Integrated Math 1 · Unit 0 · Review of the Essentials

Integers

Counting stops at zero; measuring does not. Signed numbers say both how much and which direction — and every rule in this section is a fact about one fold at zero.

bookSHelf  ·  Integrated Math 1  ·  §0.2  ·  a self-paced section

bookSHelf · Integrated Math 11 / 42
The fold-at-zero image carries the entire section — introduce it here and keep returning to it.
Integers · bookSHelf Integrated Math 1§0.2

Outline — by the end of this section you will be able to

Objectives

  1. Locate positives and negatives on a number line and compare any two §0.2.1
  2. Find the opposite of a number, and say what its absolute value measures Def 0.2.3–0.2.4
  3. Add and subtract signed numbers, and explain why subtraction is addition Def 0.2.5
  4. Multiply and divide signed numbers, predicting the sign first §0.2.4
  5. Apply the order of operations to expressions containing negatives Ex 0.2.9–0.2.10
bookSHelf · Integrated Math 12 / 42
The five section SLOs verbatim.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — Negative numbers and the number line

Counting stops at zero. Measuring does not.

A thermometer keeps reading past the freezing mark, a bank balance keeps dropping past empty, an elevation keeps descending past the shoreline.

In every one of those we need a number that says both how much and which direction — and a whole number cannot do that.

bookSHelf · Integrated Math 13 / 42
Each of these situations already has a zero built into it. That is the point of the table next.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — Each one already had a zero

Quantities that fall below a natural zero

SituationDescribed in wordsAs an integer
Lowest point in Death Valley282 feet below sea level−282-282
Home freezer setting4 degrees below zero−4-4
Overdrawn checking account65 dollars owed to the bank−65-65
Golf score for the round3 strokes under par−3-3
Parking garage level2 floors below the lobby−2-2

Table 0.2.1: Everyday quantities that fall below a natural zero.

Sea level, the freezing mark, an empty account, par, the lobby — each situation supplies its own zero, and the minus sign records that we are on the far side of it.

bookSHelf · Integrated Math 14 / 42
The number system we need is the whole numbers plus a mirror image of themselves.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — Definition

Integer

Definition 0.2.1 — Integer

The integers are the whole numbers together with their negatives:

…,−4,−3,−2,−1,0,1,2,3,4,…\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots

Every integer is either negative, zero, or positive, and no integer is more than one of those.

bookSHelf · Integrated Math 15 / 42
Three mutually exclusive categories — zero belongs to neither outer one.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — Definition

Number line

Definition 0.2.2 — Number line

A number line is a line with one point chosen as zero and a fixed unit distance marked off in both directions. The positives sit to the right of zero; the negatives sit the same distances to the left.

The spacing matters. Each step of one unit is the same size everywhere, which is what makes the picture trustworthy — the gap from −7-7 to −6-6 is exactly the gap from 40 to 41.

A number line with a chosen zero and a fixed unit distance A horizontal line with arrowheads at both ends. Ticks are marked at equal spacing and labelled from negative seven on the left to seven on the right. The tick at zero is taller and highlighted, and is labelled the chosen zero. The negative integers are labelled as sitting to the left of zero and the positive integers to the right. Two brackets, one spanning negative six to negative five and one spanning four to five, are drawn the same width and both labelled one unit, showing that the step size is the same everywhere on the line. −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 the chosen zero the negative integers sit to the left the positive integers sit to the right 1 unit 1 unit the same distance everywhere on the line

Definition 0.2.2: one point chosen as zero, one unit distance, repeated.

bookSHelf · Integrated Math 16 / 42
Uniform spacing is what lets you read order off position.
Integers · bookSHelf Integrated Math 1§0.2

Insight — a mirror at zero

Zero is the crease

Fold the number line at zero and the two halves land on each other: 3 covers −3-3, 8 covers −8-8.

