**Introduction**

Manipulation of numbers occurs in mathematics (Algebra, Geometry), i.e.

To solve the equations in algebra and working with numbers in geometrical shape etc.With help of these manipulating numbers we have many advantages in mathematics.

**Properties:**

1) Various mathematical operations can be performed easily (simplification, factorization etc).

2) To solve the simple algebraic problems.

3) Polygon numbers can be generated easily.

**Symbol manipulation**

The manipulation of characters rather than numbers, as occurs in symbolic mathematics, preparation of text, and finite-state simulation is called as Symbol manipulation.

**Algebraic symbol manipulation:**

The data of algebraic expressions in symbolic form, and the operations are the operations of algebra. The operations provided usually include multiplying out brackets, simplification, factorization, polynomial division, and differentiation with respect to one or more variables. This is a programming language.

## Types of Manipulation

The types of manipulating numbers.

1) Manipulating polygonal numbers.

2) Algebraic Manipulation.

3) Manipulations of exponents.

**1) Manipulating polygonal numbers**

Successive polygon numbers are developed by manipulation method by various patterns.

**Difference of two successive polygonal numbers is:**

To find the difference of two successive x-agonal numbers, subtract 2 from x and multiply by n and subtract (x-3).

(X-2) n - (X-3)

Where n is the largest x-agonal number

**Example:**

The difference of the 6th and 7th octagonal numbers is?

**Solution**:Given data,

Octagonal number=8=X

Formula: (X-2) n - (X-3)

(8-2) n – (8-3) =6n-5

The largest number in given data is 7

6n-5 = 6(7)-5

= 42-5

= 37

**2) Algebraic Manipulation**

Simple algebraic equation with single unknown is the equation of equality. Equality sign will not change if we perform mathematical operation such as (addition, subtraction, multiplication etc). This includes squaring both sides of the equation or taking the square root of both sides of the equation.

We can get the variable value by this method by following two laws:

1) Distributive Law

2) Associative Law

**Distributive Law**

Distributive is a property of binary operations which generalizes the distributive law from elementary algebra.

General form of representing distributive property is:

a x (b + c) = (a x b) + (a x c)

For given set of numbers “s” with two binary operations can be defined by two laws:

1) Left distributive law: x • (y + z) = (x • y) + (x • z);

2) Right distributive law: (y + z) • x = (y • x) + (z • x)

Where,

X, y, z are the variables

Mathematical operations are addition (+) and multiplication (*).

**3) Manipulations of exponents**

Scientific calculations are handled by expressing quantities in scientific notation. Such operations require simple manipulation of exponents, usually exponents of 10. When the same base is used (e.g. 10), the following rules are applied:

a) When the operation involves multiplication, add the exponents.

**Example:** 10^{3} x 10^{4}

Bases are same i.e. 10

10^{3} x 10^{4} = 10^{3+4}

= 10^{7}

b) When the operation involves division, subtract the divisor exponent from the numerator exponent.

**Example:$\frac{10^5}{10^3}$**

Bases are same i.e. 10

** ****$\frac{10^5}{10^3}$** = 10^{(5 - 3)} = 10^{2}

c) When the operation involves powers or roots, multiply the exponent by the power number or divide the exponent by the power number.

**Example:** (10^{5})^{3}

algebra Answer: (10^{5})^{3} = 10^{(5 x 3)}

= 10^{15}

## Solved Problems

1) Solve x - 12 + 20 = 37?

ALGEBRA ANSWER FOR THE EQUATION :

**Solution**:Given,x-12+20=37

We need to find x values so “x” term should be independent, Hence we will bring constants to other side.

x-12+20=37

Bringing constants to other side

x=37+12-20

x=49-20

x=29.

2) Find the difference of 9th and 10th decagonal numbers?

**Solution**:Given data,

Decagonal number=10=X

Formula: (X-2) n - (X-3)

(10-2) n – (10-3) =8n-7

The largest number in given data is 10

8n-7 = 8(10)-7

= 80-7

= 73

3) John is 10 years old. His father is 45 years old. After how many years will the father be twice as old as the son?

**Solution**:Given data,

John age=10 years

Father age= 45

2 (10 + x) = 35 + x

20+2x=45+x

2x-x=45-20

X=25

Son is 35 after father is 70.

4) Evaluate the expression using distributive property 3(1+2)?

**Solution**:Given expression, 3(1+2)

This is left distributive law over addition

The expression can be rewritten as:

3(1+2)= 3x1 + 3x2

=3+6

=9.