DIGITAL ALBUM CLASSIFICATION OF NUMBERS

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MANASI ROLL NO : 28

B.Ed MATHEMATICS KRTC, KOLLAM

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WHAT IS A NUMBER ?  A number is a mathematical value used for counting or measuring or labelling objects.

Numbers are used to performing arithmetic calculations.  A number has many other variations such as even and odd numbers, composite and prime numbers.

 In a number system, these numbers are used as digits. 0 and 1 are the most common digits in the number system, that are used to represent binary numbers. On the other hand 0 to 9 digits are also used for other number system

CLASSIFICATION OF NUMBERS

NATURAL NUMBERS All the numbers starting from 1 till infinity are natural numbers, such as 1, 2, 3, 4, 5,… infinity. These numbers lie on the right side of

the number line and are positive.

Natural numbers are again classified into prime numbers and composite numbers. 1.) Prime numbers:

Prime numbers are those natural numbers which are only divisible by 1 and the number itself. Prime numbers starts from 2, 3, 5, 7, 11,…and so on.

Note: 1 and 0 are neither the prime numbers nor the composite numbers. 2.) Composite numbers:

Composite numbers are those natural numbers which are divisible by 1, the number itself and also by the other numbers. That means they have factors other than 1 and the number itself.

WHOLE NUMBERS All the numbers starting from 0 till infinity are whole numbers such 0,1,2,3,4,5…infinity. These numbers lie on the right side of the

number line from 0 and are positive.

INTEGERS Integers are the whole numbers which can be positive, negative or

Zero. Example : 2, 33, 0, -258 etc are integers

Integers are again classified into odd numbers and even numbers. 1.) Even numbers: Even numbers are the both positive and negative integers which are divisible by 2 . For example: …-6, -4, -2, 0 , 2, 4, 6 … are the even number as these numbers are divisible by 2. 2.) Odd numbers: Odd numbers are those positive and negative integers which are not divisible by 2. For example: …-5, -3, -1, +1, +3, +5… are the odd numbers as they are not divisible by 2.

Properties of Integers There are a few properties of integers which determine its operations. These principles or properties help us to solve many equations. To recall, integers are any positive or negative numbers, including zero. Properties of these integers will help to simplify and answer a series of operations on

integers quickly. All properties and identities for addition, subtraction, multiplication and division of numbers are also applicable to all the integers. Integers include

the set of positive numbers, zero and negative numbers which are denoted with the letter Z.

Integers have 5 main properties of operation which are: •Closure Property •Associative Property •Commutative Property •Distributive Property •Identity Property Integer Property

Addition

Multiplication

Subtraction

Division

Commutative Property

x + y = y+ x

x×y=y×x

x–y≠y–x

x÷y≠y÷x

Associative Property

x + (y + z) = (x + x × (y × z) = (x × (x – y) – z ≠ x – y) +z y) × z (y – z)

Identity Property

x + 0 = x =0 + x x × 1 = x = 1 × x x – 0 = x ≠ 0 – x x ÷ 1 = x ≠ 1 ÷ x

Closure Property

x+y∈Z

Distributive Property

x×y∈Z

x–y∈Z

x × (y + z) = x × y + x × z x × (y − z) = x × y − x × z

(x ÷ y) ÷ z ≠ x ÷ (y ÷ z)

x÷y∉Z

RATIONAL NUMBERS A number which can be represented in the form p/q, where q ≠ 0 is

called a rational number. Example : 1/2, 4/5, -26/7 etc

Characteristics of a Rational numbers  Every rational number is expressible either as a terminating decimal or as a repeating decimal.

 Every terminating decimal and repeating decimal are rational numbers.

IRRATIONAL NUMBERS A number is called an irrational number if it can’t be represented in the form of a ratio. 𝟑

Example : 𝟑, 𝟏𝟏, 𝟐, 𝝅, e Characteristics of irrational numbers  The non terminating, non repeating decimals are irrational numbers. Example : 0.010010000…  If m is a positive number which isn’t a perfect square, then 𝒎 is irrational. Example : 𝟕  If m is a positive number which isn’t a perfect cube then irrational. Example :

𝟑

𝟐

Did you know ?

