Properties of Arithmetic Operations

What Are Properties of Operations?

Four statements about arithmetic. Which of them is a property of an operation?

a + b = b + a

;

a - b = b - a

;

3 + 5 = 8

; and "multiply before you add".

Only the first holds for every choice of numbers and describes how an operation behaves.

statement	true for every input?	what it is
a + b = b + a	yes	a property of addition
a − b = b − a	no: 5 − 3 = 2, 3 − 5 = −2	false in general
3 + 5 = 8	about one computation only	a single fact
multiply before you add	not a statement about results	a convention of notation

One fails for some inputs, one is about a single computation, and one is an agreement about notation.

3 + 5 = 8

is true but says nothing about other numbers. The order of operations could have been agreed differently without changing what any operation does.

The definition

A property of an operation is a statement about the operation that holds for every choice of inputs.

A property of an operation is a statement that holds for every choice of inputs. The expression

3 + 5

equals

5 + 3

, and so does

-12 + 0.7

, and so does

\frac{2}{9} + \sqrt{11}

. The pattern — that the order of two addends does not affect the sum — is universal. That universality is what makes it a property.

So

a + b = b + a

is a property of addition, the commutative property, while

a - b = b - a

is not a property of subtraction.

Properties are facts about how an operation behaves — not definitions, methods or conventions.

Properties differ from definitions and from procedural rules. A definition introduces an object: what a fraction is, what a prime number is. A procedural rule prescribes a method: how to add fractions, how to test divisibility by 9. A property states a fact that the operation itself satisfies, independently of any specific computation.

Properties are also distinct from conventions. The order of operations — multiply before add, evaluate parentheses first — is a convention agreed upon to disambiguate notation. The commutative property of addition is not a convention. It would remain true under any notational system, because it describes how addition behaves, not how it is written.

Properties are used constantly, usually without being named.

The properties on this page are the foundation of arithmetic and algebra. Every manipulation in algebra — combining like terms, factoring, solving equations — relies on them. They are usually applied without being named, but they are always present.

The rest of the page covers the two main operations, the commutative, associative and distributive properties, identity and inverse elements, the zero property, sign rules, subtraction and division as inverse operations, why the properties matter, related concepts and common pitfalls.

Two Main Operations: Addition and Multiplication

Most of the properties on this page come in pairs. There is a commutative property of addition and a commutative property of multiplication. There is an associative property of addition and an associative property of multiplication. There is an additive identity and a multiplicative identity, an additive inverse and a multiplicative inverse. The pattern is consistent and intentional.

The reason is that addition and multiplication are the two fundamental binary operations of arithmetic. Every other operation reduces to one of them. Subtraction is not a separate operation but a special case of addition:

a - b

is shorthand for

a + (-b)

, treated in detail under subtraction as inverse addition. Division is similarly a special case of multiplication:

\frac{a}{b}

is shorthand for

a \cdot \frac{1}{b}

, treated under division as inverse multiplication. Once subtraction and division are recognized as derived operations, only addition and multiplication remain as the primitive ones.

This is why the properties below appear in dual form. Each section presents the addition version and the multiplication version side by side, reflecting the symmetry of the two operations and the structural pattern that runs through arithmetic.

The pairs this page goes through, in one grid: each property checked for all four operations, with the counterexamples the sections below use.

DownloadPNG2×PDF

properties of operations

Which operation has which property?

Addition and multiplication share every property; subtraction and division lose the first two.

$+$	$\times$	$-$	$\div$
Holds $a + b = b + a$	Holds $a \cdot b = b \cdot a$	Fails $7 - 4 \neq 4 - 7$	Fails $7 \div 4 \neq 4 \div 7$

Holdsany grouping	Holdsany grouping	Failsgrouping matters	Failsgrouping matters

Holds $a + 0 = a$	Holds $a \cdot 1 = a$	not on this page	not on this page

Holds $a + (-a) = 0$	Conditional $\frac{1}{a}$ , for $a \neq 0$	not on this page	not on this page

Subtraction and division are derived operations:

a - b = a + (-b)

and

\frac{a}{b} = a \cdot \frac{1}{b}

. Distributive links the two main ones:

a(b + c) = ab + ac

.

Which operation has which property?·/arithmetic/propertiesLearn Math Class

The two right-hand columns explain themselves once subtraction is read as adding

-b

and division as multiplying by

\frac{1}{b}

: rewritten that way, they borrow every property of addition and multiplication, and the asymmetric shorthand is what loses the first two rows.

