Integrated Math 1 · Unit 0 · Review of the Essentials

Properties of Real Numbers

The list of legal moves. Every rewriting you have done since §0.1 was one of these, and Chapter 3 will ask you to name them.

bookSHelf · Integrated Math 1 · §0.5 · a self-paced section

"Why am I allowed to do that?" finally has an answer you can point to.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

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

Objectives

Use the commutative and associative properties, and say which operations lack them Def 0.5.1–0.5.2
Rearrange and regroup a calculation on purpose to make arithmetic easier Ex 0.5.1
Identify the identities, and find the opposite and the reciprocal Def 0.5.3–0.5.6
Use the distributive property forwards to expand and backwards to factor Def 0.5.7–0.5.8
Distribute a negative number without losing a sign §0.5.3
State the properties of zero, including why division by zero has no value Def 0.5.9–0.5.10

The six section SLOs verbatim.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.1 — Commutative and associative properties

Two questions organize everything

Does the order matter?

For addition and multiplication, no. For subtraction and division, yes.

Does the grouping matter?

Same answer both times — and forgetting it is a reliable way to get a wrong answer.

That freedom is worth a lot: it is what lets you hunt for the convenient arrangement before computing anything.

Order and grouping — keep the two words separate all section.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.1 — Definition

Commutative Property

Definition 0.5.1 — Commutative Property

For any real numbers $a$ and $b$ ,

a + b = b + a \qquad a \cdot b = b \cdot a

Changing the order does not change the sum or the product. Commute means to travel back and forth — the two numbers swap seats and nothing else changes.

Definition 0.5.1: swapping the two numbers leaves a sum or product alone, and changes a difference or quotient.

9 + 4 = 4 + 9 = 13

;

6 \cdot 7 = 7 \cdot 6 = 42

.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

Context Pause — one counterexample settles it

A property claims something about every pair

So a single pair where it fails is enough to sink it.

You do not need to test subtraction on a hundred pairs to know it is not commutative. One pair does the job — which is exactly why the next two lines matter so much.

A genuinely transferable idea about how mathematical claims get refuted.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.1 — The half students skip

Which operations let you swap

Operation	Commutative?	The test	What happened
Addition	yes	$9 + 4 = 13$ and $4 + 9 = 13$	same result
Multiplication	yes	$6 \cdot 7 = 42$ and $7 \cdot 6 = 42$	same result
Subtraction	no	$11 - 4 = 7$ and $4 - 11 = -7$	opposite results
Division	no	$18 \div 6 = 3$ and $6 \div 18 = \tfrac{1}{3}$	reciprocal results

Table 0.5.1: This costs students points for years.

Same warning §0.1 gave about translating words: "5 less than $n$ " is $n - 5$ and not $5 - n$ because subtraction is not commutative. If it were, the distinction would not exist.

You can get the freedom back:

11 - 4 = 11 + (-4) = -4 + 11 = 7

. Just never leave the minus sign behind.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.1 — Definition

Associative Property

Definition 0.5.2 — Associative Property

For any real numbers $a$ , $b$ and $c$ ,

(a + b) + c = a + (b + c) \qquad (ab)c = a(bc)

The order stays exactly the same in both statements. Only the parentheses move.

Definition 0.5.2: the fence moves, the numbers stay where they are, and the total holds.

$(4 \cdot 25) \cdot 17 = 100 \cdot 17 = 1700$ and $4 \cdot (25 \cdot 17) = 4 \cdot 425 = 1700$ . Both routes land in the same place — but the first is much easier to walk.

Subtraction fails it too:

(20-6)-5 = 9

but

20-(6-5) = 19

.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

Insight — swapping seats versus moving the fences

Two different freedoms

Commutative — the numbers move

$3 + 8$ becomes $8 + 3$ . They change seats.

Associative — the fences move

$(3 + 8) + 5$ becomes $3 + (8 + 5)$ . Same seats, different fence.

Because grouping is free for addition and multiplication, we can write $a + b + c$ and $abc$ with no parentheses at all. That is not true of $a - b - c$ , which the order of operations settles left to right by convention, not by property.

