Math is filled with theorems, axioms, and properties. Though the following properties may seem simplistic, the ISEE will couple these properties with difficult concepts, such as algebraic functions.

Reflexive Property

A value is equal to itself.

  • x = x

Examples

  • 17 = 17
  • y = 400

Symmetric Property

Equal values may be placed on either side of the equals sign.

  • x = y
  • y = x

Examples

  • x = 25 or 25 = x
  • 5 = y or y = 5

Transitive Property

If one value equals a second value and the second value equals a third value, then the third value equals the first.

  • x = y
  • y = z
  • x = z

Commutative Property 

When solely adding values, you may change the order without changing the result. The same is true for multiplication.

  • x + y = y + x
  • x × y = y × x

Examples

  • 1 + 2 + 3 = 6
  • 3 +1 + 2 = 6
  • 3 × 4 × 5 = 60
  • 4 × 5 × 3 = 60

Associate Property 

When solely adding or multiplying values, you may change the groupings of values without changing the result. The groups are separated by parentheses.

  •  x + (y + z) = (x + y) + z
  • x × (y × z) = (x × y) × z

Examples

  • 4 + (6 + 5) = (4 + 6) + 5
  • 4 × (6 × 5) = (4 × 6) × 5

Addition Property 

You must add the same value to both sides of the equals sign to maintain the equality of the values.

  • x + a = y + a

Examples

  • x + 2 = y + 2
  • 5 + 3 = 5 + 3

Multiplication Property 

You must multiply the same value to both sides of the equals sign to maintain the equality of the values.

  • ax = ay

Examples 

  • 4(20) = 4(20)
  • 4(3-1) = 4(2)

Distributive Property

You may multiply a value to a sum by first multiplying the value to the sum’s addends and then adding those products together.

  • x(y + z) = xy + xz

Examples

  • 5(3 + 7) = 5(3) + 5(7) = 15 + 35 = 50

Definition of Division

When dividing values, it is the same thing as multiplying the value in the numerator by the reciprocal of the value in the denominator.

  • x/y = x × 1/y
  • x/(1/y) = x × y/1

Examples

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Multiplication Property of Zero

The product of any value and zero will be zero.

  • x × 0 = 0

Examples

  • 15 × 0 = 0
  • 12 × 11 × 10 × 0 = 0

Zero Product Property

If the product of two values is zero, one of the values is zero.

  • xy = 0
  • x = 0 or y = 0

Examples 

  • 17x = 0; x = 0

Definition of Zero as a Dividend 

The quotient of zero and any value (except zero) is zero.

  • 0/x = 0

Examples 

  • 0/9 = 0

Definition of Zero as a Divisor 

The quotient of any value and zero is considered undefined.

  • x/0 = undefined 

Examples

  • 9/0 = undefined