Properties of Real Numbers

In your algebra classes, you’ll encounter the three basic properties of numbers.  Although these properties will be more relevant in matrix algebra, calculus and advanced math, the math that you’ve had thus far has obeyed these properties and you may be assured that the definitions will appear on a test.

The Commutative Property, in general, states that changing the ORDER of two numbers either being added or multiplied, does NOT change its value:

a + b = b + a   and   ab = ba

The two sides are called equivalent expressions because although they look different they have the same value.

Use the commutative property to write an equivalent expression to 2.5x + 3y.

Using the commutative property of addition, where changing the order of a sum does not change its value, we get:

2.5x + 3y = 3y + 2.5x

Use the commutative property to write an equivalent expression to 4x • 7y.

Using the communicative property of multiplication (where changing the order of a product does not change its value), we get:

4x • 7y = 7y • 4x

The Associative Property, in general, states that changing the GROUPING of numbers that are either being added or multiplied does NOT change its value:

a + (b + c) = (a + b) + c  and  a(bc) = (ab)c

Use the associative property to write an equivalent expression to (a + 5b) + 2c.

Using the associative property of addition (where changing the grouping of a sum does not change its value) we get:

(a + 5b) + 2c = a + (5b + 2c)

Use the associative property to write an equivalent expression to (1.5xy.

Using the associative property of multiplication (where changing the grouping of a product does not change its value) we get:

(1.5x)y = 1.5(xy)

IT IS IMPORTANT TO NOTE THAT COMMUTATIVE AND ASSOCIATIVE PROPERTIES DO NOT WORK FOR SUBTRACTION AND DIVISION!

Distributive Properties: When you have two or more terms being added or subtracted within parentheses and being multiplied by an outside term, multiply the outside term times EVERY term on the inside:

a(b + c) = ab + ac  and  3(y − z) = 3y − 3z

This idea can be extended to more than two terms within the parentheses.

Use the distributive property to write 2(x − y)

Multiplying every term on the inside of the parentheses by 2 we get 2x − 2y

Use the distributive property to write 3(5x + 4y − 7)

Multiplying every term on the inside of the parentheses by 3 we get 15x + 12y − 21

Other Number Properties

These two inverses will become important when you begin solving equations.  Keep them in your memory banks until that time.

Additive Inverse: For each real number a, except 0, there is a unique real number, −a, such that a + (−a) = 0.

When you add a number to its additive inverse, the result is 0.  This zero is also referred to as the additive “identity” since adding zero to any number does not change its value.  Terms that are synonymous with additive inverse are “negative” and “opposite”.

Write the additive inverse of 3.   The additive inverse of 3 is −3 since 3 + (−3) = 0

Write the additive inverse of ⅕.   The additive inverse of ⅕ is −⅕ since ⅕ + (−⅕) = 0

Multiplicative Inverse: For each real number a, except 0, there is a unique real number 1/a, such that a × 1/a = 1

When you multiply a number by its multiplicative inverse the result is 1.  The one here also refers to the multiplicative “identity” since multiplying any number by one does not change its value.  A more common term for multiplicative inverse is “reciprocal”.

Write the reciprocal of 3.   The reciprocal of 3 is ⅓ since 3 × ⅓ = 1

Write the reciprocal of .  The reciprocal of ⅕ is 5 since 5 × ⅕ = 1

As with the inverses, these properties will be important tools for solving equations.

The Property of Equality states that you may perform any operation in an equation as long as you do the same thing to both sides.  For example, in solving the equation x + 3 = 7, you would subtract 3 from both sides, x + 3 (− 3) = 7 (− 3) and the result is x = 4.  Another instance would be in solving 4x = 12.  In this case, dividing both sides by 4 gives you x = 3.

The Zero-Product Property states that, if p × q = 0 then either p = 0 or q = 0.  This will be especially useful when solving quadratics.

The Reflexive Property says anything equals itself.  This may sound silly but you will almost certainly run into equations that solve to something like −2 = 10 or x < x.  Equations of this type are called contradictions and have no solution.

There are some properties that are handy for solving word problems, especially where substitution is required.

The Symmetric Property states, if x = y, then y = x.  Again, this seems obvious but it’s good for spotting contradictions as well as solving complex word problems.

The Transitive Property: If you know x = y and y = z, you can say that x = z.

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