Number Properties

Number properties and number operations is nothing but we are going to learn about the basic number properties and number operation. We are having the three basic properties and we are having the four basic operations. In this we are going to learn about the number properties and number operations using some example. It is very useful to performing the operations on the numbers easily.

Number Properties:

Basically we are having three basic number properties. We will see all the properties using some examples.

Distributive property:

Distributive property is nothing but giving the values. We can easily remember the distributive property using the following words multiplication distributes over addition.

2 X (3 + 5) = 2 X 3 + 2 X 5

2 X (8)        = 6 + 10 = 16

We are getting the equal value.

Associative property:

Associative property is nothing but grouping the numbers. We can use this for addition and multiplication.

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

2 + 8          = 5 + 5 = 10

This is the associative property of numbers.

Commutative property:

Commutative property of numbers is nothing but the value of numbers won’t change when the place of the numbers change.

2 + 3 = 3 + 2 = 5
Number Operations:

In numbers we can perform four types of operations.  Those operations are addition operation, subtraction operation, multiplication operation and division operation.



Addition operations

Addition operation is nothing but we are adding the value of any two numbers. If we take 5 and 6 if we want to perform addition operation we have to add the quantities of the numbers. The addition operation is denoted by the symbol`+`

So 5 + 6 = 11

Subtraction operation

Subtraction operation is nothing but we are going to subtract the value of numbers. The subtraction operation is denoted by `-`

Example:

9 – 6 = 3

Multiplication operation

Multiplication operation is nothing the repeated addition. The multiplication operation is denoted by `xx`

Example: 5 X 3

5 X 3 = 5 + 5 + 5 (We have to add the value of 5 three times)

= 15

Division operation

Division operation is nothing but the repeated subtraction. The division operation is denoted by `-:`

For example take 15 `-:` 3

We have to subtract the value of 3 from 15 again and again.

So 15 – 3 = 12

12 – 3 = 9

9 – 3 = 6

6 – 3 = 3

3 – 3 = 0

So totally this operation performed in 5 steps we got the remainder as 0.

So the quotient is 5 and the remainder is 0.

Decimal Numbers

The decimal number system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by modern civilizations.

Decimal notation often refers to the base-10 positional notation such as the Hindu-Arabic numeral system; however it can also be used more generally to refer to non-positional systems such as Roman or Chinese numerals which are also based on powers of ten.

(Source – Wikipedia.)
Example Problems for Dividing Decimal Numbers:

Example problems 1 for dividing decimal numbers:

Divide 3.5 by 0.7

Solution:

Step 1: First we begin with 3.5 /0.7

Step 2: and Then we divide without decimal points, we attain

= 35 / 7

= 5

Step 3: and then we put decimal places

3.5 have one decimal place.

0.7 has one decimal place.

Step 4: Final answer has two decimal places 5

Answer 5.

Example problems 2 for dividing decimal numbers:

Divide 4.4 by 4

Solution:

Step 1: First we begin with 4.4 / 4

Step 2: and Then we divide without decimal points, we attain

= 44 / 4

= 11


Step 3: and then we put decimal places

4.4 have one decimal place.

4 have no decimal place.

Step 4: Final answer has one decimal places 1.1

Answer 1.1.

Example problems 3 for dividing decimal numbers:

Divide 1.2 by 0.2

Solution:

Step 1: First we begin with 1.2 / 0.2

Step 2: and Then we divide without decimal points, we attain

= 12 / 2

= 6

Step 3: and then we put decimal places

1.2 have one decimal place.

0.2 have one decimal place.

Step 4: Final answer has one decimal places 6.

Answer 2.

Example problems 4 for dividing decimal numbers:

Divide 6.4 by 8

Solution:

Step 1: First we begin with 6.4 / 8

Step 2: and Then we divide without decimal points, we attain

= 64 / 8

= 8

Step 3: and then we put decimal places

6.4 have one decimal place.

8 have no decimal place.

Step 4: Final answer has one decimal places 8

Answer 0.8.
Practice Problems for Dividing Decimal Numbers:

Practice problem 1 for dividing decimal numbers:

1) Divide 3.6 by 4

Ans: 0.9

Practice problem 2 for dividing decimal numbers:

2) Divide 7.2 by 0.8

Ans: 9

Practice problem 3 for dividing decimal numbers:

3) Divide 6.2 by 2

Ans: 3.1

Practice problem 4 for dividing decimal numbers:

4) Divide 42 by 0.7

Ans: 60

Looking out for more help on physics answers in algebra by visiting listed websites.