Commutative Property

The commutative property states that the order of operands does not change the result.

For Addition

Commutative Property

a + b = b + a

The expressions

7 + 4

and

4 + 7

both equal

11

. The order of the two addends is irrelevant.

For Multiplication

Commutative Property

a \cdot b = b \cdot a

The expressions

7 \cdot 4

and

4 \cdot 7

both equal

28

. Reversing the factors does not affect the product. The dot

\cdot

, the cross

\times

, and plain juxtaposition (

ab

) all denote the same multiplication — the three forms are listed among the arithmetic symbols.

Subtraction and Division Are Not Commutative

Neither subtraction nor division shares this property. The expressions

7 - 4

and

4 - 7

are different (

3

and

-3

). The expressions

7 \div 4

and

4 \div 7

are different (

1.75

and approximately

0.571

). The commutative pattern stops at addition and multiplication — for the reasons developed in subtraction as inverse addition and division as inverse multiplication.

Associative Property

The associative property states that the grouping of operands does not change the result.

For Addition

Associative Property

(a + b) + c = a + (b + c)

To compute

2 + 3 + 5

, the order in which the additions are performed makes no difference. Adding

2 + 3

first and then adding

5

gives

10

. Adding

3 + 5

first and then adding

2

gives the same. The parentheses can be moved without consequence.

For Multiplication

Associative Property

(a \cdot b) \cdot c = a \cdot (b \cdot c)

To compute

2 \cdot 3 \cdot 5

, the grouping

(2 \cdot 3) \cdot 5 = 6 \cdot 5 = 30

matches

2 \cdot (3 \cdot 5) = 2 \cdot 15 = 30

.

Subtraction and Division Are Not Associative

Subtraction fails:

(8 - 3) - 2 = 3

, but

8 - (3 - 2) = 7

. Division fails:

(12 \div 6) \div 2 = 1

, but

12 \div (6 \div 2) = 4

. Without associativity, parentheses are mandatory for these operations.

Generalization

The associative property extends to any number of terms or factors. A sum like

a + b + c + d + e

can be evaluated in any grouping order and yields the same result. This is why such expressions are written without parentheses — none are needed.

Fact: Computer sums depend on the grouping

In standard double precision,

(10^{20} + (-10^{20})) + 1 = 1

, but

10^{20} + (-10^{20} + 1) = 0

. A stored number keeps only about

16

significant digits, so

-10^{20} + 1

rounds back to

-10^{20}

and the

1

is lost.

The property is a statement about exact numbers. A calculator or a computer works with rounded numbers instead: it keeps a fixed number of significant digits and rounds the result of every single addition before the next one starts. A long sum is therefore computed as a chain of rounded partial sums, and the grouping decides which partial sums are formed along the way. Exact arithmetic, with whole numbers or fractions written out in full, involves no rounding at all.

Distributive Property

The distributive property connects addition and multiplication. It states that multiplying a sum by a number gives the same result as multiplying each addend separately and then adding.

Distributive Property

a(b + c) = ab + ac

For

3(4 + 5)

: applying the formula gives

3 \cdot 4 + 3 \cdot 5 = 12 + 15 = 27

. Computing directly gives

3 \cdot 9 = 27

. The two routes agree.

Pitfall: Distributing only to the first term

Multiplying only the first addend gives

3(4 + 5) = 3 \cdot 4 + 5 = 17

. The factor multiplies both addends:

3 \cdot 4 + 3 \cdot 5 = 27

, the same as

3 \cdot 9

.

The factor in front of the parentheses multiplies the entire sum. With more addends inside, the pattern extends term by term:

2(x + y + 3) = 2x + 2y + 6

, one product for each addend. The number of products always matches the number of addends inside the parentheses.

The right form mirrors the left:

Distributive Property

(b + c)a = ba + ca

The property also distributes over subtraction:

Distributive Property

a(b - c) = ab - ac

This works because subtraction is a special case of addition — see subtraction as inverse addition.

Factoring Reverses Distribution

Reading the equation right to left gives the basis for factoring:

ab + ac = a(b + c)

. Pulling a common factor out of a sum is the inverse direction of distribution. The two operations — distributing and factoring — are the same property applied in opposite directions.