Every fact about negatives in this section is really a fact about that fold. Zero is the crease, which is why it is neither positive nor negative.

bookSHelf · Integrated Math 17 / 42
Opposites, absolute value and the sign rules are all this one image.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — Order

Greater than everything to its left

ComparisonTrue statementWhy
3 and 83<83 < 83 is left of 8
−3-3 and 8−3<8-3 < 8every negative is left of every positive
−3-3 and −8-8−3>−8-3 > -8−3-3 is right of −8-8
−3-3 and 0−3<0-3 < 0negatives are left of zero
−100-100 and −1-1−100<−1-100 < -1−100-100 is far to the left

That single rule settles every comparison, including the ones that feel backwards.

Where intuition misfires. Five is bigger than two, so it feels like −5-5 should beat −2-2. It does not.

−2∘-2^\circ is a warmer day than −5∘-5^\circ. Owing $2 leaves you better off than owing $5.

bookSHelf · Integrated Math 18 / 42
Among negatives, the number that looks biggest is smallest. Reading a size and reading a position are different jobs.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — Worked example

Ordering a mixed list

Example 0.2.1 — arrange −9,  4,  0,  −1,  7,  −12-9,\; 4,\; 0,\; -1,\; 7,\; -12 from least to greatest, and say which two are farthest apart

Step 1 — Separate by sign. Negatives as a block, then 0, then positives.

Steps 2–3 — Order within each block. Among negatives, the larger digit sits farther left and is therefore smaller.

−12,  −9,  −1,  0,  4,  7-12,\; -9,\; -1,\; 0,\; 4,\; 7

Step 4 — Farthest apart. The extremes are −12-12 and 77: 12 units up to zero, then 7 more — 19 units.

bookSHelf · Integrated Math 19 / 42
Hold on to the 19 — absolute value reproduces it without counting, later in the deck.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — Definition

Opposite (additive inverse)

Definition 0.2.3 — Opposite (additive inverse)

The opposite of a number is the number the same distance from zero on the other side. The opposite of 6 is −6-6; the opposite of −6-6 is 6. Zero is its own opposite.

A pair of opposites always sums to zero:

a+(−a)=0a + (-a) = 0
A number and its opposite are the same distance from zero A number line with zero marked in the middle. First a point appears at six, with a bracket underneath measuring six units from zero out to it. Then a dashed arc sweeps across zero and a matching point appears at negative six, with a bracket of the same six-unit width measuring from zero out to it in the other direction. The two brackets are drawn identically, so the two distances are visibly equal. Finally the line six plus negative six equals zero appears, and below it the general statement a plus negative a equals zero. 0 6 6 units from zero the same distance, the other side of zero −6 6 units from zero 6 + (−6) = 0 a + (−a) = 0

Definition 0.2.3: a number and its opposite sit the same distance from zero, on opposite sides.

That fact is the whole engine behind subtraction — §0.2.3 spends its time on it.

bookSHelf · Integrated Math 110 / 42
"Additive inverse" is the same idea named for what it does in a sum.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — One symbol, three jobs

What the minus sign is doing

UseExampleRead as
A negative number−7-7"negative seven"
The opposite of−a-a"the opposite of aa"
Subtraction9−49 - 4"nine minus four"

The middle row is the one that trips people up.

−a-a is not necessarily a negative number. If a=−3a = -3, then −a=3-a = 3. The expression means "flip the sign of whatever aa is" — and flipping twice returns you to the start: −(−6)=6-(-6) = 6.

bookSHelf · Integrated Math 111 / 42
Reading −(−6)-(-6) aloud as "the opposite of negative six" settles it faster than any rule.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — Worked example

Reading the minus sign in three jobs

Example 0.2.2 — evaluate −a-a at a=11a = 11 and at a=−11a = -11; then simplify −(−(−4))-(-(-4))

−a=−(11)=−11−a=−(−11)=11-a = -(11) = -11 \qquad -a = -(-11) = 11

Both came from the same expression. The sign of the result depends entirely on the sign of aa — which is exactly why −a-a cannot be read as "a negative number."

