Other Number Properties: Identities,
Inverses, Symmetry, etc.
(page 2 of 2)

If your textbook gets really ornate, you may have to delve into some of the more esoteric properties of numbers. For this, you need to know that "the identity" is whatever doesn't change your number at all, and "the inverse" is whatever turns your number into the identity.

For addition, "the identity" is zero, because adding zero to anything doesn't change anything. The "inverse" is the additive inverse: it's the same number, but with the opposite sign. For instance, suppose your number is –6, and you're adding. The identity is zero, and the inverse is 6, because –6 + 6 = 0.

For multiplication, "the identity" is one, because multiplying by one doesn't change anything. The "inverse" is the multiplicative inverse: the same number, but on the opposite side of the fraction line. For instance, suppose your number is –6, and you're multiplying. The identity is one, and the inverse is
–1/6, because (–6)( –1/6 ) = 1.

You also know (if you've done any equation solving) that you can do anything you want to an equation, as long as you do the same thing to both sides. This is the "property of equality".

The basic fact that you need for solving many equations, especially quadratics, is that, if p×q = 0, then must have either p = 0 or else q = 0. The only way you can multiply two things and end up with zero is if one (or both) of those two things was zero to start with. This is the "zero-product property".

And there are some properties that you use to solve word problems, especially where substitution is required. Anything equals itself: this is the "reflexive" (reflecting onto itself) property. Also, it doesn't matter which order the equality is in; if x = y, then y = x: this is the "symmetric" (they match) property. You can "cut out the middleman", so to speak; if x = y and y = z, then you can say that x = z: this is the "transitive" (moving across) property. Two numbers are either equal to each other or unequal; this is the "trichotomy" law (so called because there are three cases for two given numbers, a < b, a = b,
or
a > b). And you can plug in for variables, so if x = 3, then 4x = 12, because 4x = 4(3): this is the "substitution" property.

Here are some examples. Note: textbooks vary somewhat in the names they give these properties; you'll need to refer to the examples in your book to know the exact format you should use.

Determine which property was used.

• 1×7 = 7

They multiplied, and they didn't change anything: the multiplicative identity.

• –7y = –7y

This is obvious: anything equals itself. They used the reflexive property.

• If 10 = y, then y = 10.

When solving an equation, I might rearrange things so I end up with the variable on the left. But I only switched sides; I didn't actually change anything: the symmetric property.

• x + 0 = x

• If 2(a + b) = 3c, and a + b = 9, then 2(9) = 3c.

You might be torn here between the transitive property and the substitution property. If you look closely, what they did was substitute "9" for "a + b", so they used the substitution property.

• 2 = x, so 2 + 5 = x + 5

They did the backwards of solving an equation, but the point is that they were working with an equation. They changed the equation by adding equal things to both sides:  the additive property of equality.   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

• If x + 2 = 10, then x + 2 + (–2) equals what, and why?

They solved the equation by getting rid of the 2 from both sides. Since they added the same thing to both sides, they got x = 8 by the additive property of equality.

• (x – 3)(x + 4) = 0, so x = 3 or x = –4.

They set the quadratic equal to zero, factored, and then solved each factor: the zero-product property.

• 4x = 8, so x = 2

They solved the equation by dividing both sides by 4, or, which is the same thing, multiplying both sides by ( 1/4 ).  In other words, they changed the equation by doing the same multiplying to both sides: the multiplicative property of equality.

• If x is not equal to y and not less than y, what must be true of x, and why?
• By the trichotomy law, there are only three possible relationships between x and y, and they've eliminated two of them. Then x > y, by the trichotomy law.

• x + (–x) = 0

• ( 3/3 )( 2/5 ) + ( 5/5 )( 4/3 ) =  6/1520/15

They converted to a common denominator by multiplying both fractions by a useful form of 1; remember that 3/3 and 5/5 are just 1! So they used the multiplicative identity.

• If 5x = 0, what is x, and why?

You can do this in either of two ways: multiply both sides by 1/5 (the multiplicative property of equality) and then get x = 0, or you could say that, since 5 doesn't equal zero, then x must equal zero (by the zero-product property).

• ( 2/3 )( 3/2 ) = 1

They multiplied, and they ended up with one: the multiplicative inverse.

• If 3x + 2 = y and y = 8, then 3x + 2 = 8.

You might be torn here between the transitive property and the substitution property. What they did here was "cut out the middleman" by removing the "y" in the middle, so they used the transitive property.

• If x = 14, what does x equal, and why?

To solve this, you would multiply both sides by a negative one, to cancel off the minus sign. So:

x = –14, by the multiplicative property of equality.

• If x = 3 and y = –4, then what does xy equal, and why?

By substitution (plugging in for the variables), you get (3)(–4). In other words:

xy = –12, by the substitution property.

• Can x < x? Why or why not?
• By the reflexive property, x = x. By the trichotomy law, if a = b then a cannot be less than b. So the answer is "no, by the reflexive property and the trichotomy law".

Don't let the seeming pointlessness of these questions bother you. Instead, view this stuff as "gimme" questions for the next test.