Order versus grouping — the distinction the whole subsection rests on.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.1 — Using both properties on purpose

Hunt for the convenient arrangement

As written	Rearranged	Why that helps	Result
$58 + 27 + 42$	$(58 + 42) + 27$	the first pair makes 100	$127$
$5 \cdot 39 \cdot 20$	$(5 \cdot 20) \cdot 39$	the first pair makes 100	$3900$
$-16 + 91 + 16$	$(-16 + 16) + 91$	the first pair cancels to 0	$91$
$\tfrac{3}{4} \cdot 23 \cdot \tfrac{4}{3}$	$\left(\tfrac{3}{4} \cdot \tfrac{4}{3}\right) \cdot 23$	the first pair multiplies to 1	$23$

Table 0.5.2: Good mental arithmetic is mostly this habit.

The last two rows preview §0.5.2. Pairing a number with its opposite, or with its reciprocal, makes the pair vanish — the most useful rearrangement of all.

The same moves work when letters are involved — which is how simplification actually gets done.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.1 — Worked example

Regrouping to combine the constants

Example 0.5.1 — simplify $\;12 + (n + 5)$ , naming the property behind each step

Commutative property of addition — move the 5 in front of the $n$ .

12 + (5 + n)

Associative property of addition — regroup so 12 and 5 share a set.

(12 + 5) + n = 17 + n

We could not have added 12 and 5 in the original, because the parentheses grouped the 5 with the $n$ . The two properties are what let us take that grouping apart and rebuild it differently.

Naming the step is the skill Chapter 3 will require.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.1 — Your turn

Try It Now 0.5.1

Try It Now 0.5.1 — simplify $\;(y + 9) + 14$ ; then compute $\;25 \cdot 17 \cdot 4$ mentally

Regroup so the numbers sit together — associative property of addition.

y + (9 + 14) = y + 23

Swap so 25 and 4 are neighbours (commutative), regroup them (associative), and take the easy pair first.

(25 \cdot 4) \cdot 17 = 100 \cdot 17 = 1700

Answers

y + 23

and 1700.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.2 — Identity and inverse properties

Additive Identity

Definition 0.5.3 — Additive Identity

The number $0$ is the additive identity: for any real number $a$ ,

a + 0 = a \qquad 0 + a = a

Some numbers do nothing at all. That sounds useless, and it is one of the most useful facts in algebra.

Definition 0.5.3: zero is the jump that goes nowhere, and the place a number and its opposite land.

An identity is a place to land. An inverse is the thing that gets you there.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.2 — Definition

Multiplicative Identity

Definition 0.5.4 — Multiplicative Identity

The number $1$ is the multiplicative identity: for any real number $a$ ,

a \cdot 1 = a \qquad 1 \cdot a = a

Each is the one number that does nothing to its own operation.

Definition 0.5.4: one is the factor that moves nothing, and where a number meets its reciprocal.

0 is not the multiplicative identity. Multiplying by zero does not leave a number alone — it wipes the number out, which is §0.5.4's business.

-14 + 0 = -14

;

62 \cdot 1 = 62

;

1 \cdot (-3.5) = -3.5

.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.2 — Definitions

The two inverses

Definition 0.5.5 — Additive Inverse

The additive inverse of $a$ is $-a$ , the number that adds to it to give the additive identity:

a + (-a) = 0

It is the opposite from §0.2 — same distance from zero, other side.

Definition 0.5.6 — Multiplicative Inverse

The multiplicative inverse of $a$ , where $a \neq 0$ , is $\dfrac{1}{a}$ :

a \cdot \dfrac{1}{a} = 1

It is the reciprocal from §0.3 — flip the fraction over.

A whole number $n$ is really $\dfrac{n}{1}$ , so its reciprocal is $\dfrac{1}{n}$ .