−(−(−4))=−(4)=−4-(-(-4)) = -(4) = -4

Three flips starting from 4 leave you on the opposite side; an odd number of flips always does.

bookSHelf · Integrated Math 112 / 42
Odd/even flip counting returns in §0.2.4 as the negative-factor count.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.1 — Your turn

Try It Now 0.2.1

Try It Now 0.2.1 — order −6,  2,  −15,  0,  −1-6,\; 2,\; -15,\; 0,\; -1; evaluate −x-x at x=−9x = -9; simplify −(−(−(−2)))-(-(-(-2)))

−15,  −6,  −1,  0,  2-15,\; -6,\; -1,\; 0,\; 2

Substitute with parentheses so the two signs stay separate: −x=−(−9)=9-x = -(-9) = 9.

Four opposite signs applied to 2 — an even count returns to the start: −(−(−(−2)))=2-(-(-(-2))) = 2.

bookSHelf · Integrated Math 113 / 42
Answers: the list; 9; 2.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.2 — Absolute value

Sometimes direction is beside the point

A hiker 400 feet below the trailhead and a drone 400 feet above it are in opposite directions but the same distance away.

A tool that reports distance only, ignoring direction, is exactly what absolute value is.

bookSHelf · Integrated Math 114 / 42
"How far" versus "which way" is the distinction to keep saying out loud.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.2 — Definition

Absolute value (magnitude)

Definition 0.2.4 — Absolute value (magnitude)

The absolute value of a number is its distance from zero, written with vertical bars. Distance is never negative, so ∣a∣≥0|a| \ge 0 always, and ∣a∣=0|a| = 0 only when a=0a = 0.

∣6∣=6∣−6∣=6∣0∣=0|6| = 6 \qquad |-6| = 6 \qquad |0| = 0
Absolute value measures the distance from zero A number line with zero marked in the middle. A point appears at negative six and a double-headed arrow measures the distance from zero out to it, labelled six. Then a point appears at six and a second double-headed arrow measures that distance, also labelled six. The two measurements are the same length. Below, the readings appear: the absolute value of negative six is six, the absolute value of six is six, and the absolute value of zero is zero. A closing line notes that a distance never runs backwards, so an absolute value is never negative. 0 −6 a distance of 6 6 a distance of 6 |−6| = 6 |6| = 6 |0| = 0 a distance never runs backwards, so an absolute value is never negative

Definition 0.2.4: absolute value reports the distance from zero, so it is never negative.

Opposites always have equal absolute values, so ∣a∣=∣−a∣|a| = |-a|. And ∣n∣=12|n| = 12 leaves two candidates, 12 and −12-12 — not a flaw, but the correct answer to a question that only asked about distance.

bookSHelf · Integrated Math 115 / 42
It strips the sign and reports only magnitude.
Integers · bookSHelf Integrated Math 1§0.2

Context Pause — size and direction are separate questions

Throwing away the sign, on purpose

A quality report might ask how far a part is from spec, not whether it is oversized or undersized.

Absolute value answers the first question and deliberately discards the second, which is why ∣−0.04∣=∣0.04∣=0.04|-0.04| = |0.04| = 0.04 treats both misses as the same size error.

bookSHelf · Integrated Math 116 / 42
Discarding information is the feature, not a limitation.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.2 — Absolute value as a grouping symbol

Placement is not cosmetic

ExpressionValueNote
∣−4∣\lvert -4 \rvert4distance from zero
−∣−4∣-\lvert -4 \rvert−4-4negate after taking absolute value
∣5−9∣\lvert 5 - 9 \rvert4simplify inside first
∣5∣−∣9∣\lvert 5 \rvert - \lvert 9 \rvert−4-4a different expression entirely
∣−3∣+∣−8∣\lvert -3 \rvert + \lvert -8 \rvert11two separate distances, then add

The middle two rows differ only in where the bars sit — and disagree by a sign.

Do not distribute across the bars. ∣3−10∣=7|3 - 10| = 7, not ∣3∣−∣10∣=−7|3| - |10| = -7.

