Understanding negative numbers

We can easily understand the concept of negative numbers through the Number Line. Let's take a look at SMP Figure 001 below:

The point "0" in the number line is referred to as the origin. Every point to the right of 0 is associated with a positive number and every point to the left of 0 with a negative number. This convention indicates that every number is associated with a directed distance from the origin. The negative numbers are preceded by the negative (-) sign and the positive numbers may or may not be preceded by the (+) sign.

Mathematical operation with negative numbers:

As an exercise problem let's solve -2 + 5 by using the number line in SMP Figure 002 below:

Solution: Starting from the origin 0, we can count  2 units to the left (representing the negative 2 value); then   from point -2 we move right 5 units (noting 5 as positive). This simple method helps us to quickly arrive at the answer which is +3 units.

What are the so called properties of the fundamental mathematical operations?

In our last post, we discussed the basic symbols +, - , x, and 4 that are used for the fundamental mathematical operations "add," "subtract," "multiply," and "divide," respectively. Today we'll look into the properties of these basic operations and also touch on inverse functions.

Commutative Property of Addition and Multiplication

Commutative Property of Addition:
The order in which the number are added does not change the sum.

4 + 7 = 7 + 4

This is applicable to any number a and b; that is, a + b = b + a.

Commutative means that we can interchange the order of the numbers without affecting the result.

On the other hand, Subtraction and Division are NONCOMMUTATIVE operations.

Obviously, 5 - 3 = 2 is not the same as 3 -5 = -2.

Also, 8/2 = 4 but 2/8 = 0.25.

Commutative Property of Multiplication:
The order in which numbers are multiplied does not change the product.

6 x 5 = 5 x 6. This is applicable to any number a and b; that is, a x b = b x a.


Associative Property of Addition and Multiplication

Associative Property of Addition:

The way in which addends are grouped does not change the sum.

(3 + 4) + 5 = 2 + (4+5)

This is applicable to any number a, b and c ; that is, (a + b) + c  = a + (b+ c).

Associative Property of Multiplication:

The way in which factors are grouped does not change the product.

(3 x 4)5 = 3(4x5).

This is applicable to any number a, b and c ; that is, (a x b)c  = a(b x c).

Let's recall that a pair of parentheses is a symbol for multiplication operation.


Distributive Property of Addition and Multiplication

Another important property of the mathematical operations relates addition and multiplication into a property called DISTRIBUTED LAW. It can be illustrated by the following equations:


5(6 + 7) = 5x6 + 5x7, applicable to any number a, b and c, that is,  a(b + c) = ab + ac.
       
(5 + 6)7 = 5x7 + 6x7, applicable to any number a, b and c, that is, (b + c)a = ba + ca.


We shall discuss the Inverse Function of Addition and Subtraction (along with the Inverse Function of Multiplication and Division) after reviewing and understanding exponential numbers and also negative numbers in the next post.