Why It Matters

Distribution is the only property linking the two main operations. Without it, addition and multiplication would be entirely independent. The distributive property is what allows expressions like

3x + 12

to be rewritten as

3(x + 4)

, and what makes algebraic manipulation possible.

Form	Statement	Example
Left distribution	a(b + c) = ab + ac	3(4 + 5) = 12 + 15 = 27
Right distribution	(b + c)a = ba + ca	(4 + 5)·3 = 12 + 15 = 27
Over subtraction	a(b − c) = ab − ac	3(7 − 2) = 21 − 6 = 15
Factoring (reverse direction)	ab + ac = a(b + c)	3x + 12 = 3(x + 4)

The distributive property as an array

Three rows of nine dots make 3 · 9 = 3 · (4 + 5) = 27. A line after the fourth column splits every row into 4 blue dots and 5 amber dots. Counting the parts separately gives 3 · 4 = 12 blue and 3 · 5 = 15 amber, and 12 + 15 = 27: the same dots counted two ways, which is exactly 3(4 + 5) = 3 · 4 + 3 · 5.

Read right to left, the same picture is factoring: 12 + 15 = 3 · (4 + 5).

Identity Elements

An identity element for an operation is a value that leaves any input unchanged. Each of the two main operations has its own.

Additive Identity

Identity Elements

a + 0 = a

Zero is the additive identity. Adding it to any number returns that number unchanged. The property holds in both directions:

0 + a = a

as well.

Multiplicative Identity

Identity Elements

a \cdot 1 = a

One is the multiplicative identity. Multiplying any number by

1

returns that number unchanged. Like the additive case, this works in both directions:

1 \cdot a = a

.

Why Identities Matter

Identity elements are not interchangeable. Adding

1

does not leave a number unchanged, and multiplying by

0

does not either. Each operation has exactly one identity, and that uniqueness is structural — it follows from the definition of the operation, not from convention.

Identities are essential for defining inverses. The next section explains how every real number has an additive inverse summing to

0

, and how every nonzero real number has a multiplicative inverse multiplying to

1

— relationships that require the identity elements as their target.

Inverse Elements

An inverse element undoes an operation. For each of the two main operations, every element has an inverse, with one exception.

Additive Inverse

Inverse Elements

a + (-a) = 0

For every real number

a

, the value

-a

is its additive inverse. The two combine to produce the additive identity. Every real number has exactly one additive inverse.

Multiplicative Inverse

Inverse Elements

a \cdot \frac{1}{a} = 1, \quad a \neq 0

For every nonzero real number

a

, the value

\frac{1}{a}

— also called its reciprocal — is its multiplicative inverse. The two combine to produce the multiplicative identity.

Why Zero Has No Multiplicative Inverse

Fact: The first rules for zero allowed

0 \div 0

Brahmagupta's rules of 628, the oldest known rules for computing with zero, declared

0 \div 0 = 0

. Every

q

satisfies

q \cdot 0 = 0

, so

0

has no better claim to be the quotient than any other number.

The number

0

is the only real number without a multiplicative inverse. The equation

0 \cdot x = 1

has no solution because, by the zero property of multiplication,

0 \cdot x = 0

for every

x

. There is no number that, when multiplied by

0

, gives

1

. This is the formal reason division by zero is undefined:

\frac{1}{0}

would be the multiplicative inverse of

0

, and that inverse does not exist.

Zero Property of Multiplication

a \cdot 0 = 0

RULEZero Property of Multiplication

Multiplying any number by zero gives zero.

The property holds in both directions:

0 \cdot a = 0

as well.

This is sometimes treated as a separate axiom, but it is actually a consequence of the distributive property. Starting from

a \cdot 0 = a \cdot (0 + 0) = a \cdot 0 + a \cdot 0

, subtracting

a \cdot 0

from both sides leaves

0 = a \cdot 0

. The result follows from the structure of arithmetic itself, not from a separate definition.

The zero property is what prevents zero from having a multiplicative inverse, as discussed in inverse elements. It is also why the product of any list of numbers is zero whenever even one of the factors is zero — a fact that underlies factoring techniques in algebra.

Sign Rules

Three rules govern signs in multiplication.

Sign Rules

(-a)(-b) = ab

The product of two negatives is positive. Multiplying

-3

by

-4

gives

12

, not

-12

.

Sign Rules

(-a)(b) = -(ab)

The product of a negative and a positive is negative. Multiplying

-3

by

4

gives

-12

.