A minus sign outside stays outside: −∣−9∣=−9-|{-9}| = -9. The bars protect only what is inside them.

bookSHelf · Integrated Math 117 / 42
Bars also measure a gap: the distance from −12-12 to 7 is ∣−12−7∣=19|-12 - 7| = 19 — the count from Example 0.2.1, without counting.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.2 — Worked example

Simplifying with bars in the way

Example 0.2.3 — simplify (a) ∣6−14∣\lvert 6 - 14 \rvert; (b) −∣6−14∣-\lvert 6 - 14 \rvert; (c) ∣6∣−∣14∣\lvert 6 \rvert - \lvert 14 \rvert; (d) 8−∣2−7∣8 - \lvert 2 - 7 \rvert

(a)    ∣6−14∣=∣−8∣=8(b)    −∣−8∣=−8\text{(a)}\;\; \lvert 6 - 14 \rvert = \lvert -8 \rvert = 8 \qquad \text{(b)}\;\; -\lvert -8 \rvert = -8 (c)    ∣6∣−∣14∣=6−14=−8(d)    8−∣−5∣=3\text{(c)}\;\; \lvert 6 \rvert - \lvert 14 \rvert = 6 - 14 = -8 \qquad \text{(d)}\;\; 8 - \lvert -5 \rvert = 3

Parts (a) and (c) use the same three symbols in a different arrangement and disagree by a sign. The bars are grouping symbols, so where they open and close changes the problem.

bookSHelf · Integrated Math 118 / 42
Answers: 8, −8-8, −8-8, 3.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.2 — Worked example

Two numbers with the same absolute value

Example 0.2.4 — find every integer nn with ∣n∣=7\lvert n \rvert = 7; then explain why none satisfies ∣n∣=−7\lvert n \rvert = -7

∣n∣=7\lvert n \rvert = 7 says nn sits exactly 7 units from zero. Going right lands on 7; going left lands on −7-7. Nothing else is 7 units away.

n=7orn=−7n = 7 \quad \text{or} \quad n = -7

∣n∣=−7\lvert n \rvert = -7 asks for a number whose distance from zero is negative. Distance counts units travelled, and travelling a negative number of units is not a thing you can do — no solution.

bookSHelf · Integrated Math 119 / 42
Two answers is the normal case; asking for a negative distance has none.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.2 — Your turn

Try It Now 0.2.2

Try It Now 0.2.2 — simplify ∣3−11∣\lvert 3 - 11 \rvert, −∣3−11∣-\lvert 3 - 11 \rvert, ∣−5∣+∣−2∣\lvert -5 \rvert + \lvert -2 \rvert; then find every nn with ∣n∣=4\lvert n \rvert = 4

∣3−11∣=8−∣3−11∣=−8∣−5∣+∣−2∣=7\lvert 3 - 11 \rvert = 8 \qquad -\lvert 3 - 11 \rvert = -8 \qquad \lvert -5 \rvert + \lvert -2 \rvert = 7

Two integers sit 4 units from zero, one on each side: n=4n = 4 or n=−4n = -4.

bookSHelf · Integrated Math 120 / 42
Part 3 is two separate distances added — not one bar around a sum.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.3 — Adding and subtracting integers

Adding is movement along the line

Start at the first number; a positive addend moves right, a negative addend moves left. Two cases cover everything.

Same signs

Add the magnitudes, keep the sign — both movements go the same way, so they accumulate.

4+7=11−4+(−7)=−114 + 7 = 11 \qquad -4 + (-7) = -11

Different signs

Subtract the smaller magnitude from the larger, take the sign of the larger — the movements partly cancel.

−9+4=−59+(−4)=5-9 + 4 = -5 \qquad 9 + (-4) = 5
bookSHelf · Integrated Math 121 / 42
A debt of $9 combined with a credit of $4 leaves you $5 in debt.
Integers · bookSHelf Integrated Math 1§0.2

Insight — a tug of war decides the sign

Two teams, pulling

Picture the two numbers as teams pulling in opposite directions. Same signs means both pull the same way, so the pulls add. Different signs means they pull against each other.

The smaller pull cancels part of the larger, and the winner's direction is the sign of the answer.

bookSHelf · Integrated Math 122 / 42
The sign records which direction you ended up from zero, not which direction you started.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.3 — Every case in one table

Which magnitude crosses zero

ProblemSignsWorkResult
−6+(−8)-6 + (-8)same6+8=146 + 8 = 14, keep negative−14-14
−6+8-6 + 8different8−6=28 - 6 = 2, take sign of 822
6+(−8)6 + (-8)different8−6=28 - 6 = 2, take sign of −8-8−2-2
6+86 + 8same6+8=146 + 8 = 14, keep positive1414
−7+7-7 + 7differentequal magnitudes cancel00

The magnitudes decide who crosses zero, and crossing zero is what decides the sign.

