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The word 'Simplification' is a noun. Simplifying makes a algebraic expressions easily understandable and solvable.

What is Simplification?

According to Cambridge Dictionary, The process of making something less complicated and therefore easier to do or understand, or the thing that results from this process:
  • The organization advises on the simplification of trade procedures.
  • This is a gross simplification of what actually happened.

Mathematical Concept:
To reduce a fraction to its lowest terms by canceling to the lowest common factor for both numerator and denominator or to condense an algebraic expressions by grouping and combining similar terms. Simplifying makes a algebraic expressions easily understandable and solvable. (from: www.splashlearn.com)

Learn how to simplify with BODMAS rule:

'BODMAS', We've named the simplification process easy to remember. By this rule (BODMAS), we can simplify mathematical expressions easily.

BODMAS Stands for letters:
B ↠ Bracket : ( ) { } [ ]

O ↠ Of or Orders

D ↠ Division

M ↠ Multiplication

A ↠ Addition

S ↠ Subtraction

According to BODMAS rule, we need to follow the following steps corresponding:
  • At first, We need to solve  brackets corresponding:  (), {}, []
  • Next: Of or Order
  • Next: Division or Multiplication
  • Finally: Addition  or Subtraction
Learn about: A Model

To know about BODMAS, follow the following photo attentively:

Rule: BODMAS

By following a few algebraic expressions, we can understand the BODMAS rule clearly. Please try to understand the below examples:

01. Example (Bracket related algebraic expressions):
[ 4 + { 3 + ( 7 + 2 ) - 5 } ]
= [ 4 + { 3 + 9 } - 5 ] (by completing First backer, according to BODMAS rule)
= [ 4 + 12 - 5 ] (by completing Second backer, according to BODMAS rule)
= [ 16 -5 ] ( Addition or Subtraction, as you with)
= 11 (Results)

02. Example:
20  ÷4 - 1 + 10✕2 - 8
= 5 - 1 + 20 - 8  (by completing Division & Multiplication)
= 5 + 20 - 1 - 8 (Rearranged: Positive and Negative values)
=  25 - 9 (by completing Addition)
= 16 (Results)

03. Example:
20 - [ { 10 + ( 12 - 6 ) - 4 } ]
= 20 - [  { 10 + 6 - 4 } ]
= 20 - [ { 16 -4 } ]
=  20 - [ 12]
= 20- 12
= 8

We have tried to show you basics form of simplification algebraic expressions, We believe that by following BODMAS rule, learner will be solved complex algebraic expressions easily. If you have a great idea, let's share to comments.

Thanks in Advance.
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Sets

A set is the representation of a collection of objects; distinct objects with one or more common properties. Grouping up the objects in a set is an act of distinguishing those objects from the members of another set. We can use the terms – elements or members of a set instead of the term objects.


How to denote Sets?

If ‘A’ is a set and ‘a’ one of its elements then: ‘a ∈ A’ denotes that element ‘a’ belongs to ‘A’ whereas, ‘a ∉ A’ denotes that ‘a’ is not an element of A. Alternatively, we can say that ‘A’ contains ‘a’. A set is usually represented by capital letters and an element of the set by the small letter.

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Cardinal Numbers

Cardinal Numbers
0
 Zero
1
 One
2
 Two
3
 Three
4
 Four
5
 Five
6
 Six
7
 Seven
8
 Eight
9
 Nine
10
 Ten
11
 Eleven
12
 Twelve
13
 Thirteen
14
 Fourteen
15
 Fifteen
16
 Sixteen
17
 Seventeen
18
 Eighteen
19
 Nineteen
20
 Twenty
21
 Twenty One
22
 Twenty Two
... .. .
... .. .
30
 Thirty
40
 Fourty
50
 Fifty
60
 Sixty
70
 Seventy
80
 Eighty
90
 Ninety
100
 One Hundred
1000
 One Thousand
10,000
 Ten Thousand
100,000
 One Hundred Thousand
1,000,000
 One Million
1,000,000,000
 One Billion



Ordinal Numbers:

Ordinal Numbers
First
1st
Second
2nd
Third
3rd
Forth
4th
Fifth
5th
Sixth
6th
Seventh
7th
Eighth
8th
Ninth
9th
Tenth
10th
Eleventh
11th
Twelfth
12th
... .. .
... .. .


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Proposition: √2 is an irrational number.

Proof:
We Know, 1 < 2 < 4
∴ √1 < √2 < √4
or, 1 < √2 < 2

Quick navigation: Geometric Figure

Here, 12 = 1, (√2)2 = 2, 22 = 4, (√2)2 = 2
∴ Therefore, the value of √2 is greater than 1 and less than 2.
∴ √2 is not an integer.

