harder than it really is. So Unit 0
does not try to look sophisticated. It tries to make the floor stable.
10
1. Order of operations without superstition 11
Note
This unit is where the book explains the arithmetic logic in full. That is intentional. Nothing here should be treated as “obvious,” because this is the floor the rest of the book stands on.
Later chapters will use these ideas constantly, but they will not stop and reteach every basic step. So if a later chapter feels mysterious because of arithmetic, signs, fractions, decimals, or percentages, this is the place to come back to.
Real life: what it’s for
These ideas are not “school-only” skills.
• Order of operations matters in calculators, spreadsheets, and formulas.
• Signed numbers show up in debt, temperature, elevation, and change in position.
• Fractions show up in cooking, medicine, construction, discounts, and probability.
• Percentages show up in taxes, tips, grade reports, interest rates, and data summaries.
If these basics are stable, algebra stops feeling like random punishment.
1. Order of operations without superstition
Key idea
The correct order is:
1. parentheses,
2. exponents,
3. multiplication and division from left to right,
4. addition and subtraction from left to right.
An exponent is repeated multiplication. For example,
23 = 2 · 2 · 2 = 8 .
You do not need advanced exponent rules yet. Here you only need to know that powers are evaluated before multiplication, division, addition, and subtraction.
The important point is not a memory slogan. The important point is this: multiplication and division share a level, and addition and subtraction share a level. So inside each level, you move from left to right.
Chapter 0. Unit 0: minimum kit before algebra
Worked example (no skipped steps)
Evaluate:
18 − 6 ÷ 3 · 2
Step 1: division and multiplication happen before subtraction. They also go from left to right. So we start with:
6 ÷ 3 = 2
Now the expression becomes:
18 − 2 · 2
Step 2: multiply.
2 · 2 = 4
Now the expression becomes:
18 − 4
Step 3: subtract.
18 − 4 = 14
Answer: 14.
Worked example (no skipped steps)
Evaluate:
(5 + 7) ÷ 3
Step 1: parentheses come first.
5 + 7 = 12
Step 2: divide.
12 ÷ 3 = 4
Answer: 4.
2. The equal sign means balance
Key idea
The equal sign does not mean “the answer is coming now.” It means the left side and the right side have the same value.
So in
4 + 3 = 5 + 2
we are saying that both sides equal 7.
3. Signed numbers in one clean picture
13
Worked example (no skipped steps)
Check whether this statement is true:
9 − 4 = 2 + 3
Step 1: compute the left side.
9 − 4 = 5
Step 2: compute the right side.
2 + 3 = 5
Step 3: compare. Both sides equal 5, so the statement is true.
3. Signed numbers in one clean picture
Key idea
Positive means right on the number line. Negative means left on the number line.
Subtracting a number means adding its opposite. So:
5 − 9 = 5 + (−9) .
That is not a trick. It is the meaning of subtraction.
Worked example (no skipped steps)
Compute:
5 − 9
Rewrite subtraction as addition of the opposite:
5 − 9 = 5 + (−9)
Start at 5 and move 9 units left. You land at:
−4
Answer: −4.
Worked example (no skipped steps)
Compute:
(−3)(−8)
Step 1: two negatives multiplied give a positive result.
Step 2: multiply the absolute values.
3 · 8 = 24
Answer:
(−3)(−8) = 24
Chapter 0. Unit 0: minimum kit before algebra
4. Fractions are division
Key idea
A fraction is a number. It is not “two numbers stacked.”
3
4
means:
3 ÷ 4 .
That is the cleanest way to think about it.
The top number is the numerator.
The bottom number is the denominator.
The
denominator tells you what you are dividing by. So:
1
7
= 1 ÷ 4 ,
= 7 ÷ 2 .
4
2
Worked example (no skipped steps)
Explain what
3
5
means.
It means:
3 ÷ 5 .
It can also mean “three parts when the whole is cut into five equal parts,” but the arithmetic meaning is simply:
3 = 3 ÷ 5 .
5
5. Fractions below 1, equal to 1, and above 1
15
Worked example (no skipped steps)
Rewrite
1
1 − 4
so both numbers are written in fourths.
Step 1: write 1 as a fraction with denominator 4. We can do that because: 4
1 = 4
Why? Because:
4 = 4 ÷ 4 = 1 .