That last row is the additive-inverse fact from §0.2.1 showing up inside the addition rule.

bookSHelf · Integrated Math 123 / 42
Beginning at −9-9 and walking 4 right leaves you at −5-5 — 4 steps were not enough to cross the 9-unit gap.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.3 — Worked example

A running account balance

Example 0.2.5 — an account starts at $120; a $200 rent payment clears Monday, a $65 deposit arrives Wednesday, a $30 fee is charged Friday

Step 1 — Signed numbers. Money in is positive, money out is negative.

120+(−200)+65+(−30)120 + (-200) + 65 + (-30)

Steps 2–4 — Combine left to right.

120+(−200)=−80→−80+65=−15→−15+(−30)=−45120 + (-200) = -80 \quad\rightarrow\quad -80 + 65 = -15 \quad\rightarrow\quad -15 + (-30) = -45

Answer: a balance of −45-45 — the account is overdrawn by $45.

bookSHelf · Integrated Math 124 / 42
Interpreting the sign at the end is part of the answer, not decoration.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.3 — Definition

Subtraction

Definition 0.2.5 — Subtraction

For any numbers aa and bb, the difference a−ba - b is the sum of aa and the opposite of bb:

a−b=a+(−b)a - b = a + (-b)

To subtract, add the opposite. This does not sit alongside the addition rules — it replaces the need for separate subtraction rules.

Subtracting is adding the opposite The expression negative three minus negative seven appears, then is rewritten as negative three plus seven, under the rule to subtract, add the opposite. On the number line below, a point appears at negative three. An arrow then draws itself rightward across seven units, labelled plus seven, and lands on four. The answer, negative three minus negative seven equals four, appears underneath. −3 − (−7) = −3 + 7 to subtract, add the opposite −3 0 4 +7, seven units to the right −3 − (−7) = −3 + 7 = 4

Definition 0.2.5: rewriting a subtraction as adding the opposite turns it into one walk.

bookSHelf · Integrated Math 125 / 42
Convert every subtraction to an addition, then apply the two addition cases.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.3 — The line worth slowing down on

Subtracting a negative adds

ExpressionRewrittenResult
12−2012 - 2012+(−20)12 + (-20)−8-8
−12−20-12 - 20−12+(−20)-12 + (-20)−32-32
−12−(−20)-12 - (-20)−12+20-12 + 2088
12−(−20)12 - (-20)12+2012 + 203232
0−(−6)0 - (-6)0+60 + 666

Rewriting before computing costs one extra line and eliminates most sign errors.

Read it as removing a debt. You are $3 in the hole; someone cancels a $7 debt. Going up $7 from $3 in the hole leaves you $4 to the good.

Or as a missing addend. −3−(−7)-3 - (-7) asks what you add to −7-7 to reach −3-3: move 4 units right, so +4+4.

bookSHelf · Integrated Math 126 / 42
The story and the number line agree — a good sign the rule is forced rather than chosen.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.3 — Worked example

A temperature swing

Example 0.2.6 — 6 a.m. is −11∘-11^\circF and 2 p.m. reaches 9∘9^\circF; that evening it falls back to −4∘-4^\circF

A change is the ending value minus the starting value.

9−(−11)=9+11=209 - (-11) = 9 + 11 = 20

A rise of 20 degrees — matching the line: 11 units up to zero, then 9 more.

−4−9=−4+(−9)=−13-4 - 9 = -4 + (-9) = -13

A change of −13-13 means it dropped, and the size of the drop is ∣−13∣=13\lvert -13 \rvert = 13 degrees.

bookSHelf · Integrated Math 127 / 42
Sign gives direction of change; absolute value gives its size. Both are needed to answer in English.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.3 — Worked example

Untangling a long chain

Example 0.2.7 — simplify   7−15−(−4)+(−6)−(−9)\;7 - 15 - (-4) + (-6) - (-9)

Step 1 — Rewrite every subtraction. Three subtraction signs, each becoming an addition of the opposite.