Sign Rules

-(-a) = a

The negative of a negative is the original number. The expression

-(-5)

equals

5

.

Pitfall: Using the product sign rule for sums

Applying "two negatives make a positive" to a sum gives

(-3) + (-4) = 7

. The rule is about products,

(-3)(-4) = 12

; adding two negatives goes further below zero,

(-3) + (-4) = -7

.

These rules are not independent axioms — they follow from the distributive property combined with the definition of additive inverses. Starting from

(-a)(b) + ab = (-a + a)(b) = 0 \cdot b = 0

, the value

(-a)(b)

must equal

-(ab)

. A similar derivation produces the other two rules.

Rule	Statement	Example
Negative × negative	(−a)(−b) = ab — product of two negatives is positive	(−3)(−4) = 12
Negative × positive	(−a)(b) = −(ab) — opposite signs yield a negative product	(−3)(4) = −12
Double negative	−(−a) = a — negating a negative returns the original	−(−5) = 5

The sign rules are not memorized conventions: distributivity and additive inverses force every one of them.

Subtraction as Inverse Addition

a - b = a + (-b)

Subtraction is defined as the addition of an additive inverse. The expression

7 - 3

is shorthand for

7 + (-3)

. There is no separate "subtraction" operation in formal arithmetic — only addition combined with negation.

This definition has consequences. Because subtraction reduces to addition, certain properties of addition transfer to subtraction in a controlled way. The distributive property distributes over subtraction:

a(b - c) = ab - ac

, since

a(b + (-c)) = ab + a(-c) = ab - ac

.

Pitfall: Grouping the terms after a minus sign

Reading

10 - 4 + 3

as

10 - (4 + 3)

gives

3

. The operations go left to right,

(10 - 4) + 3 = 9

, which is also

10 + (-4) + 3 = 9

.

The commutative and associative properties, however, do not transfer cleanly. The expression

a - b

does not equal

b - a

, and

(a - b) - c

does not equal

a - (b - c)

. The reason is that subtraction is shorthand for an asymmetric operation —

a + (-b)

treats

a

and

b

differently — so the symmetry that addition enjoys is lost in the abbreviation.

Recognizing subtraction as inverse addition simplifies algebraic manipulation. Once

a - b

is rewritten as

a + (-b)

, the addition properties apply directly, and many apparent quirks of subtraction dissolve.

Division as Inverse Multiplication

\frac{a}{b} = a \cdot \frac{1}{b}, \quad b \neq 0

Division is defined as multiplication by a multiplicative inverse. The expression

\frac{12}{4}

is shorthand for

12 \cdot \frac{1}{4}

. There is no separate "division" operation in formal arithmetic — only multiplication combined with reciprocation.

The structural parallel with subtraction is exact. Subtraction is addition of an additive inverse; division is multiplication by a multiplicative inverse. Both are derived operations built on top of their primary counterparts.

Pitfall: Taking division by zero as zero

Giving

5 \div 0

the value

0

, like

0 \div 5 = 0

, fails the check

0 \cdot 0 = 0 \neq 5

. No number times

0

gives

5

, so

5 \div 0

is undefined, while

0 \div 5 = 0

because

0 \cdot 5 = 0

.

The condition

b \neq 0

is unavoidable. Since

0

has no multiplicative inverse (see inverse elements), the expression

\frac{a}{0}

has no meaning. This is the formal reason division by zero is undefined.

Like subtraction, division does not inherit the commutative or associative properties of multiplication. The expression

\frac{a}{b}

does not equal

\frac{b}{a}

, and

\frac{a/b}{c}

does not equal

\frac{a}{b/c}

. Rewriting

\frac{a}{b}

as

a \cdot \frac{1}{b}

— moving from division to multiplication-by-reciprocal — is the standard technique for applying multiplication properties to expressions that involve division.

Why These Properties Matter

The properties on this page are the rules that make algebra possible. Every standard manipulation — combining like terms, factoring, expanding products, solving equations — depends on them.

Combining

3x + 5x

into

8x

is an application of the distributive property:

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

. Rewriting

a^2 - b^2

as

(a - b)(a + b)

uses distributive expansion in reverse. Solving

2x + 5 = 11

by subtracting

5

from both sides relies on the additive inverse and the principle that adding the same quantity to both sides of an equation preserves equality.