∴ √2 is either a rational number or a irrational number. If √2 is a rational number
Let, √2 =p/q; where p and q are natural numbers and co-prime to each other and q > 1
or, 2 = p2/q2; squaring
or, 2 = 2p2/q; multiplying both sides by q.

Clearly 2q is an integer but p2/q is not an integer because p and q are co-prime natural numbers and q > 1
Clearly 2q is an integer but p2/q2 is not an integer because p and q are co-prime natural numbers and q > 1

∴ 2q and p2/q2 cannot be equal, i.e., 2q≠ p2/q
∴ Value of √2 cannot be equal to any number with the form p/q , i.e., √2≠ p/q

∴ √2 is an irrational number. (Proved)
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Basic Rules of Logarithm

» loga (1) = 0
Because, a 0 = 1 and given that a>0

» logaa = 1
Because, a 1 = a and given that a>0

» logaa (x) = n => an = x

» loga (xn) = n loga (x)

» loga (x.y) = loga (x) + loga (y)

» loga (x/y) = loga (x) - loga (y)

» loga (x) = logb (x) / logb (a)

» logaa = logab . logba

Note:
» loga (x+y) ≠ loga (x) + loga (y)

» loga (x-y) ≠ loga (x) - loga (y)

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Basic Properties of Numbers in Mathematics

There are four following basic properties of numbers in Mathematics:
          1. Commutative
          2. Associative
          3. Distributive
          4. Identity


It is especially important to understand these properties once we reach advanced mathematics such as Algebra & Calculus. Since, we should be familiar with each of these.

1. Commutative Property:

a. Addition: When two numbers are added, the sum is the same regardless of the order in which the numbers are added.
          p+q=c or, q+p=c
For example:
          3+4=7 or, 4+3=7

b. Multiplication: When two numbers are multiplied together, the product is the same regardless of the order in which the numbers are multiplied.
          p x q = c or, q x p = c
For example:
          3 x 4 = 12 or, 4 x 3 = 12

2. Associative Property:

a. Addition: When three or more numbers are added, the sum is the same regardless of the way in which the numbers are grouped.
          p + (q + r) = c or, (p + q) + r = c
For example:
          5 + (7 + 2) = 14 or, (5 + 7) + 2 = 14

b. Multiplication: When three or more numbers are multiplied, the product is the same regardless of the way in which the numbers are grouped.
          p x (q x r) = c or, (p x q) x r = c
For example:
          5 x (7 x 2) = 70 or, (5 x 7) x 2 = 70

3.Distributive Property:

The sum of two numbers times a third number is equal to the sum of each addend times the third number.
          p x (q + r) = c or, p x q + p x r = c
For example:
          3 x (5 + 6) = 33 or, 3 x 5 + 3 x 6 = 33

4. Identity Property:

a. Addition: The sum of any number and zero is that number.
          p+0=p or, 0+p=p
For example:
          3 + 0 = 3 or, 0+3=3

b. Multiplication: The product of any number and one is that number.
          p x 1 = p or, 1 x p = p
For example:
          6 x 1 = 6 or, 1 x 6 = 6


You can also practice Algebra Equations.
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It is an exclusive e-Learning Blog that has been dedicated to help keen learn students to boost their knowledge in different subjects.
There are a lot of equations in Algebra. We must practice the following Algebra Equations:

Algebra Equations

» (a+b+c)2=a2+b2+c2+2ab+2bc+2ca
» a2-b2=(a+b)(a-b)
» (a+b)2=(a-b)2+4ab
» (a-b)2=(a+b)2-4ab
» ab={(a+b)2-(a-b)2}/4
» 2(a2+b2)=(a+b)2+(a-b)2
» (a2+b2)={(a+b)2+(a-b)2}/2
» (a+b)3=a3+3a2b+3ab2+b3
» (a-b)3=a3-3a2b+3ab2-b3
» (a+b)3=a3+b3+3ab(a+b)
» (a-b)3=a3-b3-3ab(a-b)
» a3+b3=(a+b)(a2-ab+b2)
» a3-b3=(a-b)(a2+ab-b2)
» a3+b3=(a-b)3+3ab(a-b)
» a3-b3=(a+b)3-3ab(a+b)
» (a+b+c)3=a3+b3+c3+3(a-b)(b-c)(c-a)
» a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca)
» a3+b3+c3-3abc=[(a+b+c){(a-b)2+(b-c)2+(c-a)2}]/2
» (x+a)(x+b)=x2+(a+b)x+ab
» (x+a)(x+b)(x+c)=x3+(a+b+c)x2+(ab+bc+ca)x+abc
By visiting our site, you can practice: Basic Properties of Numbers in Mathematics.

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