4
Step 2: substitute that form of 1.
1
4
1
1 −
=
−
4
4
4
Now both numbers are written in fourths.
5. Fractions below 1, equal to 1, and above 1
Key idea
Once you remember that a fraction is a division, three basic cases become clear:
• If the numerator is smaller than the denominator, the fraction is less than 1.
• If the numerator is equal to the denominator, the fraction is exactly 1.
• If the numerator is bigger than the denominator, the fraction is greater than 1.
Examples:
1
4
7
< 1 ,
= 1 ,
> 1 .
2
4
2
Worked example (no skipped steps)
Explain why
7
2
is greater than 1.
Because:
7 = 7 ÷ 2 .
2
If you divide 7 by 2, you get more than 1. Another way to see it is that: 2 = 1 .
2
So seven halves are more than one whole.
Chapter 0. Unit 0: minimum kit before algebra
Worked example (no skipped steps)
Rewrite
7
2
as a mixed number.
Step 1: break the fraction into full groups of
2 .
2
Since
2 = 1 ,
2
this will tell us how many whole numbers are inside.
Step 2: split the seven halves.
7
2
2
2
1
=
+
+
+
2
2
2
2
2
Step 3: replace each
2
2
with 1.
7
1
= 1 + 1 + 1 +
2
2
Step 4: add the whole numbers.
1 + 1 + 1 = 3
So:
7
1
= 3 +
2
2
Step 5: write the result as a mixed number.
7
1
= 3
2
2
The 3 comes from the three full groups of
2 ,
2
and the 1 is what is left over.
2
6. Equivalent fractions and why they matter
Key idea
Equivalent fractions are different-looking fractions with the same value.
For example:
1
2
3
4
=
=
=
.
2
4
6
8
Why are they equal? Because multiplying the numerator and denominator by the same nonzero number does not change the value. That is really multiplying by 1 in disguise. For
6. Equivalent fractions and why they matter 17
example:
1
2
2
·
=
.
2
2
4
And:
2 = 1 .
2
So the value stays the same.
Recipe (steps that always work)
To make equivalent fractions:
1. choose the factor you want,
2. multiply the numerator by that factor,
3. multiply the denominator by that factor,
4. keep in mind that you changed the look, not the value.
Worked example (no skipped steps)
Rewrite
1
4
with denominator 8.
Step 1: ask what happens to 4 to become 8.
4 · 2 = 8
So we must multiply the denominator by 2.
Step 2: multiply the numerator by the same factor.
1
1 · 2
2
=
=
4
4 · 2
8
Answer:
1
2
=
.
4
8
Worked example (no skipped steps)
Simplify:
30
60
Step 1: ask whether the numerator and denominator share a common factor. They do. Both are divisible by 30.
Step 2: divide the numerator and denominator by the same nonzero number.
30
30 ÷ 30
1
=
=
60
60 ÷ 30
2
Chapter 0. Unit 0: minimum kit before algebra
Answer:
30
1
=
.
60
2
You could also divide both by 10 first:
30
3
=
60
6
and then divide both by 3:
3
1
=
.
6
2
Both roads are legal because both roads divide top and bottom by the same nonzero number.
7. Adding and subtracting fractions without magic
Key idea
You can add or subtract fractions directly only when they are talking about the same-size pieces. That is what the common denominator does. It rewrites both fractions so they are measured in the same kind of pieces.
Recipe (steps that always work)
To add or subtract fractions:
1. check whether the denominators already match,
2. if they do not, find a common denominator,
3. rewrite each fraction with that denominator,
4. add or subtract the numerators,
5. keep the denominator,
6. simplify if possible.
Worked example (no skipped steps)
Compute:
1
3
+
4
8
Step 1: ask whether the denominators already match. They do not. One denominator is 4
and the other is 8.
Step 2: ask whether 4 can be the common denominator. No. Why not? Because the fraction 3
8
would have to be rewritten with denominator 4. That would require changing 8 into 4 by multiplying by a whole-number factor, and that is impossible. There is no whole number 𝑘
7. Adding and subtracting fractions without magic 19
such that:
8 · 𝑘 = 4 .
So 4 cannot be the common denominator.