Mental arithmetic also relies on these properties. Computing

25 \cdot 4 \cdot 7

as

100 \cdot 7 = 700

uses associativity to regroup the factors. Computing

8 \cdot 17

as

8(20 - 3) = 160 - 24 = 136

uses distributivity to break a hard multiplication into easier pieces.

The properties are usually applied without being named. The point of stating them explicitly is to clarify why standard techniques work and to provide a foundation that generalizes — the same properties hold for the rational numbers, the real numbers, and many algebraic structures encountered later in mathematics.

Technique	Example	Property used
Combining like terms	3x + 5x = (3 + 5)x = 8x	distributive (reverse)
Factoring	a² − b² = (a − b)(a + b)	distributive (reverse)
Solving by inverse	2x + 5 = 11 → subtract 5 from both sides → 2x = 6	additive inverse
Regrouping for mental math	25 · 4 · 7 = (25 · 4) · 7 = 100 · 7 = 700	associative
Breaking a product	8 · 17 = 8(20 − 3) = 160 − 24 = 136	distributive (over subtraction)

Common Pitfalls

Five mistakes account for many of the wrong answers when these properties are used. Four are marked where they happen in the sections above; mixing up the two identities is the one the identity section already points out. Here each is worked through in full, with the wrong and the right working side by side.

DownloadPNG2×PDF

1Distributing only to the first term

A factor in front of parentheses is multiplied into the first addend only, and the rest of the sum is copied unchanged.

Wrong turn

Error here

3(4 + 5) = 3 \cdot 4 + 5 = 12 + 5 = 17

. In algebra the same slip turns

2(x + 3)

into

2x + 3

.

Right path

The factor multiplies every addend, so

3(4 + 5) = 3 \cdot 4 + 3 \cdot 5 = 12 + 15 = 27

and

2(x + 3) = 2x + 6

.

Why it happensThe factor stands right next to the first addend and seems to belong to it alone, while the parentheses that extend it to the whole sum are easy to overlook.

How to checkWhen the parentheses hold numbers, add them first:

3(4 + 5) = 3 \cdot 9 = 27

, not

17

. With a variable, substitute a value: for

x = 1

,

2(1 + 3) = 8

, while

2 \cdot 1 + 3 = 5

.

2Mixing up the two identities

The identity of addition is used for multiplication, so multiplying by

0

is taken to leave a number unchanged, just as adding

0

does.

Wrong turn

Error here

7 + 0 = 7

, so

7 \cdot 0 = 7

as well.

Right path

0

leaves a number unchanged only under addition,

7 + 0 = 7

. The identity of multiplication is

1

, so

7 \cdot 1 = 7

, and by the zero property

7 \cdot 0 = 0

.

Why it happensAdding nothing changes nothing, and multiplying by zero feels like the same kind of "doing nothing" to the number.

How to checkRead the product as repeated addition.

7 \cdot 1

is one seven, which is

7

;

7 \cdot 0

is no sevens at all, which is

0

.

3Using the product sign rule for sums

The rule that two negatives give a positive is applied to addition, so the sum of two negative numbers comes out positive.

Wrong turn

Error here

Two negatives make a positive, so

(-3) + (-4) = 7

.

Right path

Adding a negative number moves down the number line. Starting at

-3

and moving

4

further down gives

(-3) + (-4) = -7

. The positive result belongs to the product:

(-3)(-4) = 12

.

Why it happensThe rule is often memorized as a short slogan without the word "multiplying", and the slogan is then used on any expression with two minus signs.

How to checkThe sum of two numbers below zero is also below zero. A debt of

3

plus a debt of

4

is a debt of

7

, never a gain.

4Grouping the terms after a minus sign

In a chain of subtractions and additions, the terms after a minus sign are added together first and their total is subtracted, as if the minus applied to everything that follows it.

Wrong turn

Error here

10 - 4 + 3 = 10 - (4 + 3) = 10 - 7 = 3

.

Right path

The operations go left to right,

(10 - 4) + 3 = 6 + 3 = 9

. Rewriting the subtraction as an addition gives the same:

10 + (-4) + 3 = 9

, and now the terms may be grouped in any way, for example

10 + (-4 + 3) = 10 + (-1) = 9

.

Why it happensSums can be regrouped freely, and the same freedom is used on an expression that contains a minus sign, where the associative property does not hold.

How to checkRewrite every subtraction as adding the opposite before regrouping. Each minus sign then stays attached to the number right after it:

10 + (-4) + 3

has the terms

10

,

-4

and

3

, which add to

9

.

