Complex Numbers + Properties

What’s the hierarchy of important number sets (smallest to biggest)?

Naturals ⊂ Integers ⊂ Rationals ⊂ Reals ⊂ Complex

What are natural numbers?

Counting numbers: 0, 1, 2, 3, …

What are integers?

Whole numbers + negatives: … -3, -2, -1, 0, 1, 2, 3,

What are rationals?

Numbers you can write as a fraction

What are irrationals?

Real numbers you can’t write as a fraction. They go on forever, no repeats, Ex: √2, π

What are reals?

All rationals + all irrationals. Basically everything on the number line

What are complex numbers?

Numbers of the form a + bi, where a and b are reals and i = √(-1).
Ex: 3 + 2i, -5i, 7.

What does closure mean?

If you do an operation (like +, -, ×) on numbers in a set, you stay in the set.

What’s the commutative property?

Order doesn’t matter; you can change the order and the answer will not change: a + b = b + a, a × b = b × a.
⚠ Doesn’t apply to subtraction or division.

What’s the associative property?

Grouping doesn’t matter: (a + b) + c = a + (b + c).
⚠ Doesn’t apply to subtraction or division.

What’s the distributive property?

a(b + c) = ab + ac. Always true for real and complex numbers.

Which sets are closed under multiplication?

Naturals, integers, rationals, reals, and complex. All good.

Which sets are not closed under subtraction?

Naturals (5 – 8 = -3, which isn’t natural).

Which sets are not closed under division?

Naturals, integers, and rationals (division by 0 breaks closure).

Complex numbers closure

Closed under +, -, ×, ÷ (except dividing by 0).