NumberA number is a mathematical object used to count, label, and measure. In mathematics, the definition of number has been extended over the years to include such numbers as 0, negative numbers, rational numbers, irrational numbers, and complex numbers.

Mathematical operations are certain procedures that take one or more numbers as input and produce a number as output. Unary operations take a single input number and produce a single output number. For example, the successor operation adds 1 to an integer, thus the successor of 4 is 5. Binary operations take two input numbers and produce a single output number. Examples of binary operations include addition, subtraction, multiplication, division, and exponentiation. The study of numerical operations is called arithmetic.

A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels (telephone numbers), for ordering (serial numbers), and for codes (e.g., ISBNs).

1. Whole Number

Definition of Whole Numbers

The numbers in the set {0, 1, 2, 3, 4, 5, 6, 7, . . . . } are called whole numbers. In other words, whole numbers is the set of all counting numbers plus zero. Whole numbers are not fractions, not decimals. Whole numbers are nonnegative integers.

For example:

95,163,578,1250,2500,…

2. real numbers

Definition of Real Numbers

Real Numbers include all the rational and irrational numbers. Real numbers are denoted by the letter R. Real numbers consist of the natural numbers, whole numbers, integers, rational, and

irrational numbers.

Examples of Real Numbers

Natural numbers, whole numbers, integers, decimal numbers, rational numbers, and irrational numbers are the examples of real numbers.

Natural Numbers = {1, 2, 3,...} Whole Numbers = {0, 1, 2, 3,...} Integers = {..., -2, -1, 0, 1, 2,...}

, 10.3, 0.6, , , 3.46466466646666..., , are few more examples.

3. rational numbers

Definition of Rational Numbers

A Rational Number is a real number written as a ratio of integers with a non-zero denominator.

Rational numbers are indicated by the symbol .

Rational number is written in form, where p and q are integers and q is a non-zero denominator.

All the repeating or terminating decimal numbers are rational numbers. Rational numbers are the subset of real numbers.

Examples of Rational Numbers

, 10.3, 0.6, , - All these are examples of rational numbers as they terminate.

.

4. irrational number

Definition of Irrational Number

Irrational numbers are real numbers that cannot be expressed as fractions, terminating decimals, or repeating decimals.

Examples of Irrational Number

are few examples of irrational numbers.

5. prime number

Definition of a Prime Number

A prime number is a positive integer that has exactly two factors, 1 and the number itself.

Examples of Prime Number

2, 3, 5, 7, 11, 13, 17, 19, etc. are all prime numbers.

There are infinitely many prime numbers.

6. fraction

Definition of Fraction

A fraction is a number that represents part of a whole.

A fraction is written in the form , where q ≠ 0. A fraction is a way of writing numbers that are not whole numbers. The top number of a fraction is called the numerator. The bottom number of a fraction is called the denominator.

Examples of Fraction

32

256 32 numerator, 256 denominator

a. decimal fraction

Definition of Decimal Fraction

A Decimal Fraction is a fraction in which the denominator is a power of ten. Decimal fraction can be easily converted into decimal by inserting a decimal separator in

the value of the numerator at the position from the right in accordance with the power of ten of the denominator.

Examples of Decimal Fraction

are the examples of decimal fractions. These decimal fractions can be expressed as 0.089 without the denominators.

256

5120=0.05

21601620

=1,3333333 ≈ 1,34

b. mixed fraction

Definition of Mixed Fraction

A Mixed Fraction is a number with a combination of an integer and a proper fraction. Mixed fraction is also called as mixed number. An improper fraction can be converted into a mixed fraction and vice versa.

Examples of Mixed Fraction

5367

is a mixed number, in which 53 is an integer and 67

is a fraction.

−3234

is a mixed number, in which – 32 is an integer and 34

is a fraction.

Mathematical operations

The most common are add, subtract, multiply and divide (+, -, ×, ÷ ). But there are many more, such as squaring, square root, etc.If it isn't a number it is probably an operation.

1. addition

1. Definition of Addition

Addition is an operation that finds the total number when two or more numbers are put together.

In other words, addition is the process to find the sum of two or more numbers.

Addition is written using the plus sign "+" between the terms; that is, in infix notation. Sum is the result that you get when you addition one number from another.

For example,

1. 3100 + 573 = 3673 Sum

2. 5125 + 670 = 5795 Sum

3. 150000 + 95000 = 245000 Sum

2. Propertie

Commutativity

Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result is the same as the last one. Symbolically, if a and b are any two numbers, then

a + b = b + a.

The fact that addition is commutative is known as the "commutative law of addition". This phrase suggests that there are other commutative laws: for example, there is a commutative law of multiplication. However, many binary operations are not commutative, such as subtraction and division, so it is misleading to speak of an unqualified "commutative law".

