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
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
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