5Taking division by zero as zero

A number divided by zero is given the value

0

, by analogy with zero divided by a number.

Wrong turn

Error here

0 \div 5 = 0

, so

5 \div 0 = 0

as well.

Right path

A quotient

q = a \div b

must satisfy

q \cdot b = a

. For

0 \div 5

, the value

q = 0

works, since

0 \cdot 5 = 0

. For

5 \div 0

, the quotient would need

q \cdot 0 = 5

, but

q \cdot 0 = 0

for every

q

by the zero property, so no value works and

5 \div 0

is undefined. In the terms of this page,

\frac{5}{0} = 5 \cdot \frac{1}{0}

would need the multiplicative inverse of

0

, which does not exist.

Why it happensBoth expressions contain the same two numbers and one of them really is

0

. Division is not commutative, so swapping the two numbers changes the question.

How to checkMultiply back. The quotient times the divisor must give the dividend, and

0 \cdot 0 = 0

, not

5

.

Common pitfalls with the properties of arithmetic·/arithmetic/propertiesLearn Math Class

Each slip applies a property where it does not hold. The distributive law multiplies every addend,

0

is the identity for addition and

1

for multiplication, two negatives make a positive only in a product, a subtraction cannot be regrouped unless each minus sign stays with its number, and division by zero has no value at all. Working the numbers out directly, or multiplying back, catches each slip.

Related Concepts

The properties on this page connect to several adjacent topics on the site.

Fractions inherit every property of multiplication and addition. The reciprocal of a fraction is its multiplicative inverse, and adding fractions with a common denominator uses the same commutative and associative rules as adding integers.

Modular arithmetic extends these properties to a finite cyclic system. Addition and multiplication remain commutative, associative, and distributive within any modulus, but inverses behave differently — an element has a multiplicative inverse modulo

n

only when it is coprime to

n

.

Algebraic equations are solved by applying these properties step by step. Equivalent equations are produced by adding the same quantity to both sides (additive inverse), multiplying both sides by a nonzero constant (multiplicative inverse), or distributing across parentheses. The solving methods catalogued under linear, quadratic, and polynomial equations all rest on the rules listed here.

Powers and exponents extend multiplication into repeated multiplication, and the exponent rules —

a^m \cdot a^n = a^{m+n}

,

(ab)^n = a^n b^n

— are direct consequences of the associativity and commutativity of multiplication.

Properties at a Glance

The properties of arithmetic come in matched pairs — one version for addition, one for multiplication — with the distributive property bridging the two and the zero property breaking the symmetry on the multiplication side. The table below presents the duality in one view, alongside whether each property carries over to subtraction or division (which are themselves derived from the two primary operations).

Property	For addition	For multiplication	Inherits to sub / div?
Commutative	a + b = b + a	a·b = b·a	✗
Associative	(a + b) + c = a + (b + c)	(a·b)·c = a·(b·c)	✗
Identity element	a + 0 = a (identity is 0)	a · 1 = a (identity is 1)	via inverse form
Inverse element	a + (−a) = 0 for every a	a · (1⁄a) = 1 for every a ≠ 0	defines sub, div
Distributive (bridge)	a(b + c) = ab + ac — the only property linking the two operations; distributes over subtraction too		✓
Zero property	— (no analog)	a · 0 = 0; consequence of distributivity; reason 0 has no multiplicative inverse	n/a

DownloadPNG2×PDF

Before you read

Properties in matched pairs

The properties of arithmetic come in matched pairs for addition and multiplication, with the distributive property bridging the two.

L0Properties of operations
- L1Addition
  - L2Commutativea + b = b + a
  - L2Associative(a + b) + c = a + (b + c)
  - L2Identity0
  - L2Inverse−a
- L1Multiplication
  - L2Commutativea · b = b · a
  - L2Associative(ab)c = a(bc)
  - L2Identity1
  - L2Inverserequires a ≠ 01/a
  - L2Zero propertyno additive analoga · 0 = 0
- L1Bridge
  - L2Distributivea(b + c) = ab + ac

Takeaway

One property breaks the symmetry

The zero property,

a \cdot 0 = 0

, has no counterpart for addition.

Properties of operations·/arithmetic/propertiesLearn Math Class

Arithmetic Properties FAQ

Why is a negative times a negative a positive?