Step 3: ask whether 8 can be the common denominator. Yes. Why? Because a common denominator must be a denominator that both fractions can use without changing their value.
The fraction
1
4
can be rewritten with denominator 8 because 4 can be turned into 8 by multiplying by 2: 4 · 2 = 8 .
The second fraction already has denominator 8. So 8 is a common denominator. In fact it is the least common denominator.
Step 4: rewrite the first fraction in eighths.
1
1 · 2
2
=
=
.
4
4 · 2
8
The second fraction stays the same:
3
3
=
.
8
8
Step 5: add the numerators because the denominators now match.
2
3
5
+
=
.
8
8
8
Answer:
1
3
5
+
=
.
4
8
8
Worked example (no skipped steps)
Compute:
1
1 − 4
Step 1: write 1 in fourths.
4
1 = 4
Step 2: subtract.
4
1
3
−
=
4
4
4
Answer:
1
3
1 −
=
.
4
4
Chapter 0. Unit 0: minimum kit before algebra
Worked example (no skipped steps)
Compute:
5
1
−
6
4
Step 1: find a common denominator. A convenient choice is 12 because: 6 · 2 = 12
and
4 · 3 = 12 .
Step 2: rewrite both fractions with denominator 12.
5
5 · 2
10
=
=
6
6 · 2
12
1
1 · 3
3
=
=
4
4 · 3
12
Step 3: subtract the numerators.
10
3
7
−
=
12
12
12
Answer:
5
1
7
−
=
.
6
4
12
8. Multiplying and dividing fractions
Key idea
Because a fraction is already a division, fraction multiplication and division are not mysterious once you stay organized.
For multiplication:
𝑎
𝑐
𝑎𝑐
·
=
.
𝑏
𝑑
𝑏𝑑
For division, dividing by a fraction means multiplying by its reciprocal: 𝑎
𝑐
𝑎
𝑑
÷
=
·
.
𝑏
𝑑
𝑏
𝑐
Worked example (no skipped steps)
Compute:
3
10
·
5
12
Step 1: multiply the numerators.
3 · 10 = 30
Step 2: multiply the denominators.
5 · 12 = 60
So:
3
10
30
·
=
5
12
60
8. Multiplying and dividing fractions
21
Step 3: simplify. Both 30 and 60 are divisible by 30. So: 30
30 ÷ 30
1
=
=
60
60 ÷ 30
2
Answer:
3
10
1
·
=
.
5
12
2
Chapter 0. Unit 0: minimum kit before algebra
Worked example (no skipped steps)
Compute:
7
1
÷
8
4
Step 1: keep the first fraction.
7
8
Step 2: change division to multiplication by the reciprocal of the second fraction. The reciprocal of
1
4
is
4 .
1
So:
7
1
7
4
÷
=
·
8
4
8
1
Step 3: multiply numerators and denominators.
7 · 4
28
=
8 · 1
8
Step 4: simplify. Both 28 and 8 are divisible by 4.
28
28 ÷ 4
7
=
=
8
8 ÷ 4
2
Step 5: if you want a mixed number, use the idea from Section 5.
7
1
= 3 .
2
2
Answer:
7
1
7
1
÷
=
= 3 .
8
4
2
2
9. Decimals and percentages without fog
Key idea
A decimal is another way to write a fraction. A percent means “out of 100.”
So:
35
35% =
= 0 . 35 .
100
That is the whole idea. A percentage is just a fraction with denominator 100, written in a shorter way.
9. Decimals and percentages without fog
23
Recipe (steps that always work)
Useful translations:
1. To go from percent to fraction, put the number over 100.
2. To go from percent to decimal, divide by 100.
3. To go from decimal to percent, multiply by 100.
4. In word problems, “of” usually means multiply.
Worked example (no skipped steps)
Convert:
18%
to a decimal.
Step 1: write percent as “out of 100.”
18
18% = 100
Step 2: divide by 100.
18 = 0 . 18
100
Answer:
18% = 0 . 18 .
Worked example (no skipped steps)
Convert:
0 . 45
to a percent.
Step 1: understand what percent means. A percent is “out of 100.” So we want the number written with denominator 100.
Step 2: multiply by 100.
0 . 45 · 100 = 45
Step 3: attach the percent sign.