Associativity

A somewhat subtler property of addition is associativity, which comes up when one tries to define repeated addition. Should the expression

"a + b + c"

be defined to mean (a + b) + c or a + (b + c)? That addition is associative tells us that the choice of definition is irrelevant. For any three numbers a, b, and c, it is true that

(a + b) + c = a + (b + c).

For example:

(211 + 92) + 310 = 303 + 310 = 613 = 211 +402 = 211 + (92 + 310).

Not all operations are associative, so in expressions with other operations like subtraction, it is important to specify the order of operations.

Identity element

When adding zero to any number, the quantity does not change; zero is the identity element for addition, also known as the additive identity. In symbols, for any a,

a + 0 = 0 + a = a.

2. SubtractionDefinition of Subtraction

Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign (-).

In other words, subtraction is the process of finding how many are left when some are taken away.

Subtraction is the opposite or inverse operation of addition. Minuend: The number that is to be subtracted from.

Subtrahend: The number that is to be subtracted.Difference: The result of subtracting one number from another

For example:

1. 1300 – 570 = 730

1300 minuend

570 subtrahend

730 difference

2. 5530 – 2890 = 2640

3. 350000 – 153000 = 197000

3. Multiplication 1. Definition of Multiplication

Multiplication of two numbers is the repeated addition of one number to the number of times equal to the other number.

Multiplication is often written using the multiplication sign "×" between the terms; that is, in infix notation. The result is expressed with an equals sign.

Multiplication of numbers is commutative, associative, and distributive. The result of a multiplication is called a product, and is a multiple of each factor if the other

factor is an integer.

For example :

1. Calculate 765 × 9.

Solution:

Write the smaller number, 9, under the larger number, 765, and then calculate the multiplication.

6885 is the product of 765 and 9, and is both a multiple of 765 and a multiple of 9.

2. 65 x 78 = 5070

5070 is the product of 65 and 78, and is both a multiple of 65 and a multiple of 78.

3. 156 x 138 = 21528

4. 5000 x 23 =115000

2. Properties

Multiplication of numbers 0-10. Line labels = multiplicand. X axis = multiplier. Y axis = product.

For the real and complex numbers, which includes for example natural numbers, integers and fractions, multiplication has certain properties:

Commutative property

The order in which two numbers are multiplied does not matter:

.

Ex: 210 x 75 = 75x 210

Associative property

Expressions solely involving multiplication or addition are invariant with respect to order of operations:

Ex : (135 x 33) 4=135(33 x $)

Distributive property

Holds with respect to multiplication over addition. This identity is of prime importance in simplifying algebraic expressions:

674 (234 + 121) = 674 ∙ 234 + 674 ∙ 121

Identity element

The multiplicative identity is 1; anything multiplied by one is itself. This is known as the identity property:

3245 ∙ 1 = 3245

Zero element

Any number multiplied by zero is zero. This is known as the zero property of multiplication:

Ex: 586 ∙ 0 = 0

Zero is sometimes not included amongst the natural numbers.

There are a number of further properties of multiplication not satisfied by all types of numbers.

Negation

Negative one times any number is equal to the additive inverse of that number.

Negative one times negative one is positive one.

The natural numbers do not include negative numbers.

Inverse element

Every number x, except zero, has a multiplicative inverse, , such that .

4. division1. Definition of Division

Division is an operation that tells us the number of groups that can be made out of a number of items or the number of items that should be there in a group.The symbol ‘÷’ denotes division.

Dividend: In a division problem, the number that is to be divided is called the dividend In the division number sentence 8 ÷ 2 = 4, 8 is the dividend.

Divisor: In a division number sentence, the number that divides the dividend is called the divisor.In the division number sentence 8 ÷ 2 = 4, 2 is the divisor.

Quotient: Upon division, the number obtained other than the remainder is called the quotient.   In the division number sentence 8 ÷ 2 = 4, 4 is the quotient.

For example:

1. 3600 ÷ 40 = 90

3600 : dividend

40 : divisor

90 : quotient

2.

2. Notation

Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a vinculum or fraction bar, between them. For example, a divided by b is written

This can be read out loud as "a divided by b", "a by b" or "a over b". A way to express division all on one line is to write the dividend (or numerator), then a slash, then the divisor (or denominator), like this:

This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of ASCII characters.

A typographical variation halfway between these two forms uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:

a⁄b

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the numerator and denominator), and there is no implication that the division must be evaluated further. A second

way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:

Example

1. 2250

5=450

2. 6500

25=260

3. 875000 ÷ 25 = 35000