Number System:
The number system is a fundamental concept in mathematics and forms the basis for various calculations and problem-solving techniques. Mastering the number system is crucial for excelling in competitive exams, as it helps improve accuracy and speed. In this study guide, we will explore important concepts, tips, and tricks related to the number system that will aid you in your exam preparation.
Classification of Numbers:
Numbers can be classified into various categories based on their properties and characteristics. Let's explore the main classifications of numbers:
A. Natural Numbers (N): Natural numbers are the set of positive integers, starting from 1 and extending infinitely. They are used for counting and do not include zero or negative numbers.
Example: 1, 2, 3, 4, 5, ...
B. Whole Numbers (W): Whole numbers include all the natural numbers along with zero. They form the set of non-negative integers.
Example: 0, 1, 2, 3, 4, ...
C. Integers (Z): Integers encompass both positive and negative whole numbers, along with zero.
Example: ..., -3, -2, -1, 0, 1, 2, 3, ...
D. Rational Numbers (Q): Rational numbers are numbers that can be expressed as a fraction of two integers. They can be terminating or repeating decimals.
Example: 1/2, -3/5, 0.75, 2.333..., ...
E. Irrational Numbers (I): Irrational numbers cannot be expressed as a fraction and have non-repeating, non-terminating decimal expansions. They are typically represented using radical symbols.
Example: √2, π (pi), e (Euler's number), ...
F. Real Numbers (R): Real numbers encompass both rational and irrational numbers, forming the set of all possible numbers on the number line.
Example: All rational and irrational numbers together.
Notable Properties of Number Systems: Each classification of numbers has its own unique properties:
A. Closure Property: The closure property states that the sum or product of two numbers from the same set will always belong to that set.
Example: The sum of two natural numbers is always a natural number (e.g., 3 + 5 = 8).
B. Commutative Property: The commutative property holds for addition and multiplication, stating that changing the order of numbers in an operation does not affect the result.
Example: 4 + 2 = 2 + 4 and 3 * 6 = 6 * 3.
C. Associative Property: The associative property holds for addition and multiplication, stating that the grouping of numbers in an operation does not affect the result.
Example: (2 + 3) + 4 = 2 + (3 + 4) and (5 * 6) * 2 = 5 * (6 * 2).
D. Distributive Property: The distributive property links addition and multiplication, stating that multiplying a number by a sum is the same as multiplying the number by each addend separately and then adding the results.
Example: 2 * (3 + 4) = (2 * 3) + (2 * 4).
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0 is known as fraction.