Number and Algebraic Foundations
The Field Laws of Real Number Arithmetic
The commutative, associative, distributive, identity and inverse laws, what each one licenses, and how they justify every step of routine algebraic simplification.
What this page covers
- State the five laws governing addition and multiplication of real numbers
- Name the law that justifies each step of a simplification
- Apply the distributive law forwards to expand and backwards to factor
- Explain why a · b = 0 forces a = 0 or b = 0, and why that matters
What the laws are for
Everyone rearranges algebra without thinking about it. The laws below are the reason the rearrangements are legal. Naming them is worth the effort for two reasons: it makes each step in a long simplification defensible, and it shows exactly which steps stop working when the setting changes — matrices, for instance, obey four of these five and not the fifth.
All five are stated for the real numbers under addition and multiplication. Subtraction and division are not separate operations here: a - b means a + (-b), and a ÷ b means a · b-1. That reduction is what lets five laws cover all four arithmetic operations.
The commutative laws
The source notes illustrate with 3 + 5 = 5 + 3 and 3 · 5 = 5 · 3. The content of the law is that this holds for every pair, not just the ones anyone has checked.
Neither subtraction nor division commutes. 8 - 3 = 5 but 3 - 8 = -5; 8 ÷ 2 = 4 but 2 ÷ 8 = 0.25. Matrix multiplication does not commute either, which is one of the first genuinely surprising facts in linear algebra.
The associative laws
The notes give (3 + 5) + 2 = 3 + (5 + 2). Both sides equal 10. Because grouping is irrelevant, the brackets can be dropped entirely and a + b + c written without ambiguity — which is the practical payoff of the law.
Commutativity and associativity together allow any rearrangement of a sum or a product. This is why collecting like terms is legitimate: 3x + 7 + 5x can be reordered and regrouped as (3x + 5x) + 7 without changing its value.
The distributive law
This is the only law connecting the two operations, and it is by some distance the most heavily used. The source notes open with it, using arithmetic to make the claim checkable before it is generalised.
Worked example — the source's own check
Expand 2(3 + 7) two ways and compare.
A second check with subtraction, which is covered because 7 - 3 means 7 + (-3):
Running the law backwards
Read right to left, ab + ac = a(b + c) is the whole of common-factor factoring. Expansion and factoring are not two techniques; they are one law used in two directions.
| Direction | Name | Example |
|---|---|---|
| Left to right | Expanding, removing brackets | 3x(x2 - 2y) = 3x3 - 6xy |
| Right to left | Factoring, taking out a common factor | 24 + 36 = 12(2 + 3) |
Applying it twice: the origin of FOIL
Multiplying two binomials needs the law twice. Treat (a + b) as a single quantity, distribute it over (c + d), then distribute each part.
The four products are conventionally remembered as First, Outer, Inner, Last. The mnemonic is a bookkeeping aid, not a separate rule, and it only applies to two binomials. The distributive law itself handles any number of terms.
Identities and inverses
The source notes make the additive case concrete on the number line: the inverse of 3 is -3, and 3 + (-3) = 0. For multiplication the inverse of 3 is 3-1 = 1/3, and 3 · 13 = 1.
- Additive identity
- 0 — adding it changes nothing
- Additive inverse of a
- -a — the reflection of a through 0
- Multiplicative identity
- 1 — multiplying by it changes nothing
- Multiplicative inverse of a
- a-1 = 1a, defined only when a ≠ 0
Zero has no multiplicative inverse. There is no real number z with 0 · z = 1, because 0 · z = 0 for every z. This single exception is the reason division by zero is undefined, and the reason q ≠ 0 is stipulated in the definition of a rational number.
The zero-product property
One consequence of the laws deserves separate billing because every method of solving equations by factoring depends on it.
The proof uses the multiplicative inverse. Suppose ab = 0 and a ≠ 0. Then a-1 exists, so
So if a is not zero, b must be. The source notes state the conclusion directly and immediately apply it: (x + 3)(x + 2) = 0 means x + 3 = 0 or x + 2 = 0, hence x = -3 or x = -2.