Practice problem 5 for dividing decimal numbers:

5) Divide 72 by 0.8

Ans: 90.

Factoring Polynomials

Polynomials:

A polynomial is called an algebraic expression having the power of each variable is a only positive integral numbers.

For example,  8x2 -  16x3y2 + 24 y4 + 10, 9y2 - 16, y2 + 20x + 100.

These examples are a polynomial expression in two variables x and y and polynomial expression in one variable y.

Let us discuss about factoring polynomial .
Factoring Polynomials:

Factoring Polynomials:

Factoring a polynomial is the reverse process of multiplying polynomials.

When we factor a real number, First we are getting for prime factors for real number which multiply together to give the same real number.

For example,   28 = 7 x 2 x 2

When we are going to factor a polynomial,  First we need  to look for simpler polynomial expressions which are multiplied each other and give the same polynomial expression what we started with.

For example,  y5x2 + 25xy = 5xy(x + 5)

Let us solve sample problems on factoring polynomials.
Sample Problems: Factoring Polynomials

Steps to factoring Polynomials:

Step 1: First we factor the polynomial

Step 2: Expand the factorized polynomial using algebraic identities formula.

Step 3: Solve it if possible.

Problem 1:

Factor the polynomial expression: 25m2 – 64

Solution:

The given polynomial expression is 25m2 – 64.

Step 1: Factoring it,

= 52m2 – 82

= (5m)2 - 82

Step 2: Expand it

This is in the form of a2 – b2 = (a + b) (a – b)

So,

= (5x + 8)(5x – 8)

Therefore, the factors are (5x + 8)(5x – 8).

Step 3: Solving it,

5x + 8 = 0

5x = - 8

x = - `8/5`

5x - 8 = 0

5x = 8

x = `8/5`

Therefore, Solutions are x = `8/5` , `- 8/5` .

Problem 2:

Factor the polynomial expression: 9p2 – 81

Solution:

The given polynomial expression is 9p2 – 81.

Step 1: Factoring it,

= 32p2 – 92

= (3p)2 - 92

Step 2: Expand it

This is in the form of a2 – b2 = (a + b) (a – b)

So,

= (3p + 9)(3p – 9)

Therefore, the factors are (3p + 9)(3p – 9).

My forthcoming post is on algebraic expression simplifier, simplify the algebraic expression will give you more understanding about Algebra.

Step 3: Solving it,

3p + 9 = 0

3p = - 9

p = -` 9/3` = - 3

3p - 9 = 0

3p = 9

p = `9/3` = 3

Therefore, Solutions are p = 3 , - 3.
Problem 3:

Factor the polynomial expression: A2 – `1/49`

Solution:

The given polynomial expression is A2 – `1/49` .

Step 1: Factoring it,

= A2 – `1/7^2`

= A2 - `(1/7)^2`

Step 2 : Expand it,

This is in the form of a2 – b2 = (a + b) (a – b)

So,

= (A + `1/7` )(A – `1/7` )

Therefore, the factors are (A + `1/7` )(A – `1/7` ).

Step 3 : Solving it,

(A + `1/7` )(A - `1/7` ) = 0

A + `1/7` = 0

A = - `1/7`

X - `1/7` = 0

X = `1/7`

Therefore, Solutions are A = `1/7` , - `1/7` .

Problem 4:

Factor the polynomial expression: y2 + 2y - 15

Solution:

The given polynomial expression is y2 + 2y  - 15.

Step 1 : Factoring it,

y2 + 2y - 15

Step 2 : Expand the polynomial expression

= y2 - 3y + 5y - 15

= y(y - 3) + 5(y - 3)

Taking out common factor,

= (y - 3)(y + 5)

Therefore, the factors are (y + 5)(y – 3).

Step 3 : Solving it,

(y + 5)(y - 3) = 0

y + 5 = 0

y = - 5

y - 3 = 0

y = 3

Therefore, Solutions are y = 3 , - 5.

Problem 5:

Solve for x in the perfect square trinomial: 9x2 + 24x + 16 = 0

Solution:

The given perfect square trinomial is 9x2 + 24x + 16 =0

Step 1: First factoring this perfect square trinomial

(3X)2 + 2·3.4x + 42 = 0

(3x + 4)2 = 0

Step 2: Expand it

This is in the form (ax + b)2 = (ax + b) (ax + b)

(3x + 4)(3x + 4) = 0

Step 3: Solving x

3X + 4 = 0

3X = - 4

Divide by 3 each side.

X = `- 4/3` .