11 + (-11) = 0

;

\tfrac{4}{9} \cdot \tfrac{9}{4} = 1

.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.2 — Read across each row and check

Both inverses for the same number

Number	Additive inverse	Multiplicative inverse	Check
$8$	$-8$	$\dfrac{1}{8}$	$8 + (-8) = 0$ ; $8 \cdot \tfrac{1}{8} = 1$
$-5$	$5$	$-\dfrac{1}{5}$	$-5 \cdot \left(-\tfrac{1}{5}\right) = 1$
$\dfrac{4}{9}$	$-\dfrac{4}{9}$	$\dfrac{9}{4}$	$\tfrac{4}{9} \cdot \tfrac{9}{4} = 1$
$-\dfrac{2}{7}$	$\dfrac{2}{7}$	$-\dfrac{7}{2}$	$-\tfrac{2}{7} \cdot \left(-\tfrac{7}{2}\right) = 1$
$1$	$-1$	$1$	1 is its own reciprocal
$0$	$0$	none	no number times 0 gives 1

Table 0.5.3: A reciprocal always keeps the sign — a negative times a positive could never give $+1$ . And 0 is its own opposite, which is another way of saying zero is neither positive nor negative.

The last row is not an oversight — zero's reciprocal would have to be 1/0, which §0.2 already showed is undefined.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

Context Pause — two ways to say the same thing

One fact seen from two sides

Division by zero is undefined

Because no number times 0 gives a nonzero result.

Zero has no reciprocal

Because no number times 0 gives 1.

That is why the multiplicative inverse carries the condition $a \neq 0$ . Watch for that condition — the rest of algebra is full of it, and every appearance traces back to this one place.

Not two rules to memorize — one fact.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

Insight — a 1 in disguise

The move behind every common denominator

Any fraction with matching top and bottom equals 1: $\dfrac{5}{5}$ , $\dfrac{7}{7}$ , $\dfrac{12}{12}$ . Multiplying by one of those changes how a fraction looks without changing what it is worth — like trading four quarters for a dollar.

\dfrac{3}{8} = \dfrac{3}{8} \cdot 1 = \dfrac{3}{8} \cdot \dfrac{5}{5} = \dfrac{15}{40}

The value never moved — it got multiplied by 1. That single line justifies every common denominator you have ever found, and reducing is the same move run backwards: $\dfrac{15}{40} = \dfrac{3}{8} \cdot \dfrac{5}{5}$ , peeling off a factor of 1 rather than cancelling anything mysterious.

This is §0.3's Equivalent Fractions Property, finally explained.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.2 — Choosing the right disguised 1

Pick the one that lands the denominator you want

Goal	Disguised 1	Result
Write $\dfrac{5}{6}$ with denominator 42	$\dfrac{7}{7}$	$\dfrac{35}{42}$
Write $\dfrac{2}{9}$ with denominator 45	$\dfrac{5}{5}$	$\dfrac{10}{45}$
Write $7$ as a fraction with denominator 3	$\dfrac{3}{3}$	$\dfrac{21}{3}$
Add $\dfrac{1}{4} + \dfrac{1}{6}$	$\dfrac{3}{3}$ and $\dfrac{2}{2}$	$\dfrac{5}{12}$

Table 0.5.4: Ask what turns the denominator you have into the one you want.

Row 4 is the LCD procedure from §0.3, restated as a choice of two disguised 1s.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.2 — Worked example

Finding both inverses

Example 0.5.2 — find and verify both inverses of $-\dfrac{3}{10}$

Additive inverse — flip the sign, then check it lands on the additive identity.

-\dfrac{3}{10} + \dfrac{3}{10} = 0

Multiplicative inverse — turn the fraction over, sign stays negative.

-\dfrac{3}{10} \cdot \left(-\dfrac{10}{3}\right) = \dfrac{30}{30} = 1

Verifying against the identity is what makes it an inverse, not just a rearrangement.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.2 — Your turn

Try It Now 0.5.2

Try It Now 0.5.2 — find the reciprocal of $-6$ ; rewrite $\dfrac{4}{7}$ with a denominator of 63

-6 = -\dfrac{6}{1} \longrightarrow -\dfrac{1}{6}, \quad \text{check } -6 \cdot \left(-\dfrac{1}{6}\right) = 1

Since $7 \cdot 9 = 63$ , the disguised 1 to use is $\dfrac{9}{9}$ .

