Essay on Number System

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he number theory or number systems happens to be the back bone for CAT preparation. Number systems not only form the basis of most calculations and other systems in mathematics, but also it forms a major percentage of the CAT quantitative section. The reason for that is the ability of examiner to formulate tough conceptual questions and puzzles from this section. In number systems there are hundreds of concepts and variations, along with various logics attached to them, which makes this seemingly easy looking topic most complex in preparation for the CAT examination. The students while going through these topics should be careful in capturing the concept correctly, as it’s not the speed but the concept that will solve the question here. The correct understanding of concept is the only way to solve complex questions based on this section.

Real numbers: The numbers that can represent physical quantities in a complete manner. All real numbers can be measured and can be represented on a number line. They are of two types: Rational numbers: A number that can be represented in the form p/q where p and q are integers and q is not zero. Example: 2/3, 1/10, 8/3 etc. They can be finite decimal numbers, whole numbers, integers, fractions. Irrational numbers: A number that cannot be represented in the form p/q where p and q are integers and q is not zero. An infinite non recurring decimal is an irrational number. Example: √2, √5 , √7 and Π(pie)=3.1416. The rational numbers are classified into Integers and fractions Integers: The set of numbers on the number line, with the natural numbers, zero and the negative numbers are called integers, I = {…..-3, -2, -1, 0, 1, 2, 3…….}

Fractions:
A fraction denotes part or parts of an integer. For example 1/6, which can represent 1/6th part of the whole, the type of fractions are: 1. Common fractions: The fractions where the denominator is not 10 or a multiple of it. Example: 2/3, 4/5 etc. 2. Decimal fractions: The fractions where the denominator is 10 or a multiple of 10. Example 7/10, 9/100 etc. 3. Proper fractions: The fractions where the numerator is less than the denominator. Example ¾, 2/5 etc. its value is always less than 1. 4. Improper fractions: The fractions where the numerator is greater than or equal to the denominator. Example 4/3, 5/3 etc. Its value is always greater than or equal to 1. 5. Compound fraction: A fraction of a fraction is called a compound fraction

Example 3/5 of 7/9 = 3/5 x 7/9 = 21/45

6. Complex fractions: The combination of fractions is called a complex fraction.

Example (3/5)/ (2/9)

7. Mixed fractions: A fraction which consists of two parts, an integer and a fraction. Example 3 ½, 6 ¾ Example: Express 27/8 as a mixed fraction

Ans. Divide the numerator by denominator; note the multiplier, whatever remainder is left divide it with the original denominator. For 27/8, 24/8 = 3, and remainder left is 3, therefore 3 3/8 is the mixed fraction Example: Express 35 7/17as an improper fraction.

Ans. Here we need to multiply the denominator with the non-fraction part and add it to numerator and using same denominator. For 35 7/17= = 602/17
The integers are classified into negative numbers and whole numbers Negative numbers: All the negative numbers on the number line, {…..-3, -2, -1} Whole numbers: The set of all positive numbers and 0 are called whole numbers, W = {0, 1, 2, 3, 4…….}. Natural numbers: The counting numbers 1, 2, 3, 4, 5……. are known as natural numbers, N = {1, 2, 3, 4, 5…..}. The natural numbers along with zero make the set of the whole numbers.

Even numbers: The numbers divisible by 2 are even numbers. e.g., 2, 4, 6,8,10 etc. Even numbers can be expressed in the form 2n where n is an integer other than 0. Odd numbers: The numbers not divisible by 2 are odd numbers. e.g. 1, 3, 5, 7, 9 etc. Odd numbers are expressible in the form (2n + 1) where n is an integer other than 0. Composite numbers: A composite number has other...
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