𝟑

𝒎 is

Properties of Irrational numbers  These satisfy the commutative, associative and distributive laws for addition and multiplication.

 Sum of irrationals need not be irrational.  Difference of two irrationals need not be irrational.

 Product of two irrationals need not be irrational.  The quotient of two irrationals need not be irrational.  Sum of rational and irrational is irrational.

 Difference of rational and irrational is irrational.  Product of rational and irrational is irrational.

 Quotient of rational and irrational is irrational.

Irrational numbers can be Surds and Transcendental numbers

Surds Surds are the square roots (√) of numbers that cannot be simplified into

a whole or rational number. It cannot be accurately represented in a fraction. In other words, a surd is a root of the whole number that has an irrational value. Consider an example, √2 ≈ 1.4142

Transcendental numbers We can define a transcendental number as a real number that is not algebraic as well as is not the solution of any single-variable polynomial equation whose

coefficients are known to be all integers (basically whole numbers) Examples : 𝝅, e

RATIONALISATION If we have an irrational number, then the process of converting the denominator to a rational number by multiplying the numerator and denominator by a suitable number is called rationalisation.

Laws of radicals Let a > 0 be a real number, and let p and q be rational numbers then we have,

i.

(𝒂𝒑 )𝒂𝒒 = 𝒂(𝒑+𝒒)

ii.

(𝒂𝒑 )𝒒

= 𝒂𝒑𝒒

iii. 𝒂𝒑 /𝒂𝒒 = 𝒂(𝒑−𝒒) iv.

𝒂𝒑 × 𝒃𝒑 = (𝒂𝒃𝒑 )

REAL NUMBERS The collection of all rational and irrational numbers is called real numbers. Real numbers are denoted by R.

Every real number is a unique point on the number line and also every point on the number line represents a unique real number. Properties of Real numbers

 Real numbers follow closure property, associative law, commutative law, the existence of an additive identity, existence of additive inverse of addition.

 Real numbers follow closure property, associative law, commutative law, the existence of a multiplicative identity, existence of multiplicative

inverse and distributive laws of multiplication over addition for multiplication.

COMPLEX NUMBERS The numbers which are not real are imaginary numbers. Complex numbers are the numbers that are expressed in the form of a + ib where a,b are real numbers and i is an imaginary number

called “iota”. The value of i = −𝟏. An equation of the form z = a+ib, then real part denoted as Re z = a Imaginary part is denoted as Im z =ib.

ARGAND PLANE Similar to the XY plane , the Argand or complex plane is a system of rectangular coordinates in which the complex number a+ib is

represented by the point whose coordinates are a and b. The axis which is horizontal is called the real axis and axis which is vertical is called the imaginary axis.

Properties of complex numbers

 The addition of two conjugate complex numbers will result in a real number.  The multiplication of two conjugate complex number will also result in a real number.  If x and y are the real numbers and x+iy = 0 then x = 0 and y = 0.  If p, q, r and s are the real numbers and p+qi = r+si , then p = r and q = s.  The complex number obeys the commutative law of addition and multiplication  The complex number obeys the associative law of addition and multiplication.

 The complex numbers obeys the distributive law.  If the sum of two complex number is real and also the product of two complex number is also real, then these complex numbers are conjugate

to each other.  For any two complex numbers, say 𝒛𝟏 and 𝒛𝟐 then,

ǀ 𝒛𝟏 + 𝒛𝟐 ǀ ≤ ǀ𝒛𝟏 ǀ + ǀ𝒛𝟐 ǀ  The result of the multiplication of two complex numbers and its conjugate value should result in a complex number and it should be positive value.  Modulus and conjugate let z = a+ib be a complex number. Then modulus of z is given by

ǀzǀ = 𝑎2 + 𝑏 2 The conjugate of z is denoted by 𝒛ത = a-ib.

PICTORIAL REPRESENTATION