Directed Numbers

Directed Numbers

Directed numbers are numbers which have a direction and a size. Once a direction is chosen as positive (+), the opposite direction is taken as negative (-).

A negative number is a real number that is less than zero. Whilst a positive number is a real number greater than zero. If positive represents a movement to the right, negative represents a movement to the left.

In everyday life, if positive represents height above sea level, then negative represents depth below sea level.

In banking and finance if positive represents a deposit then negative represents a withdrawal.

Negative numbers are also used to describe values on a scale that goes below zero. Negative numbers are usually written with a minus sign in front. For example,−3 degress celsius means 3 degress celcius below zero.

−3 is pronounced "minus three" or "negative three".

Zero is neither positive nor negative.

The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. Lack of a sign in front of number also means its a positive number, e.g. 3 = +3.

Every real number other than zero is either positive or negative and the non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3…), while the positive and negative whole numbers (together with zero) are referred to as integers.

Operations on directed numbers

Directed numbers can be added, subtracted, multiplied or divide as shown in the following examples.

Adding negative numbers

When adding negative numbers, you add their magnitudes and keep the answer negative.

  • (-2)+(-3) = -5
  • (-10)+(-8) = -18
  • (-1)+(-2) = -3

As you can see, if you add two negative numbers the answer is always negative.

Same applies even if you add more than two numbers.

  • (-1)+(-1)+(-1) = -3
  • (-2)+(-2)+(-2)+(-2) = -8

In some cases you might find (-2)+(-3) written as -2+-3 without the brackets. Don’t worry about that, its the same thing.

Two consecutive opposite signs resolve to a negative sign. For example (-7)+(-3) is the same as -7+-3 or (-7)-(+3) or -7-3 because consecutive opposite signs (+ and -) resolve to -.

  • -2-3 = -5
  • -10-8 = -18
  • -1-2 = -3

Subtracting negative numbers

When you subtract two negative numbers, the easiest way is to deal the consecutive negative sign formed. For example, (-2)-(-8) without brackets is -2 8. Notice those consecutive negative signs.

Two consecutive negative signs resolve to a positive sign. This means -2–8 resolves to -2+8.

Examples:

  • (-2)-(-8) = -2+8 which when rearranged becomes 8-2 = 2.
  • (-3)-(-7) = -3+7 = 7-3 = 4
  • (-1)-(-1) = -1+1 = 1-1 = 0

Smaller number subtracting bigger number

A smaller number minus a bigger number = negative number. For example: for 3-8, just do 8-3 and make the answer negative. So 3-8 = -5.

  • 2-23 (Again just do 23-2, get 21 and make it negative) = -21.
  • 5-8 = -3
  • 1-2 = -1

In addition and subtraction, the "Commutative Law" say we can swap numbers over and still get the same answer, i.e. 1+2 = 2+1:

Which means -8+5 = 5-8 = -3.

This simplifies a lot of calculations which involve long chains of directed numbers. For example:

-3+5-6-10+6–5
First resolve consecutive negative signs to get:
= -3+5-6-10+6+5
Then rearrange the numbers and start with positive numbers.
= 6+5+5-10-6-3
Add the positive numbers:
= 16-10-6-3
Add the negative numbers and subtract their total from the first number:
= 16-(10+6+3)
= 16-19
= -3

Multiplying and dividing directed numbers

Multiplying or dividing numbers with the same signs gives a positive answer:

  • 3 × 4 = 12
  • -3 × -4 = 12
  • 10 ÷ 2 = 5
  • -10 ÷ -2 = 5

Multiplying or dividing numbers with the different signs gives a negative answer:

  • 3 × -4 = -12
  • -3 × 4 = -12
  • 10 ÷ -2 = -5
  • -10 ÷ 2 = -5