7+(−15)+4+(−6)+97 + (-15) + 4 + (-6) + 9

Steps 2–5 — Left to right.

−8  →  −4  →  −10  →  −1-8 \;\rightarrow\; -4 \;\rightarrow\; -10 \;\rightarrow\; -1

Check by grouping. Positives total 7+4+9=207 + 4 + 9 = 20; negatives total 15+6=2115 + 6 = 21. One unit onto the negative side — agrees.

bookSHelf · Integrated Math 128 / 42
Rewriting first is what keeps a stray minus sign from attaching to the wrong number partway through.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.3 — Your turn

Try It Now 0.2.3

Try It Now 0.2.3 — simplify −14+6-14 + 6, −14−6-14 - 6, −14−(−6)-14 - (-6), and 5−12−(−8)+(−3)5 - 12 - (-8) + (-3)

−14+6=−8−14+(−6)=−20−14+6=−8-14 + 6 = -8 \qquad -14 + (-6) = -20 \qquad -14 + 6 = -8

The first and third land on the same answer by different routes — one is already an addition, the other becomes one.

5+(−12)+8+(−3)=−7  →  1  →  −25 + (-12) + 8 + (-3) = -7 \;\rightarrow\; 1 \;\rightarrow\; -2
bookSHelf · Integrated Math 129 / 42
Answers: −8-8, −20-20, −8-8, −2-2.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.4 — Multiplying and dividing integers

Size and sign are two separate jobs

First factorSecond factorSign of productExample
positivepositivepositive4⋅6=244 \cdot 6 = 24
negativenegativepositive(−4)(−6)=24(-4)(-6) = 24
positivenegativenegative4(−6)=−244(-6) = -24
negativepositivenegative(−4)(6)=−24(-4)(6) = -24

Like signs give a positive result; unlike signs give a negative one.

The magnitude never depends on the signs — in every row the digits are the same and only the sign changes.

Division obeys the identical table, and not by coincidence: −204=−5\dfrac{-20}{4} = -5 is right precisely because (−5)(4)=−20(-5)(4) = -20.

bookSHelf · Integrated Math 130 / 42
Computing size and sign separately is faster and safer than tracking both at once.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.4 — Why two negatives make a positive

The pattern forces the answer

ProductValueChange from the row above
(−3)(2)(-3)(2)−6-6—
(−3)(1)(-3)(1)−3-3up 3
(−3)(0)(-3)(0)00up 3
(−3)(−1)(-3)(-1)33up 3
(−3)(−2)(-3)(-2)66up 3

Table 0.2.2: Products of −3-3 with a factor decreasing by 1 each row.

Continuing the pattern past zero forces (−3)(−1)(-3)(-1) to be +3+3. It is not a convention chosen for convenience.

A second argument uses only the distributive property:

0=(−3)(5+(−5))=−15+(−3)(−5)0 = (-3)\bigl(5 + (-5)\bigr) = -15 + (-3)(-5)

Only +15+15 cancels the −15-15.

bookSHelf · Integrated Math 131 / 42
Any other value would break the distributive property — the sign rule is the price of keeping the rest of arithmetic intact.
Integers · bookSHelf Integrated Math 1§0.2

Context Pause — reversing a reversal

Run the tape backwards twice

Multiplying by a negative flips direction on the number line, the way playing a video in reverse flips the action. Do it twice and you are running forward again.

That is the whole content of "a negative times a negative is a positive."

bookSHelf · Integrated Math 132 / 42
Or: (−3)(−5)(-3)(-5) is removing five groups of a $3 debt — wiping out $15 you owe leaves you $15 better off.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.4 — Counting the negative factors

Even pairs off; odd leaves one behind

Even count — positive

Each pair of negatives cancels, so an even count pairs off completely.

(−2)4=16(-2)^4 = 16

Odd count — negative

One unmatched negative is left to set the sign.

