"How Numbers Behave When Added or Multiplied"

 

      How many basic operations are there in arithmetic? To cut the long story short, there are four : addition, subtraction, multiplication, and division.

     Have you realized that numbers give same results when combined by two of these operations? Okay, do a try and see. The results you just got, form laws or properties over the operations used. 

     Do you agree that adding 3 white balls to 4 white ball will always give you the same result irrespective of how the digits are placed in the sum? What I mean is writing the sum 3+4 or 4+3 gives the same result 7.

     Furthermore, if you have two groups of the balls above, you can write them as 3+4, 3+4 and 4+3, 4+3. To economize space, you can introduce brackets and each of the pairs of numbers written as 2(3+4) and 2(4+3).

     Suppose a group of seven white balls are included to the ones above. We now get a new sum written as 3+4+7. You can group the numbers here into pairs as (3+4) +7 or 3+ (4+7)

 

The behavior of the sums in giving the same results irrespective of the position of these digits will give you to what is called laws or properties of numbers. These properties will ease your work in algebra especially where expansions are involved.

    Remember wherever you find calculations involving brackets, the numbers in the brackets are evaluated first before moving to those outside. That is, (3+4) +7 = (7) +7, evaluating the digits in the brackets first. Similarly, 3+ (4+7) = 3+ (11).

    The approach to building the properties of numbers will also ease your work in algebra especially in factorization and expansion of algebraic expressions.

There are 4 basic properties of numbers that you are going to learn here. Bur before I continue, I will like to know what is meant by properties of numbers.

  Properties of numbers are the common behavior the share when added or multiplied here. They do not share this behavior when subtracted or divided.

Now, you will learn how to use ADDITION and MULTIPLICATION to derive these properties.

 

A)     USING ADDITON TO DERIVE THE FOUR LAWS OR PROPERTIES OF NUMBERS

i)                    From above, you saw that 3+4, and 4+3 will give the same result irrespective of their positions, right!  This means 3+4 = 4+3=7. 

Furthermore, -3+4 = 4+ -3 =1. This means adding any of these pairs separately will give you the same result. That is 3+4=7, 4+3=7, and -3+4 =1, 4+ -3 =1.  When numbers behave like this when added, they are said to have the commutative property or respect the commutative law of addition.  As a reminder, the commutative law is also known as the commutative property. This law or property applies to addition of integers.

 It is not hold in the subtraction

Conclusion: Let a and b be two whole numbers. The sums a+ b and b+ a are said to be commutative if a + b= b + a.  But if a + b = b+a commutative law does not hold.

 

ii)                   From above, brackets were introduced to summarize the pairs 3+4, 3+4, and  4+3, 4+3 as 2(3+4) and 2(4+3) respectively. Instead of brackets, we can use braces [  ] or curly brackets {  } .

To remove the brackets, the digit outside multiplies the ones inside separately as follows;

2(3+4) = 2x3 + 2x4

2(4+3) = 2x4 + 2x3

  The multiplication sign replaces the brackets.

In removing this bracket leads us to another property called the Distributive Law or the Distributive Property. This will help you in the expansion of algebraic expressions.

Conclusion: Let m, y, z be three whole numbers, then, m(y + z) = m  x y + m x  z.

iii)                 In the statements (3+4)+7 and  3+(4+7) are grouped .As above, if evaluated, the same results will be obtained. But remember that you first solve what you find in the brackets before what is outside.

 For example: (3+4) +7 = (7) +7

                         (3+4) +7 = 7+7

                         (3+4) + 7 = 14 (left hand side)

Similarly,   3+ (4 + 7) = 3 + (11)

                   3+ (4 +7) = 3+ 11

                   3 + (4 +7) = 14 (Right hand side)

Adding numbers the way we have done above gives another property of numbers called the Associative law of addition or Associative property of addition.

Like above you can also use braces [   ], or curly brackets {  } to write these numbers. You will have something like [3+4] +7,   {3+4} + 7 or 3+ [4+7], 3+ {4+ 7}. Follow the same procedure for adding these numbers as above.

Conclusion. Let a, b, and c be any three numbers.

 If (b+a) +c = a+ (b+a) represents the associative law or associative property of addition.

 

 

iv)                 When zero is added to any number, the result id the number added to zero. For example, 5+0 =5, -7+0 =-7, 2+0=2 and so on.  Because of this behavior, the number, 0, is called the Additive Identity.

 

From above, you it is clear that you can identify the four laws or properties of numbers under addition.

  The laws are: Commutative Law, Distributive Law, Associative Law, and Additive identity.  None of these laws is true under subtraction.

   For example, 3 – 4 = -1, and 4-3 = 1.  3 – 4 = 4 – 3.  Meaning the numbers are not commutative under subtraction. The numbers are not commutative under subtraction .The commutative law does not hold. Note that the law is derived from the operation being used.

 

A)     USING MULTIPLICATIOM TO DERIVE THE FOUR LAWS OR PROPERTIES OF NUMBERS

It will just be the same thing as above. The only difference will be in the operation which will change from addition (+) to multiplication (x). Rushing through this will be as follows:

 

i)                    The Commutative Law or Property under Multiplication

3x4=12, and 4x3=12.

Then 3x4 = 4x3

Conclusion. If a and b are any pair of whole numbers where a x b = b x a then the numbers are said to be commutative under multiplication. So the commutative law holds.

 

ii)                  The Distributive Law or Property under Multiplication

From addition above change the operation; 2(3x4) = 2x3 x 2x4, brackets removed.

Conclusion: If a, b, and c are three whole numbers where a (b x c) = a x b x a x c . The numbers a, b, and c are distributive over multiplication.

 

 

iii)                The Associative Law or Property under Multiplication.

From addition above, change the operation; (3 x 4) x 7 = 3 x (4 x 7)

Conclusion:  If a, b, and c are whole numbers, then

 (a x b) x c = a x (b x c). The numbers are said to be associative over multiplication. This called the associative law or associative property.

 

iv)                The multiplicative Identity. When any number is multiplied by 1, the result  remain the same. That is to say 4 x 1 =4, 6 x 1 =6, -9 x 1=-9. Because of this behavior, 1 is called the multiplicative identity.

                  You will discover that the laws or properties above do not work under division. You can choose a group of any three numbers and test to show that these laws do not hold in division.

 

        The four main properties or laws of whole numbers are the commutative property, the distributive property, the associative property, and the identity property. Each of these properties enable us to see how numbers behave when combined by ADDITION or MULTIPLICATION.

This  article was written by Awah Aweh who is an expert in writing articles for the internet at here See sample of my blog posts above link. I write marketing content. This involves articles, blog posts, sales letters, email campaigns, ads and white papers for owners of small and mid-size businesses in the self-help, education, and finance industries. Read samples of my blog posts on personal finance at here.    I write e-books on self-improvement at self-help e-book store . I also make designs that can be printed on your products to boost sales  on promotion here

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