\dfrac{4}{7} \cdot \dfrac{9}{9} = \dfrac{36}{63}

Answers

-\tfrac{1}{6}

and

\tfrac{36}{63}

.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.3 — The distributive property

Distributive Property

Definition 0.5.7 — Distributive Property

For any real numbers $a$ , $b$ , $c$ ,

a(b + c) = ab + ac

The outside factor gets handed out to everything inside.

6(x + 4) = 6x + 24

Definition 0.5.7: one rectangle cut in two, so the outside factor reaches both terms.

The bridge between multiplication and addition — the property you will use more than any other. Test it: $9(20 + 3) = 9 \cdot 23 = 207$ , and $9 \cdot 20 + 9 \cdot 3 = 207$ . That second route is how most people multiply in their heads, so you have been distributing for years without a name for it.

Subtraction needs no new rule:

a(b-c) = ab - ac

, since subtracting is adding the opposite.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

Insight — chopping one rectangle into two

Same rectangle, so the two must be equal

Picture a rectangle 6 units tall whose width splits into a piece of length $x$ and a piece of length 4. Measure its area as one rectangle, $6(x + 4)$ , or as two added together, $6x + 24$ .

It also explains the most common way to get this wrong. Writing $6(x + 4) = 6x + 4$ measures the second rectangle as 4 square units instead of $6 \cdot 4 = 24$ . The height applies to both pieces — every term inside gets multiplied, no exceptions.

Worth carrying around — it diagnoses the error, not just the rule.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

Context Pause — the minus sign is part of the factor

Circle the sign with the number

When you distribute $-3$ , the thing being handed out is $-3$ , not 3.

-3(m - 8) = -3 \cdot m + (-3)(-8) = -3m + 24

The second term comes out positive, because a negative times a negative is positive — §0.2. The single most common error in beginning algebra is distributing the 3 and leaving the minus behind, producing $-3m - 24$ . Most sign errors here are bookkeeping errors, not arithmetic errors.

A minus in front of a parenthesis is a multiplication by

-1

:

-(4k - 9) = -4k + 9

.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.3 — Four ways this goes wrong

Correct against what students write

Expression	Correct	Common error	What went wrong
$-3(m - 8)$	$-3m + 24$	$-3m - 24$	the minus sign was not distributed
$-5(2k + 3)$	$-10k - 15$	$-10k + 15$	sign dropped from the second term
$-(p - 6)$	$-p + 6$	$-p - 6$	the invisible $-1$ was not distributed
$4 - 2(x + 1)$	$4 - 2x - 2$	$2(x + 1)$	subtracted before distributing

Table 0.5.5: Three sign slips and one order-of-operations slip.

That last row is a different mistake, and a sneaky one. The $4 -$ out front does not combine with the 2 first — multiplication outranks subtraction, so the $-2$ has to be distributed before anything gets combined.

Rewriting

7(2y-5)

as

7(2y + (-5))

costs one line and prevents most of these.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.3 — Reading it backwards

Common Factor

Definition 0.5.8 — Common Factor

A common factor appears in every term. Rewriting $ab + ac$ as $a(b + c)$ is factoring out that factor.

12n + 18 = 6 \cdot 2n + 6 \cdot 3 = 6(2n + 3)

Every property is an equation, and an equation reads in either direction.

Definition 0.5.8: the shared factor lifts out of every term and stands in front.

Factoring is one of the two directions you travel constantly; Chapter 7 builds a unit on it.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.3 — Why like terms combine

The promise from §0.1, kept

Expression	Common factor	Factored form
$12n + 18$	$6$	$6(2n + 3)$
$15t - 25$	$5$	$5(3t - 5)$
$8h + 8$	$8$	$8(h + 1)$
$-6r - 21$	$-3$	$-3(2r + 7)$

Table 0.5.6: Row 3 leaves a 1 behind, not nothing — that second term is $8 \cdot 1$ .

5w + 8w = (5 + 8)w = 13w

That middle step is the distributive property read right to left. Combining like terms is not a separate rule — it is factoring out the variable part, adding the coefficients, and putting the variable back.

And $9x^2 + 8x$ gives $x(9x + 8)$ — the parenthesis still has a variable, so nothing collapses.

