Understanding the Properties of Numbers

By Yang Kuang, Elleyne Kase

Remembering the properties of numbers is important because you use them consistently in pre-calculus. The properties aren’t often used by name in pre-calculus, but you’re supposed to know when you need to utilize them. The following list presents the properties of numbers:

  • Reflexive property. a = a. For example, 10 = 10.

  • Symmetric property. If a = b, then b = a. For example, if 5 + 3 = 8, then 8 = 5 + 3.

  • Transitive property. If a = b and b = c, then a = c. For example, if 5 + 3 = 8 and

    If 8 = 4 * 2, then 5 + 3 = 4 * 2

  • Commutative property of addition. a + b = b + a. For example, 2 + 3 = 3 + 2.

  • Commutative property of multiplication.

    a * b = b * a, so 2 * 3 = 3 * 2

  • Associative property of addition. (a + b) + c = a + (b + c). For example, (2 + 3) + 4 = 2 + (3 + 4).

  • Associative property of multiplication.

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

  • Additive identity. a + 0 = a. For example, –3 + 0 = –3.

  • Multiplicative identity.

    a times 1 equals a

  • Additive inverse property. a + (–a) = 0. For example, 2 + (–2) = 0.

  • Multiplicative inverse property.

     a * 1/a = 1

  • Distributive property.

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

  • Multiplicative property of zero.

    a multiplied by zero equals zero.

  • Zero-product property.

    If a * b = 0, then a or b is zero.

    For example, if x(x + 2) = 0, then x = 0 or x + 2 = 0.

If you’re trying to perform an operation that isn’t on the previous list, then the operation probably isn’t correct. After all, algebra has been around since 1600 BC, and if a property exists, someone has probably already discovered it. For example, it may look inviting to say that

10(2 + 3) does not equal 23.

but that’s incorrect. The correct answer is

10(2 + 3) = 50

Knowing what you can’t do is just as important as knowing what you can do.