(−2)(−3)(−4)=−24(−2)3=−8(-2)(-3)(-4) = -24 \qquad (-2)^3 = -8

† The placement caution from §0.1.4 still applies: (−2)4=16(-2)^4 = 16 but −24=−16-2^4 = -16, because in the second the exponent attaches only to the 2.

bookSHelf · Integrated Math 133 / 42
The counting rule is just the pairing rule applied repeatedly.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.4 — Worked example

Predicting the sign before computing

Example 0.2.8 — predict the sign, then compute: (a) (−2)(3)(−5)(-2)(3)(-5); (b) (−1)(−2)(−3)(−4)(−5)(-1)(-2)(-3)(-4)(-5); (c) −72−9\dfrac{-72}{-9}; (d) (−3)4(-3)^4; (e) −34-3^4

(a) two negatives, even  →  30(b) five negatives, odd  →  −120\text{(a) two negatives, even} \;\rightarrow\; 30 \qquad \text{(b) five negatives, odd} \;\rightarrow\; -120 (c) like pair  →  8(d) base −3, even  →  81(e) base 3  →  −81\text{(c) like pair} \;\rightarrow\; 8 \qquad \text{(d) base } -3,\ \text{even} \;\rightarrow\; 81 \qquad \text{(e) base } 3 \;\rightarrow\; -81

Parts (d) and (e) use the same digits and disagree by a sign, exactly as −42-4^2 and (−4)2(-4)^2 did in §0.1.

bookSHelf · Integrated Math 134 / 42
Predicting first turns the sign into a check on the arithmetic rather than an afterthought.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.4 — Definition

Division by zero is undefined

Definition 0.2.6 — Division by zero is undefined

For every number aa, the quotient a0\dfrac{a}{0} is undefined — it names no number at all. An expression is undefined when no value satisfies what it asks for, or when more than one does.

50\dfrac{5}{0} asks what times 0 gives 5 — nothing does. 00\dfrac{0}{0} asks what times 0 gives 0 — everything does.

Why dividing by zero is undefined, in two cases Two panels side by side. The left panel asks what five divided by zero is, which means asking which number times zero gives five. Three candidates are tested in turn: three times zero is zero, negative seventeen times zero is zero, and one thousand times zero is zero. Each is marked with a cross, because none of them gives five. The verdict is that no candidate works, so the quotient names no number. The right panel asks what zero divided by zero is, which means asking which number times zero gives zero. The same three candidates are tested and each is marked with a check, because every one of them works. The verdict is that every candidate works, so the quotient cannot name one number. Both cases are undefined, for opposite reasons. 5 ÷ 0 = ? which number times 0 gives 5? 0 ÷ 0 = ? which number times 0 gives 0? 3 × 0 = 0 3 × 0 = 0 −17 × 0 = 0 −17 × 0 = 0 1,000 × 0 = 0 1,000 × 0 = 0 no candidate works undefined every candidate works undefined

Definition 0.2.6: both 5/05/0 and 0/00/0 are undefined, for opposite reasons.

One case has no candidates, the other has too many. Zero divided by a nonzero number is fine: 05=0\dfrac{0}{5} = 0, since 0⋅5=00 \cdot 5 = 0.

bookSHelf · Integrated Math 135 / 42
Not a rule to be worked around. Recognizing a possible zero denominator matters in Chapter 3.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.4 — Order of operations with negatives

Nothing changes but the care

StepExpressionReason
1−2(−3)2+3-2(-3)^2 + 3grouping first: 5−8=−35 - 8 = -3
2−2(9)+3-2(9) + 3exponent: (−3)2=9(-3)^2 = 9
3−18+3-18 + 3multiplication
4−15-15addition

Evaluating   −2(5−8)2+3\;-2(5 - 8)^2 + 3, one rule per line.

Substituting keeps the parentheses habit from §0.1.4. Evaluate x2−4xx^2 - 4x at x=−5x = -5:

(−5)2−4(−5)=25+20=45(-5)^2 - 4(-5) = 25 + 20 = 45

Two places to lose a sign — the squaring, and subtracting a negative. Parentheses protect both.

bookSHelf · Integrated Math 136 / 42
Only the arithmetic gets more careful; the order itself is unchanged.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.4 — Worked example

Order of operations with a fraction bar

Example 0.2.9 — simplify   −4(3−7)−2−(−5)\;\dfrac{-4(3 - 7)}{-2} - (-5)

The fraction bar groups. Simplify numerator and denominator separately before dividing.

