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Operation - Identity - Associative - Commutative - Distributive - Closure - Inverse - Equality

An operation works to change numbers. (The word operate comes from Latin operari, "to work.") There are six operations in arithmetic that "work on" numbers: addition, subtraction, multiplication, division, raising to powers, and taking roots.

A binary operation requires two numbers. Addition is a binary operation, because "5 +" doesn't mean anything by itself. Multiplication is another binary operation.

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Identity

An identity is a special kind of number. When you use an operation to combine an identity with another number, that number stays the same. Zero is called the additive identity, because adding zero to a number will not change it: the number stays the same.

    0 + a = a = a + 0.

Since any number multiplied by one remains constant, the multiplicative identity is 1.

    1 * a = a = a * 1.

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Associative Property

An operation is associative if you can group numbers in any way without changing the answer. It doesn't matter how you combine them, the answer will always be the same. Addition and multiplication are both associative.

    Here are some addition examples:

    1 + (2 + 3) = (1 + 2) + 3
    1 + (5) = (3) + 3
    6 = 6.
    (-1 + 66) + 14 = -1 + (66 + 14)
    (65) + 14 = -1 + (80)
    79 = 79.

    More generally,

      a + (b + c) = (a + b) + c.

    Here is a multiplication example:

    2 * (4 * 3) = (2 * 4) * 3
    2 * (12) = (8) * 3
    24 = 24.

    In general terms, that's

      a * (b * c) = (a * b) * c.

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Commutative Property

An operation is commutative if you can change the order of the numbers involved without changing the result. Addition and multiplication are both commutative. Subtraction is not commutative: 2 - 1 is not equal to 1 - 2.

    Here are some examples of the commutative properties of addition and multiplication:

    88 + 65 = 65 + 88
    153 = 153.

    12 * 13 = 13 * 12
    156 = 156.

    More generally,

      a + b = b + a, and
      a * b = b * a.

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Distributive Property

When you distribute something, you give pieces of it to many different people. One example of distributing objects is handing out papers in class. In math, people usually talk about the distributive property of one operation over another.

The most common distributive property is the distribution of multiplication over addition. It says that when a number is multiplied by the sum of two other numbers, the first number can be handed out or distributed to both of those two numbers and multiplied by each of them separately. Here's the distributive property in symbols:

    a * (b + c) = a * b + a * c.

Here's an example:

    5 * (2 + 8) = 5 * 2 + 5 * 8
    5 * (10) = 10 + 40
    50 = 50.

Not all operations are distributive. For instance, you cannot distribute division over addition. Let's try an example:

    14 / (5 + 2)
    = 14 / (7)
    = 2,

    but

    14/5 + 14/2
    = 2.8 + 7
    = 9.8.

    Clearly, 2 is not equal to 9.8.

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Property of Closure

If we take two real numbers and multiply them together, we get another real number. (The real numbers are all the rational numbers and all the irrational numbers.) Because this is always true, we say that the real numbers are "closed under the operation of multiplication": there is no way to escape the set. When you combine any two elements of the set, the result is also included in the set.

Real numbers are also closed under addition and subtraction. They are not closed under the square root operation, because the square root of -1 is not a real number.

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Inverse

The inverse of something is that thing turned inside out or upside down. The inverse of an operation undoes the operation: division undoes multiplication.

A number's additive inverse is another number that you can add to the original number to get the additive identity. For example, the additive inverse of 67 is -67, because 67 + -67 = 0, the additive identity.

Similarly, if the product of two numbers is the multiplicative identity, the numbers are multiplicative inverses. Since 6 * 1/6 = 1 (the multiplicative identity), the multiplicative inverse of 6 is 1/6.

Zero does not have a multiplicative inverse, since no matter what you multiply it by, the answer is always 0, not 1.

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Equality

The equals sign in an equation is like a scale: both sides, left and right, must be the same in order for the scale to stay in balance and the equation to be true.

The addition property of equality says that if a = b, then a + c = b + c: if you add the same number to (or subtract the same number from) both sides of an equation, the equation continues to be true.

The multiplication property of equality says that if a = b, then a * c = b * c: if you multiply (or divide) by the same number on both sides of an equation, the equation continues to be true.

The reflexive property of equality just says that a = a: anything is congruent to itself: the equals sign is like a mirror, and the image it "reflects" is the same as the original.

The symmetric property of equality says that if a = b, then b = a.

The transitive property of equality says that if a = b and b = c, then a = c.

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- Ursula Whitcher, for the Math Forum

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