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.

The 4 basic mathematical operations

The 4 basic mathematical operations are:

1) Addition
2) Subtraction
3) Multiplication
4) Division

Addition:
In Addition we add up numbers using the symbol (+), which is called the plus sign.

For example, 3 + 4 = 7 which is read as "three plus four equals seven."

The whole statement of the mathematical operation is called an EQUATION. We will encounter various kinds of equations in our subsequent math works.

When we add two numbers together, those two numbers are called addends,and the resulting number is called the SUM. In our example, the addends are 3 and 4, and the sum is 7.

Subtraction:
In mathematics, Subtraction is the process of deducting one number or quantity from another, and we use the minus symbol (-). We find this symbol in our computer keyboard as the hyphen below the underscore key.

As an example, the equation 6 – 2 = 4 is read as "six minus two equals four".

In subtraction, the first number is called the SUBTRAHEND, and the second number is called the MINUEND. The resulting number in a subtraction operation is called DIFFERENCE. It is obvious in our example that 6 is the subtrahend and 2 is the minuend with the resulting number 4 as the difference.

Multiplication:
By definition to multiply is to increase by a considerable number, amount, or make a certain number increase by several times as desired. Thus, Multiplication is often described as a shortcut process for repeated addition.

For example,
5 × 4 means add 5 to itself 4 times: 5 + 5 + 5 + 5 = 20
8 × 7 means add 8 to itself 7 times: 8 + 8 + 8 + 8 + 8 + 8 + 8 = 56
100 × 3 means add 100 to itself 3 times: 100 + 100 + 100 = 300

In Multiplication, we use the times sign (×) or the dot (.) symbol to signify the multiplication operation.
We can use also the pair of parenthesis ( ) symbols. Let's take note of the following examples:

6 x 7 means  6·7 also means 6 (7) or  (6)7  or  (6)(7)

8 x 10 means  8·10 also means 8 (10) or  (8)10  or  (8)(10)

60 x 25 =  60·25 = 60 (25) = (60)25  or  (60)(25)

The first number in a multiplication operation is called MULTIPLICAND and the second is MULTIPLIER. Both multiplicand and multiplier are referred to as the FACTORS. When the factors are multiplied, the resulting number we get is called the PRODUCT.

Division
In the real sense, Division is just the opposite of the Multiplication operation. For in Multiplication, we add up or increase the number itself to several times, whereas, in the Division process we split things or break down a number into several parts.

There a number of symbols used in the Division operation, they are:

When you divide one number by another, the first number is called the DIVIDEND, the second is called the DIVISOR. We call the result a QUOTIENT. Let's consider the following examples:

In our next post, we shall talk about the PROPERTIES of the 4 mathematical operations we just discussed.

Let's start by reviewing Basic Math concepts

First, Let's have a review of Basic Math or the Pre-Algebra Math. Let's have a solid foundational understanding of the basics then move on until we’re ready to enter the algebra subjects. We'll discuss the Hindu-Arabic number system. The Hindu-Arabic number contains ten digits:

1   2   3   4   5   6   7   8   9   0

The ten digits in Hindu-Arabic number system (which we use today) allow us to count from 0 to 9. All higher numbers are produced using place value.

Let' have an example to understand how a whole number gets its place value. Suppose we write the number 90,514 all the way to the right in BM Chart 001 below, one digit per cell, and we'll add up the numbers.

We have 9 ten thousands, 0 thousands, 5 hundreds, 1 ten, and 4 ones. The chart shows you that the number breaks down as follows:

90,514 = 90,000 + 0 + 500 + 10 + 4

Please note the presence of the digit 0 in the thousands place which means that zero thousands are added to the total sum of 90,514.

We will review the four (4) mathematical operations in the next post.