The property has no analogue for other constants. ab = 6 tells you nothing about a and b individually. This is why an equation must be rearranged to equal zero before factoring is any use — a step that is skipped surprisingly often.
The property can fail in other number systems. Among 2 · 2 matrices there are non-zero A and B with AB = 0, because a non-zero matrix need not have an inverse. The proof above shows precisely where the argument breaks.
Using the laws deliberately
In routine work the laws are applied silently. When a simplification goes wrong, applying them one at a time and naming each is the fastest way to find the bad step.
Worked example — simplify (3 + x) + 4
| Step | Result | Law used |
|---|---|---|
| Start | (3 + x) + 4 | — |
| Swap inside the bracket | (x + 3) + 4 | Commutative (addition) |
| Regroup | x + (3 + 4) | Associative (addition) |
| Evaluate | x + 7 | Arithmetic |
Source example, Week 1, page 5. The rearrangement is not cosmetic: it brings the two constants together so they can be combined, which is the entire purpose.
Worked example — expand (2x + 3)(x - 5)
Substitute x = 1. Left side: (2 + 3)(1 - 5) = 5 · (-4) = -20. Right side: 2 - 7 - 15 = -20. Agreement at one value is not a proof, but disagreement is an immediate refutation, so the check is worth the few seconds.
Which laws survive elsewhere
The laws are stated for real numbers, but they recur throughout mathematics, and knowing which ones fail in a new setting is often the fastest way to understand it.
| Law | Real numbers | Matrix addition | Matrix multiplication |
|---|---|---|---|
| Commutative | Holds | Holds | Fails — AB ≠ BA in general |
| Associative | Holds | Holds | Holds |
| Distributive | Holds | — | Holds over addition |
| Identity | 0 and 1 | The zero matrix | The identity matrix I |
| Inverse | All except 0 | -A always exists | Only when the determinant is non-zero |
The two failures in the right-hand column are the source of most early difficulty with matrices, and both are covered in Matrix Multiplication and Matrix Algebra.
Common mistakes
| Mistake | What went wrong | Correct form |
|---|---|---|
| -(x - 3) = -x - 3 | The -1 was distributed over only the first term | -(x - 3) = -x + 3 |
| (a + b)2 = a2 + b2 | Squaring was treated as distributing | (a + b)2 = a2 + 2ab + b2 |
| 1a + b = 1a + 1b | Division does not distribute over addition in the denominator | No simplification; combine over a common denominator instead |
| From ab = 6, concluding a = 6 or b = 6 | The zero-product property applied to a non-zero constant | Rearrange to ab - 6 = 0 and factor properly |
| Cancelling a from ab = ac without checking a ≠ 0 | The multiplicative inverse was assumed to exist | If a = 0 the equation holds for all b and c; state the case |
Frequently asked questions
Why does the order of the laws matter when simplifying?
It usually does not change the answer, but it does change how much work there is. Rearranging (3 + x) + 4 to x + (3 + 4) groups the two constants so they collapse to 7 in one step. The commutative and associative laws are what permit the rearrangement.
Is subtraction associative?
No. (8 - 3) - 2 = 3 but 8 - (3 - 2) = 7. Subtraction and division are neither commutative nor associative. The laws are stated for addition and multiplication only, and subtraction is handled as adding the inverse.
What is the difference between an identity and an inverse?
An identity leaves a number unchanged: a + 0 = a and a · 1 = a. An inverse cancels a number back to the identity: a + (-a) = 0 and a · a-1 = 1. Every real has an additive inverse; every real except zero has a multiplicative inverse.
Why does a·b = 0 imply one of them is zero?
Suppose a ≠ 0. Then a-1 exists, so multiplying both sides gives b = a-1 · 0 = 0. The argument needs a-1 to exist, which is why the property holds over the reals but fails in systems where non-zero elements can multiply to zero.
Source. Handwritten teaching notes, Week 1, pages 3-5.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.