Additional Number Properties: Identities, Inverses, Symmetry, etc.
Purplemath
If your textbook gets really ornate, you may have to delve into some of the more esoteric properties of numbers. Two of these properties are the identity property and the inverse property.
What is the identity property?
The identity property says that, for a given number (and operation), there is another number that doesn't change the original number at all (under that operation, being addition or multiplication).
For the numbers (and variables) you've always worked with, there are two versions of the identity: one for addition and one for multiplication.
For addition, the identity is zero, because adding zero to a number or variable doesn't change the original expression. The statement 6 + 0 = 6 is true "by the identity property" (of addition).
For multiplication, the identity is the number 1, because multiplying by 1 doesn't change the original expression. The statement 6 × 1 = 6 is true "by the identity property" (of multiplication).
What is the inverse property?
The inverse property says that, for a given number (and operation), there is another number which will take the original number and convert it to that operation's identity.
For the numbers (and variables) you've always worked with, there are two versions of the identity: one for addition and one for multiplication.
For addition, the inverse is the same value but with the opposite sign, because adding these two values together will result in zero. The statement 6 + (−6) = 0 is true "by the inverse property" (of addition), because −6 is the "additive inverse" of 6.
For multiplication, the inverse is the reciprocal (that is, the flipped-over version) of the original number. The statement is true "by the inverse property" (of multiplication), because is the "multiplicative inverse" of 6.
What is the equality property?
The "property of equality" says that, given a true equation, that equation remains true (that is, the two sides remain equal to each other) when you add the same thing to both side, subtract the same thing from both sides, multiply both sides by the same thing, or divide both sides by the same thing. In particular, this is the property that lets you solve equations.
What is the Zero-Product Property?
The Zero-Product Property says that, if p×q = 0, then either p = 0 or else q = 0 (or maybe both are zero). The only way you can multiply two things and end up with zero is if one (or maybe both) of those two things was zero to start with.
This, by the way, is why you have to have your factored quadratic on one side of the "equals" sign, with zero alone on the other side, in order to solve the factors.
What is the reflexive property?
The reflexive property says that anything is equal to itself; its value "reflects" back on itself. It's like looking in a mirror. This is useful when you need to substitute from one equation into another.
For instance, they might give you two equations, y = 2x + 3 and y = −3x, and ask you to find where their graphs intersect. Where the graphs intersect, they share a point; in particular, the x- and y-values are the same. Since y = y at the intersection, then you can use the reflexive property to substitute, getting the linear equation 2x + 3 = −3x. This works because, when a thing is equal to itself, it doesn't matter (mathematically) which form of the thing you use.
What is the symmetric property?
The symmetric property says that equality has no preferred direction or location. If x = y, then y = x. The location of equal things does not change their equal-ness.
What is the transitive property?
To "transit" is to cross from one location to another; the transitive property lets you go from the start of one equality and cross to the end of another equality, as long as there are equations that pass through from start to end. If x = y and y = z, then you can say that x = z: this is due to the "transitive" (moving across) property.
What is the trichotomy law?
The trichotomy law says that, given two numbers a and b, exactly and only one of the following is true:
- a < b
- a = b
- a > b
There are no other options, and no two options can be true at the same time.
(Yes, there are "or equal to" inequalities, but these involve variables that can take on many different values. With numbers [or with variables where you've plugged in a particular value], the values are either equal to each other, or else one is greater and the other is lesser. No two numbers can be both equal to each other and also not equal to each other at the same time.)
What is the substitution property?
The substitution property says that, if you are given numerical values for variables, you can plug these values into the expression in place of those variables. So, for instance, if x = 3 and the expression 4x, then you can plug the 3 in for the x and simplify the expression to get 4x = 4(3) = 12. (You may also see this referred to as "evaluation".)
You can also plug expressions in for other expressions. For instance, suppose you know the formula for the circumference of a circle is C = 2πr, you are told that the circumference of a given circle is C = 12π, and you are asked to find the value of the radius. You can substitute the 12π for the C, getting the equation 12π = 2πr. Then you can divide through by 2π to find that the radius r is equal to 6.
Below are some worked 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 name / title / format you should use.
In the following exercises, determine which property was used.
- 1×7 = 7
They didn't change anything, so this is an identity. They multiplied, so this is:
the multiplicative identity
- −7y = −7y
This is obvious: anything equals itself. They used:
the reflexive property
- If 10 = y, then y = 10.
They only switched sides; they didn't actually change anything. All they did was swap across the "equals" sign. They used:
the symmetric property
- x + 0 = x
They didn't change anything, so this is an identity. They added, so they used:
the additive identity
- 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, they didn't say that 2(a + b) equalled something, which then equalled 3c. Instead, what they did was substitute the 9 from the second equation for the a + b in the first equation, so they used:
the substitution property
- 2 = x, so 2 + 5 = x + 5
Usually I'd do something to both sides of an equation in order to get the variable by itself. Here, they made the equation more complicated by adding to both sides, so now the variable is not by itself. But the point here is that all they did was add the same thing to both sides. That is, they used:
the additive property of equality
- If x + 2 = 10, then x + 2 + (−2) equals what, and why?
If they'd just given me the first equation and asked me to solve it, I'd have subtracted 2 from both sides, and simplified to get x = 8. Here, they've made things complicated. But all they've really done is show only the left-hand side of the subtraction (or, to be technical, the addition of the "minus" number), leaving me to figure out that they're wanting me to picture the right-hand side and given them their answer.
Since all they really did was add 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. This is:
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 . In other words, they changed the equation by doing the same multiplying to both sides. They used:
the multiplicative property of equality
- If x is not equal to y and is 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 (namely, to be greater than, to be equal to, or to be less than), and they've eliminated two of these options. Then:
x > y, by the trichotomy law.
- x + (−x) = 0
They added, and they ended up with zero. Zero is the thing which, when added to other things, doesn't change anything. So zero is the additive identity. Since they ended up with the additive identity, then they used:
the additive inverse
They converted to a common denominator by multiplying both fractions by a useful form of 1; remember that and are just useful forms of 1. By multiplying by 1, they didn't actually change the numbers; they only changed how the numbers are being stated. Since they didn't really change anything, this is an identity. Since they multiplied by something that didn't actually change the value, they used:
the multiplicative identity
Yes; any time you convert to a common denominator, you are multiplying by a useful form of 1, so you are applying the multiplicative identity — just in a lumpy form.
- If 5x = 0, what is x, and why?
I can do this in either of two ways: multiply both sides by (the multiplicative property of equality) and then get x = 0, or I could say that, since 5 doesn't equal zero, then x must equal zero (by the zero-product property).
They multiplied, and they ended up with one. The number 1 is the thing that, when multiplied on another number, doesn't change anything, so it's the multiplicative identity. Since they ended up with the multiplicative identity, then they used:
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. But what they did here was "cut out the middleman" by removing the y in the middle; they said that 3x + 2 = y = 8, and then deleted the y in the middle, transiting right over it to create a new equation. In other words, they used:
the transitive property
- If −x = 14, what does x equal, and why?
To solve the equation, I would multiply both sides by the same value (in this case, a negative one), to cancel off the minus sign from the variable. So:
x = −14, by the multiplicative property of equality
- If x = 3 and y = −4, then what does xy equal, and why?
By substitution (that is, by plugging the numbers in for the variables), I 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.
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