Axioms

Axioms

Axiom is a mathematical statement that is assumed to be true. There are five axioms of algebra. These are reflexive axiom, symmetric axiom, transitive axiom, additive axiom, and multiplicative axiom.

Reflexive Axiom

A number is equal to itself. (e.g a = a). This is known as the first axiom of equality.

Symmetric Axiom

Numbers are symmetric around the equals sign. If a = b then b = a. This is known as the second axiom of equality.

Transitive Axiom

If a = b and b = c then a = c. This is the third axiom of equality and this is related to Euclid’s Common Notion One: “Things equal to the same thing are equal to each other.”

Additive Axiom

If a = b then a + c = b + c. If two quantities are equal and an equal amount is added to each then they are still equal. This is known as “the addition property of equality.”

Multiplicative Axiom

If a = b then ac = bc. Multiplication is just repeated addition and the multiplicative axiom follows from the additive axiom. It is known as “the multiplication property of equality.”

 

There are two rearrangement properties of algebra.

1. Addition has the commutative axiom, associative axiom, and rearrangement property.

Commutative Axiom for Addition

 The order of addends in an addition expression is switched.

 For example x + y = y + x

Associative Axiom for Addition

 In an addition expression, it does not matter how the addends are grouped.

For example (x + y) + z = x + (y + z)

Rearrangement Property of Addition

The addends in an addition expression are arranged and grouped in any order and this is a combination of the associative and commutative axioms.

For example x + y + z = x + (y + z) = (x + y) + z = z + (y + x) = y + (z + x)

 

2. Multiplication has the commutative axiom, associative axiom, and rearrangement property.

Commutative Axiom for Multiplication

 The order of factors in a multiplication expression is switched.

 For example x * y = y * x

Associative Axiom for Multiplication

 In a multiplication expression, it does not matter how the factors are grouped.

For example (x * y) * z = x * (y * z)

Rearrangement Property of Multiplication

The factors in a multiplication expression are arranged and grouped in any order and this is a combination of the associative and commutative axioms.

For example xyz = x(yz) = z(yx) = y(zx)

 

Also, read the Operations with negative numbers