1
A property is a licence, not an observation
3 + 5 = 5 + 3 is arithmetic; you can check it. a + b = b + a for every pair of reals is the property, and it is what lets you reorder x + 7 into 7 + x when you have no idea what x is.
Lesson · Foundations of Algebra · The number system
Properties of the Real Numbers
Five kinds of permission — commutativity, associativity, distribution, identities and inverses — are all the licence you have to rearrange an expression. Everything later in this path is one of them applied to a longer expression. Subtraction and division have none of them, and that omission is where most sign errors come from.
a + b = b + a commutative
(a + b) + c = a + (b + c) associative
a(b + c) = ab + ac distributive
a + 0 = a, a · 1 = a identities
The properties you are allowed to use
Core ideas
What this lesson establishes
A property is not a fact about one sum. It is a permission, granted for every pair of real numbers at once, and that is what makes it usable on a letter.
2
Distribution is the only one that mixes the two operations
Commutativity and associativity each talk about one operation on its own. Distribution is the bridge between addition and multiplication, and every expansion, every collection of like terms and every sign that reaches through a bracket is that one property.
3
Subtraction and division have none of these
7 − 3 is not 3 − 7, and (24 / 4) / 2 is 3 while 24 / (4 / 2) is 12. The repair is to stop treating them as operations: subtracting is adding the negative, dividing is multiplying by the reciprocal. Once rewritten, every property applies again.
The material
The rules you are permitted to use
Five kinds of property, stated precisely, and then the ones people invent because they look similar.
These properties are assumed. This course states them and uses them; building the real numbers so that they hold is a different subject. What matters here is knowing precisely what each one says, because anything beyond that is not licensed.
Definition · Commutative properties
For all reals a and b: a + b = b + a and ab = ba.
Order does not matter — for addition and for multiplication, and for nothing else.
Definition · Associative properties
For all reals a, b, c: (a + b) + c = a + (b + c) and (ab)c = a(bc).
Grouping does not matter, which is why a + b + c can be written with no brackets at all and still name one number.
Taken together those two say that a sum of any number of terms may be added in any order and any grouping. That is the entire justification for "collect the x terms" in “Combining Like Terms” — you are moving terms past each other, and commutativity and associativity are the permission to do it.
Definition · Distributive property
For all reals a, b, c: a(b + c) = ab + ac, and equally (b + c)a = ba + ca.
Read left to right it expands a bracket. Read right to left it factors one out. Both directions are used constantly, and they are the same statement.
This is also the property that makes −(x − 3) equal −x + 3. Writing the minus as (−1) · turns it into a distribution: (−1)(x) + (−1)(−3) = −x + 3. Nothing is remembered; both signs are computed.
Identities and inverses
Definition · Identities
a + 0 = a for every real a, and a · 1 = a for every real a.
0 is the additive identity and 1 the multiplicative identity. Multiplying by 1 in a chosen disguise — by 3/3, say — is how every common denominator in Linear Equations and Inequalities is built.
Definition · Inverses
Every real a has an additive inverse −a with a + (−a) = 0. Every real a except 0 has a multiplicative inverse 1/a with a · (1/a) = 1.
That single exclusion is the reason division by zero is undefined. It is not a rule imposed on top; there is simply no number that works.
Theorem · Multiplying by zero
a · 0 = 0 for every real a.
This is not one of the assumed properties. It follows from them, which is worth seeing once.
0 = 0 + 0, by the additive identity. So a · 0 = a · (0 + 0), and by distribution a · 0 = a · 0 + a · 0.
Add the additive inverse of a · 0 to both sides. The left becomes 0 and the right becomes a · 0. So a · 0 = 0.
□
That theorem is why 0 has no reciprocal: 0 · x is 0 for every x and so is never 1. It is also the engine of every quadratic you solve in Quadratics and Complex Numbers, where a product being zero forces one of its factors to be zero.
Properties that do not exist
The errors in this course are almost all invented properties — rules that have the shape of a real one. Each is killed by a single pair of numbers:
- a(bc) = (ab)(ac) is false. At a = 2, b = 3, c = 4 the left side is 24 and the right side is 48. Multiplication does not distribute over multiplication.