Like terms combine exactly when pulling out the variable part leaves a plain numerical sum. Check every factoring by distributing back.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.3 — Worked example

Distributing, then combining

Example 0.5.3 — simplify $\;-2(5a - 3) + 7a$ , naming the property behind each step

Distributive — hand out the whole factor, sign included.

-10a + 6 + 7a

Commutative — move the like terms together.

-10a + 7a + 6

Distributive, backwards — factor out the $a$ , then add the coefficients.

(-10 + 7)a + 6 = -3a + 6

Answer

-3a + 6

. Every line had a reason, and every reason was a property.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.3 — Worked example

Two distributions in one expression

Example 0.5.4 — simplify $\;4(2x + 5) - 3(x - 6)$

Distribute the 4, then the $-3$ — the factor is $-3$ , not 3, so $-3 \cdot (-6) = +18$ .

8x + 20 - 3x + 18

Group the like terms (commutative), factor out the $x$ (distributive backwards), finish the arithmetic.

(8 - 3)x + 38 = 5x + 38

† That is the format Chapter 3 will ask you to produce for equations — every line justified by a named property.

Answer

5x + 38

.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.3 — Your turn

Try It Now 0.5.3

Try It Now 0.5.3 — simplify $\;-4(3b - 2) + 5b$ ; then factor $\;21c - 14$

Distribute $-4$ keeping the sign attached, reorder, factor out the $b$ .

-12b + 8 + 5b \;\rightarrow\; (-12 + 5)b + 8 = -7b + 8

Both terms are divisible by 7 — check by distributing back.

21c - 14 = 7(3c - 2)

Answers

-7b + 8

and

7(3c - 2)

.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.4 — Properties of zero

Multiplication Property of Zero

Definition 0.5.9 — Multiplication Property of Zero

For any real number $a$ ,

a \cdot 0 = 0 \qquad 0 \cdot a = 0

Fifteen groups of nothing is nothing, and nothing groups of fifteen is also nothing. A single zero factor collapses a whole product:

(-8)(93)(0)(41) = 0

Definition 0.5.9: one zero factor collapses the whole product.

A second property runs the other way, and it is a separate fact: if $ab = 0$ , then $a = 0$ or $b = 0$ — because two nonzero reals always have a nonzero product. That sentence is the engine behind solving $(x-4)(x+9) = 0$ in Chapter 7.

Zero is where zero stops being polite. Adding and subtracting it change nothing.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

Insight — zero is the only product that tells on its factors

What a product reveals

Product is 12

You learn almost nothing about the factors — there are infinitely many pairs.

Product is 0

You learn a great deal: at least one of them was 0. There is no other way to reach zero by multiplying.

That asymmetry is exactly why factoring an equation to zero is worth doing — it converts one hard question into a short list of easy ones.

The mechanics come in Chapter 7; the property is here.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.4 — Definition

Division Involving Zero

Definition 0.5.10 — Division Involving Zero

For any real $a$ with $a \neq 0$ ,

\dfrac{0}{a} = 0 \qquad \dfrac{a}{0} \text{ is undefined}

Division by zero is undefined for every numerator, including $\dfrac{0}{0}$ .

Definition 0.5.10: zero on top is fine, zero on the bottom is not.

Every division question is really a multiplication question, and that is the fastest way to see why the two cases split apart.

Two very different situations that get confused with each other.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.4 — Read the third column as the real question

Every division involving zero

Expression	Value	The multiplication question
$\dfrac{0}{13}$	$0$	what times 13 gives 0? Only 0.
$\dfrac{13}{0}$	undefined	what times 0 gives 13? Nothing.
$\dfrac{0}{0}$	undefined	what times 0 gives 0? Everything.
$\dfrac{13}{13}$	$1$	what times 13 gives 13? Only 1.

Table 0.5.7: One case has no candidates; the other has too many. An expression must name exactly one value.

The rule of thumb is short: zero on top is fine, zero on the bottom is not.

Splitting nothing among thirteen people gives each of them nothing — a perfectly good answer.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

Context Pause — undefined does not mean zero

It does not mean "very large" either

It means there is no such number, so an expression containing it has no value to report.

