Unit 1: Real And Complex Numbers - Mathematics For Class IX (Science Group) - Explanation Of Exercise 1.1

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Unit 1: Real And Complex Numbers
Explanation Of Exercise 1.1

SOME IMPORTANT SETS OF NUMBERS:

Following notations will be used for sets of numbers:

• Set of Natural Numbers: N = {1, 2, 3, . . .}
• Set of Whole Numbers: W = {0, 1, 2, 3, . . .}
• Set of Integers: Z = {0, ±1, ±2, ±3, . . }
• Set of Positive Integers: Z+ = {0, +1, +2, +3, . . }
• Set of Negative Integers: Z¯ = {0, -1, -2, -3, . . }
• Set of Positive Prime Numbers: P = {2, 3, 5, 7, 11, . . . }
• Set of Odd Numbers: O = {±1, ±3, ±5, . . . .}
• Set of Even Numbers: E = {0, ±2, ±4, ±6, . . . .}
• Set of Rational Numbers: Q = {x|x = p/q ; p, q ∈ Z, q ≠ 0}
• Set of Irrational Numbers: Q' = {x|x ≠ p/q ; p, q ∈ Z, q ≠ 0}
• Set of Real Numbers: R = Q U Q'
• Also, R⁺ and R^- will denote the set of all positive and negative real numbers, respectively.

All above sets are contained in the set of real numbers. Hence real numbers are classified as:

CLASSIFICATION OF REAL NUMBERS:

REAL NUMBERS:
The set of real numbers is the union of the set of rational and irrational numbers, i.e.

R = Q U Q'

Also, R⁺ and R^- will denote the set of all positive and negative real numbers, respectively.

RATIONAL NUMBERS:
Rational number is a number that can be written as quotient of two integers (i.e. expressed in the form of fraction ^p/_q) are called rational numbers.
In rational numbers q or denominator is not equal to zero

Q = { x | x ; x = ^p/_q ; p,q ∈ Z and q ≠ 0}

All rational numbers contain terminating and non-terminating decimal fractions.
For example:
⁰/₂ is a rational number but ²/₀ is not a rational number because denominator of rational number is never zero.
Similarly 2, -3 are also rational number because the denominator in each number is 1 i.e.

•  ²/₁ = 2 and
•  ^-3/₁ = -3

NOTE: 
Every natural number and integer is also a rational number.
There are infinite rational numbers between any two numbers.

IRRATIONAL NUMBERS:
Numbers that can not be written as quotient of integers are called irrational numbers.
For Example:

• √ 2 = 1.14142135 ....
• √ 3 = 1.7320508 ....
• π = 3.1415926 ..........
• 0.02002002000200002.......

Irrational numbers contain non-terminating decimal fractions only.

DISTINGUISHING DECIMAL REPRESENTATION OF RATIONAL AND IRRATIONAL NUMBERS

1. Terminating Decimal Fractions:
A decimal fraction in which the decimal part contains only a finite number of digits is called a Terminating Decimal Fraction.
All terminating decimal fractions represent rational numbers.
Example:

• (i) ²⁵⁰¹/₁₀₀ = 25.01
• (ii) ²⁴⁵⁸/₁₀₀₀ = 0.2458
• (iii) ⁵/₂ = 2.5
• (iv) ¹/₂ = 0.5

2. Recurring Or Non Terminating Decimal Fractions:
A non-terminating decimal fraction whose decimal parts contain some digits which are repeated again and again in the same order is called a Recurring Or Non Terminating Decimal Fraction.
All non-terminating decimal fractions represent both rational and irrational numbers.
Example:

• √ 2 = 1.14142135 ....
• √ 3 = 1.7320508 ....
• π = 3.1415926 ..........
• 0.02002002000200002.......

NUMBER LINE

It is a line on which we represent the real number at both sides of zero. The numbers are whole number and also integers.

REPRESENT RATIONAL NUMBER (Terminating And Non-Terminating Recurring Decimal) ON THE NUMBER LINE

In order to locate number with terminating and non-terminating recurring decimal on the number line:
Suppose:
The points associated with the rational number ^a/_b, where a, b are positive integers.

• Step 1: Draw a line show whole parts of the fraction or mixed fraction.
• Step 2: Sub-divide the each unit length (points between two numbers) into b equal parts (i.e according to value of denominator).
• Step 3: Select the a^th of division (i.e. point according to numerator and start counting these on number line.)
• Step 4: The right of the origin represent positive value (i.e. ^a/_b) and the left of the origin at the same distance represents negative value (i.e. - ^a/_b)