- √(a + b) = √a + √b is false. At a = 4, b = 9 the left side is √13 and the right side is 5, and 5² = 25, not 13.
- 1/(a + b) = 1/a + 1/b is false. At a = b = 1 the left side is 1/2 and the right side is 2.
One counterexample is a complete disproof, and it costs about ten seconds. Whenever you are unsure whether a rearrangement is legal, put small numbers in both sides. It will not tell you that a rule is true, but it will tell you when one is false, and that is the direction the errors run.
Example · Every step named
2(x + 5) + 3x becomes (2x + 10) + 3x by distribution, then 10 + (2x + 3x) by commutativity and associativity of addition, then 10 + (2 + 3)x by distribution read backwards, and finally 10 + 5x. Four lines, four named properties, no invention.
Interactive
The named properties
Tested on your numbers, then searched for a counterexample
Method
Justifying a rearrangement
The goal is not the answer. It is being able to say why each line follows from the one above.
1
Rewrite subtraction and division out of the way
a − b becomes a + (−b); a / b becomes a · (1/b) for b ≠ 0. Until you do this, no commutative or associative property applies, and the temptation to use one anyway is where the sign errors start.
2
Change one thing per line
Reorder, or regroup, or distribute — not two at once. A line that does three things is a line you cannot check and cannot name.
3
Name the property
Say which of the five you used. If you cannot name it, you have either invented a rule or done arithmetic, and it matters which.
4
Test on numbers when unsure
Substitute small values into the expression you started with and the one you finished with. They must agree. Disagreement locates the illegal step; agreement is evidence, not proof.
Name the property
Score 0 / 0Choose the answer you can justify.
Simplifying, with every property named
5 − 2(x − 3). The answer is short; the point is the column on the right.
5 − 2(x − 3)
= 5 + (−2)(x − 3) subtraction is adding the negative
= 5 + [(−2)(x) + (−2)(−3)] distributive
= 5 + (−2x + 6) arithmetic: (−2)(−3) = 6
= 5 + (6 + (−2x)) commutative (addition)
= (5 + 6) + (−2x) associative (addition)
= 11 − 2x arithmetic
The sign on the 6 was computed by distribution, not recalled. That is the difference between this and the common wrong answer −1 − 2x, which comes from taking (−2)(−3) to be −6 and finishing 5 − 6 − 2x. At x = 4 that gives −9 rather than 3.
Check by substitution. At x = 4 the original is 5 − 2(4 − 3) = 3, and the answer is 11 − 8 = 3. Agreement at one value does not prove the two expressions are equal everywhere — the named steps do that — but disagreement would have proved they are not.
For a faded rehearsal, start 4 − 3(y − 2) by rewriting it as 4 + (−3)(y + (−2)). Finish one change per line and name each change. The result is 10 − 3y; −2 − 3y means the two negative factors in the constant term were multiplied as though only one were negative.
Common mistakes
1
Reordering across a subtraction sign5 − x is not x − 5. Commutativity is a property of addition, and subtraction does not have it. Write 5 + (−x) and then the reordering to (−x) + 5 is legal, which is −x + 5.
2
Distributing over a producta(bc) is abc, not (ab)(ac). At a = 2, b = 3, c = 4 those are 24 and 48. Distribution crosses from multiplication to addition and nowhere else.
3
Letting a minus sign reach only the first term−(x − 3) is −x + 3, not −x − 3. The −1 multiplies every term inside the bracket, and (−1)(−3) = 3. This one error will outnumber all the others in the next two courses.
Check your understanding
✓
Finish when you can name the property behind every line of a rearrangement.
Simplify 7 − 4(2 − z) and annotate it, one property per line, with "arithmetic" allowed for steps that are just numbers. Then disprove 1/(a + b) = 1/a + 1/b with values for which every denominator is defined. If a line has no property name, it is the line to inspect.
Note: “The Distributive Law” does nothing but the distributive property, at length, and “Combining Like Terms” does it backwards. It is worth being fluent here first, because an error made in a property is invisible — the result still looks like algebra, and only substituting a number reveals it.