Later, denominators will contain variables, and part of your job will be spotting which values would make a denominator zero and ruling them out first.

Same conclusion §0.2, §0.3 and §0.4 each reached from their own direction.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.4 — Every property in this section

The whole section in two tables

Property	Addition	Multiplication
Commutative	$a + b = b + a$	$ab = ba$
Associative	$(a + b) + c = a + (b + c)$	$(ab)c = a(bc)$
Identity	$a + 0 = a$	$a \cdot 1 = a$
Inverse	$a + (-a) = 0$	$a \cdot \dfrac{1}{a} = 1$ , $a \neq 0$

Table 0.5.8: The four that come in pairs.

Property	Statement
Distributive	$a(b + c) = ab + ac$
Multiplication by zero	$a \cdot 0 = 0$
Division into zero	$\dfrac{0}{a} = 0$ , $a \neq 0$
Division by zero	$\dfrac{a}{0}$ undefined

Table 0.5.9: The four that stand alone.

Assume

a

,

b

,

c

are real numbers throughout.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.4 — Worked example

Deciding whether an expression has a value

Example 0.5.5 — evaluate $\;\dfrac{n - 5}{n + 2}\;$ at $n = 5$ , then at $n = -2$

\dfrac{5 - 5}{5 + 2} = \dfrac{0}{7} = 0

Numerator zero, denominator not — the safe case. What times 7 gives 0? Only 0.

\dfrac{-2 - 5}{-2 + 2} = \dfrac{-7}{0}

Now the denominator is zero: what times 0 gives $-7$ ? Nothing does, so this has no value.

0 when

n = 5

; undefined when

n = -2

.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5.4 — Your turn

Try It Now 0.5.4

Try It Now 0.5.4 — simplify $\;(14)(-77)(0)(3)$ ; state $\dfrac{0}{9}$ and $\dfrac{9}{0}$

(14)(-77)(0)(3) = 0

One factor is 0, so the product collapses without any other arithmetic.

\dfrac{0}{9} = 0 \qquad \dfrac{9}{0} \text{ is undefined}

What times 9 gives 0? Only 0. What times 0 gives 9? Nothing.

Answers 0; 0; undefined.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

Key Terminology — the nine words this section defines

Key terms

commutative property — changing the order does not change a sum or product.

associative property — changing the grouping does not change a sum or product.

additive identity — the number 0, which leaves any number unchanged when added.

multiplicative identity — the number 1, which leaves any number unchanged when multiplied.

additive inverse — what adds to a number to give 0; its opposite.

multiplicative inverse — what multiplies a number to give 1; its reciprocal.

distributive property — multiplying a sum equals multiplying each term and adding.

common factor — a factor shared by every term, which can be pulled out front.

undefined — describes an expression that names no number at all.

Second column reveals on click.

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

The headline result

These properties are the list of legal moves

Solving an equation means producing a chain of equivalent equations, each a small rewriting of the last, ending with the variable alone. Every link has to be legal — and this section is the list.

Up to now the properties simplified expressions. Chapter 3 puts them to a second use, and asks you to name them: distributive property, commutative property, additive inverse, multiplicative identity.

† The question "why am I allowed to do that?" now has an answer you can point to — which is the difference between following a procedure and doing algebra.

The closing idea of the whole unit, not just this section.

0.5

Properties of Real Numbers · bookSHelf Integrated Math 1§0.5

§0.5 — Conclusions

What to carry forward

The one idea

Order and grouping are free for addition and multiplication and not for subtraction and division. The distributive property bridges the two operations, and read backwards it explains why like terms combine at all. Zero and one are the numbers that do nothing — which is what makes them useful.

Where it goes wrong

Swapping the operands of a subtraction, leaving a minus sign behind when distributing, forgetting the invisible $-1$ in front of a parenthesis, combining before distributing, dropping the 1 when factoring $8h + 8$ , and reading "undefined" as zero.

Next: Chapter 1 — where these properties stop being facts to state and become steps to cite. Back to start.

Closes Unit 0. Every failure listed is a bookkeeping error with a named rule behind it.