+

It follows from the distributive property rather than being a separate rule. Start from (−a)(b) + ab, which factors as (−a + a)(b) = 0 · b = 0, forcing (−a)(b) to equal −(ab). Applying the same argument again to (−a)(−b) forces it to equal ab. So (−3)(−4) = 12: the result is fixed by the structure of arithmetic, not chosen by convention.Read more →

Why does any number times zero equal zero?

+

It is a consequence of the distributive property, not a separate axiom. Since 0 = 0 + 0, we get a · 0 = a · (0 + 0) = a · 0 + a · 0. Subtracting a · 0 from both sides leaves 0 = a · 0. This result is also what prevents zero from having a reciprocal, and it underlies factoring: a product is zero whenever any factor is zero.Read more →

Why does zero have no multiplicative inverse?

+

The number 0 is the only real number without a multiplicative inverse. The equation 0 · x = 1 has no solution because, by the zero property of multiplication, 0 times any number equals 0. There is no value that turns 0 into 1. This is the formal reason division by zero is undefined: 1/0 would name an inverse that does not exist.Read more →

How are subtraction and division related to addition and multiplication?

+

Subtraction is defined as addition of an additive inverse: a − b is shorthand for a + (−b). Division is defined as multiplication by a multiplicative inverse: a ÷ b is shorthand for a · (1/b). Both are derived operations rather than primitive ones, which is why neither inherits commutativity or associativity — a − b is not b − a.Read more →

Can changing the grouping change a sum on a computer?

+

Yes. The associative property holds for exact numbers, but a computer rounds after every single addition, so the grouping decides which rounded partial sums are formed. In double precision

(10^{20} + (-10^{20})) + 1 = 1

, while

10^{20} + (-10^{20} + 1) = 0

: a stored number keeps only about

16

significant digits, so

-10^{20} + 1

rounds back to

-10^{20}

and the

1

is lost. Whole numbers or fractions written out in full have no such problem.Read more →

What is 0 divided by 0?

+

It is undefined, for a different reason than

5 \div 0

. A quotient

q = 0 \div 0

would have to satisfy

q \cdot 0 = 0

, and every number does, so no single value can be chosen; for

5 \div 0

, no number works at all. Brahmagupta's rules of 628, the oldest known rules for computing with zero, declared

0 \div 0 = 0

, but

0

has no better claim to be the quotient than any other number.Read more →

Do two negatives make a positive when you add them?

+

No, that rule is about products.

(-3)(-4) = 12

, but

(-3) + (-4) = -7

: adding a negative number moves further down the number line, so the sum of two numbers below zero is also below zero. A debt of

3

plus a debt of

4

is a debt of

7

, never a gain. The slip comes from remembering the rule as a slogan without the word multiplying.Read more →

Is 10 − 4 + 3 equal to 3 or 9?

+

It is

9

. Subtraction and addition are done left to right,

(10 - 4) + 3 = 6 + 3 = 9

. Getting

3

means computing

10 - (4 + 3)

, a regrouping that subtraction does not allow because it is not associative. Rewriting the subtraction as adding the opposite removes the trap:

10 + (-4) + 3

has the terms

10

,

-4

and

3

, which can be grouped in any way and still add to

9

.Read more →

Arithmetic Properties

The Rules That Make Arithmetic Work

Key Terms

Properties (umbrella entities)

Related Terms

What Are Properties of Operations?

Two Main Operations: Addition and Multiplication

Which operation has which property?

Commutative Property

For Addition

For Multiplication

Subtraction and Division Are Not Commutative

Associative Property

For Addition

For Multiplication

Subtraction and Division Are Not Associative

Generalization

Distributive Property

Factoring Reverses Distribution

Why It Matters

Identity Elements

Additive Identity

Multiplicative Identity

Why Identities Matter

Inverse Elements

Additive Inverse

Multiplicative Inverse

Why Zero Has No Multiplicative Inverse

Zero Property of Multiplication

Sign Rules

Subtraction as Inverse Addition

Division as Inverse Multiplication

Why These Properties Matter

Common Pitfalls

Related Concepts

Properties at a Glance

Arithmetic Properties FAQ

Why is a negative times a negative a positive?

Why does any number times zero equal zero?

Why does zero have no multiplicative inverse?

How are subtraction and division related to addition and multiplication?

Can changing the grouping change a sum on a computer?

What is 0 divided by 0?

Do two negatives make a positive when you add them?

Is 10 − 4 + 3 equal to 3 or 9?