Introduction
The number system is the way we represent and use numbers in mathematics. It includes a wide variety of number types including integers, fractions, decimals, exponents, and roots, each with its own set of rules and operations.
Importance in SSC, RRB, and Banking Exams
Knowledge of the number system is foundational for anyone preparing for SSC, RRB, and banking examinations. Many questions in these exams, especially those in the Quantitative Aptitude or Mathematics sections, are based on the concepts of the number system. Hence, having a thorough understanding of the number system is critical.
Basic Concepts and Definitions
Integers: These are all the whole numbers, including zero, and their negatives. For example, …, -3, -2, -1, 0, 1, 2, 3, … are all integers.
Fractions: A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). The numerator represents how many parts we have, and the denominator represents how many equal parts the whole is divided into.
Decimals: Decimal numbers are another way of representing fractions. They use a decimal point to separate the whole number part from the fractional part.
Exponents: An exponent refers to the number of times a number, the base, is multiplied by itself. For example, in 2^3, 2 is the base and 3 is the exponent, which means 2 is multiplied by itself 3 times.
Roots: A root of a number is a value that, when multiplied by itself a certain number of times, gives the original number. For example, the square root of 9 is 3 because 3*3 equals 9.
Basic Operations
Each of these number types has its own set of operations.
Integers: The basic operations with integers are addition, subtraction, multiplication, and division.
Fractions: With fractions, you can add, subtract, multiply, and divide, but the operations are a little more complex than with integers.
Decimals: Operations with decimals are similar to those with integers, but you need to take care of the decimal point’s position.
Exponents: When dealing with exponents, the rules of power multiplication, division, and power of power come into play.
Roots: Square roots, cube roots, and higher roots are computed differently than basic operations, often requiring special techniques or calculator use.
Advanced Concepts and Problem Types
More advanced number system concepts can include rational and irrational numbers, prime numbers, least common multiple (LCM), and greatest common divisor (GCD). Problems can range from simple operations to solving equations or interpreting word problems.
Problem-Solving Techniques and Examples
Example 1: Addition with decimals.
Add 3.4 and 2.7. You line up the decimal points and add as with integers, giving you 6.1.
Example 2: Multiplication with exponents.
Calculate 2^3 * 2^4. With the same base, you add the exponents, giving you 2^7.
Example 3: Simplify a fraction.
Simplify 20/45. The GCD of 20 and 45 is 5, so when both numbers are divided by 5, the fraction simplifies to 4/9.
Practice Problems
- Subtract: 7.8 – 4.3.
- Calculate: 3^2 * 3^3.
- Simplify the fraction: 30/42.
Summary
The number system forms the foundation of mathematics. Understanding integers, fractions, decimals, exponents, and roots, as well as their operations, is essential not only for mathematical proficiency but also for successful performance on competitive exams like the SSC, RRB, and banking exams. Mastering the number system opens the door to more advanced mathematical topics and problem-solving techniques.