3−7=−4→−4(−4)=16→16−2=−83 - 7 = -4 \quad\rightarrow\quad -4(-4) = 16 \quad\rightarrow\quad \frac{16}{-2} = -8

Then the remaining subtraction. Subtracting a negative adds.

−8−(−5)=−8+5=−3-8 - (-5) = -8 + 5 = -3
bookSHelf · Integrated Math 137 / 42
Answer −3-3. The bar is a grouping symbol, exactly like parentheses.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.4 — Worked example

Evaluating at a negative value

Example 0.2.10 — evaluate   −x2+3x−8\;-x^2 + 3x - 8 when x=−4x = -4

Substitute with parentheses. The leading minus sign stays exactly where it was.

−(−4)2+3(−4)−8-(-4)^2 + 3(-4) - 8

Exponent first. The parentheses make the base −4-4, and an even power of a negative is positive, so (−4)2=16(-4)^2 = 16 and the first term is −16-16.

−16+(−12)−8=−36-16 + (-12) - 8 = -36
bookSHelf · Integrated Math 138 / 42
The minus sign in front was never part of the base — it is applied after squaring.
Integers · bookSHelf Integrated Math 1§0.2

§0.2.4 — Your turn

Try It Now 0.2.4

Try It Now 0.2.4 — simplify (−4)(−3)(−2)(-4)(-3)(-2), −459\dfrac{-45}{9}, −3(2−6)2+10-3(2 - 6)^2 + 10; then evaluate x2−5xx^2 - 5x at x=−3x = -3

(−4)(−3)(−2)=−24−459=−5(-4)(-3)(-2) = -24 \qquad \frac{-45}{9} = -5 −3(−4)2+10=−48+10=−38(−3)2−5(−3)=9+15=24-3(-4)^2 + 10 = -48 + 10 = -38 \qquad (-3)^2 - 5(-3) = 9 + 15 = 24

Three negative factors is odd, so part 1 is negative. Part 4 is the parentheses habit again.

bookSHelf · Integrated Math 139 / 42
Answers: −24-24, −5-5, −38-38, 24.
Integers · bookSHelf Integrated Math 1§0.2

Key Terminology — the seven words this section defines

Key terms

integer — a whole number or the negative of a whole number.

number line — a line with a marked zero and a fixed unit distance, positives right, negatives left.

opposite — the number the same distance from zero but on the other side.

additive inverse — another name for the opposite; the two sum to zero.

absolute value — the distance a number sits from zero, written ∣a∣\lvert a \rvert, never negative.

magnitude — the size of a number with its sign ignored.

undefined — describes an expression that names no number, as division by zero does.

bookSHelf · Integrated Math 140 / 42
Second column reveals on click.
Integers · bookSHelf Integrated Math 1§0.2

The headline result

Subtraction is not a second operation

a−b=a+(−b)a - b = a + (-b) — every subtraction is an addition once you take the opposite, so the two addition cases are the only rules you need.

And the opposite is just the fold at zero. One picture generates the order rule, absolute value, the addition cases, and the sign rules.

† This is why rewriting before computing is worth its extra line: it converts every problem into the one case you have already practised, instead of asking you to hold two rule sets at once.

bookSHelf · Integrated Math 141 / 42
One real headline result per deck — the others fold into the conclusions cards.
0.2
Integers · bookSHelf Integrated Math 1§0.2

§0.2 — Conclusions

What to carry forward

The one idea

A signed number carries size and direction at once. Absolute value asks only for the size; the sign rules track only the direction. Separating the two questions is what makes every computation in this section routine.

Where it goes wrong

Reading −100-100 as larger than −1-1, distributing across absolute-value bars, dropping a sign in a chain instead of rewriting first, and letting an exponent reach a minus sign it was never attached to.

Next: §0.3 Fractions — where the same "rewrite it into a form you already know" move does the work again. Back to start.

bookSHelf · Integrated Math 142 / 42
Every failure listed is a reading error, matching §0.1's closing card.