0 . 45 = 45%
Answer: 45%.
Chapter 0. Unit 0: minimum kit before algebra
Worked example (no skipped steps)
Find 25% of 60.
Step 1: turn the percent into a decimal or fraction.
25
1
25% =
=
= 0 . 25
100
4
Step 2: “of” means multiply.
25% of 60 = 0 . 25 · 60
Step 3: multiply.
0 . 25 · 60 = 15
Answer: 15.
Common mistakes (and how to avoid them)
• doing addition before multiplication just because it appears first,
• reading the equal sign as decoration instead of balance,
• forgetting that subtraction means adding the opposite,
• thinking a fraction is not a number,
• adding denominators when adding fractions,
• choosing a “common denominator” without checking that both fractions can actually be rewritten with it,
• simplifying a fraction without saying what number divides both the numerator and denominator,
• writing a mixed number without showing where the whole-number part came from,
• treating 20% as 20 instead of 20 or 0 . 20.
100
Bridge (why this matters later)
You will use this unit everywhere later.
• In Math Functions, algebraic expressions break when sign and fraction moves are weak.
• In Trigonometry, angle formulas and identities still depend on clean algebra.
• In Calculus, weak fraction control makes limits, derivatives, and integrals look much harder than they are.
• In Statistics and Probability, formulas, percentages, and ratios show up constantly.
Unit 0 is not glamorous, but it is structural.
25
Recap: what you’re taking away
If Unit 0 did its job, then these ideas should now feel less mystical:
• a fraction is a division,
• fractions can be less than 1, equal to 1, or greater than 1,
• equivalent fractions are the same value in different clothes,
• common denominators are about matching the size of the pieces,
• percentages are just fractions out of 100,
• simplifying means dividing top and bottom by the same nonzero number.
Those are not side notes. They are the floor under the rest of the book.
Problems
0.1. Evaluate: 20 − 4 · 3.
0.2. Evaluate: 24 ÷ 6 · 2.
0.3. Evaluate: (18 − 6) ÷ 3.
0.4. Evaluate: 7 + 3 · 4.
0.5. Decide whether this statement is true: 8 + 5 = 10 + 3.
0.6. Decide whether this statement is true: 9 − 2 = 3 + 5.
0.7. Compute: 5 − 9.
0.8. Compute: −8 − 5.
0.9. Compute: −7 + 12.
0.10. Compute: (−6)(+2).
0.11. Compute: (−3)(−8).
0.12. Explain in words what the equal sign means.
0.13. Write 3 as a division statement.
5
0.14. Write 1 as a fraction with denominator 4.
0.15. Rewrite 1 − 1 so both numbers are written in fourths.
4
0.16. Explain why 7 is greater than 1.
2
0.17. Rewrite 7 as a mixed number.
2
0.18. Rewrite 1 as a fraction with denominator 10.
2
0.19. Rewrite 3 as a fraction with denominator 12.
4
Chapter 0. Unit 0: minimum kit before algebra
0.20. Simplify 30 .
60
0.21. Simplify 12 .
18
0.22. Compute: 2 + 5 .
9
9
0.23. Compute: 1 + 1 .
3
6
0.24. Compute: 1 + 3 .
4
8
0.25. Compute: 1 − 1 .
4
0.26. Compute: 5 − 1 .
6
4
0.27. Compute: 3 · 10 .
5
12
0.28. Compute: 7 ÷ 1 .
8
4
0.29. Convert 0 . 45 to a percent.
0.30. Convert 0 . 125 to a percent.
0.31. Convert 18% to a decimal.
0.32. Convert 35% to a decimal.
0.33. Find 25% of 60.
0.34. Find 12% of 250.
Quick checklist (if you miss it, reread and try again)
• I can follow order of operations without inventing new rules.
• I understand that the equal sign means both sides have the same value.
• I can handle basic signed-number movement on the number line.
• I understand that a fraction is a division.
• I can explain why a fraction is less than 1, equal to 1, or greater than 1.
• I can create equivalent fractions and explain why they keep the same value.
• I can add, subtract, multiply, and divide basic fractions without magical jumps.
• I understand that a percent means “out of 100.”
What algebra actually is (and why
people think it is harder than it is)