Numbers And Negative Numbers

Number Sets

Number sets you want to know about:

• Real numbers (R)

  • Irrational numbers (R-Q): all numbers that can't be represented by a fraction (e.g. pi)

  • Rational numbers (Q): all numbers that can be represented by a fraction (e.g. 7/2)

    • Integers (Z): all numbers of the set { ..., -2, -1, 0, 1, 2, 3, ... }

      • Whole numbers (W): from 0 and up { 0, 1, 2, 3, ... }

        • Natural numbers (N): From 1 and up { 1, 2, 3, ... }

Each nested number set is a subset of the previous!

Note: The curly braces are part of set notation: the syntax for representing a set.

Identity Numbers

Identity numbers are numbers for operations that don't change the initial value.

0 is the identity number of addition: 4 + 0 = 4.

1 is the identity number of multiplication: 4 * 1 = 4.

Opposite of A Number

The opposite of a number is its distance away from 0. Think of a number line to understand this:

So, the opposite of 4 is -4.

Bonus: 0 is the opposite of itself because it is 0 units away from 0!

Pro tip: Think of addition and subtraction as indicators of direction:

• Subtracting a number is saying to go in the opposite direction of the number being subtracted. So 7 - 5 means we start at 7 on the number line, and then we move 5 units in the opposite direction: 5 to the left.

• Conversely, addition is about going the same direction of the being number added. So 7 + -5 means go left.

Adding and Subtracting Signed Numbers

Signed numbers are just numbers with a positive or negative sign. There are 8 combinations when adding and subtracting them:

• Positive + Positive = Positive

• Negative + Negative = Negative

• Positive + Negative = Positive or Negative depending on which has higher absolute value

  • The sign is with the higher absolute value

• Negative + Positive = Positive or Negative depending on which has higher absolute value

  • The sign is with the higher absolute value

• Positive - Positive = Positive if first number higher, Negative if second number higher

• Negative - Negative = Positive if first number has lower absolute value, Negative if second number has lower absolute value

• Positive - Negative = Positive

• Negative - Positive = Negative

Multiplying Signed Numbers

There are 4 possible combinations:

• Positive * Positive = Positive

• Negative * Negative = Positive

• Positive * Negative = Negative

• Negative * Positive = Negative

When multiplying more than 2 numbers, an even number of negative signs leads to a positive result, and an odd number of negative signs leads to a negative result.

Note: Multiplying anything by 0 will equal 0.

Pro tip: If you're curious how Negative * Negative = Positive makes sense, check out these proofs.

Dividing Signed Numbers

There are also 4 possible combinations:

• Positive / Positive = Positive

• Negative / Negative = Positive

• Positive / Negative = Negative

• Negative / Positive = Negative

When dividing more than 2 numbers, an even number of negative signs leads to a positive result, and an odd number of negative signs leads to a negative result.

On dividing with zero:

• 0 divided by any number equals 0 because division is the inverse of multiplication, and since 0 * 5 = 0, 0 / 5 = 0.

• However, any number divided by 0 is undefined because there is no x where since 5 / 0 = x, x * 0 = 5. x * 0 = 0, not anything else!

Note: For some intuitive explanations of why the above division combinations work the way they do, check out this Stack Overflow question.

Absolute Value

The absolute value of a number is its distance away from 0 in a number line. For example, the distance away from 2 and -2 from 0 is 2.

To denote an absolute value, you wrap the number of mathematical expression in |x|.

|2| = 2
|-2| = 2
|4 - 5| = 1