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Number System

UPSC CSAT Maths

Table of contents

Core Concepts

  1. Number line: It is one of the most crucial concepts in Basic Numeracy. The number line is a straight line between negative infinity on the left to positive infinity to the right.

  2. Important Mathematical Representations

  • Even numbers: 2n
  • Odd Numbers: 2n-1
  • Numbers multiple of 4: 4n, and so on
  1. Prime Factors: Representation of numbers in the standard form.

  2. Standard Form or Canonical Form: eg. Finding standard form of 144

Therefore, Prime factors of 144= 2 * 2 * 2 * 2 * 3 * 3 = 2^4 * 3^2

  1. General Form for Writing 2-3 Digit Numbers: In mathematics many a time we have to use algebraic equations in order to solve questions. In such cases an important concept is the way we represent two or three-digit numbers in equation form. For instance, suppose we have a 2-digit number with the digits ‘AB’. In order to write this in the form of an equation, we have to use:
    10A + B. This is because in the number ‘AB’ the digit A is occupying the tens place. Hence, in order to represent the value of the number ‘AB’ in the form of an equation — we can write 10A + B.
    Thus, the number 29 = 2 * 10 + 9 * 1

  2. Similarly, for a three-digit number with the digits A, B and C respectively — the number ‘ABC’ can be represented as below:
    ABC = 100A+10B+C
    Thus, 243 = 2 * 100 + 4 * 10 + 3 * 1

  3. The BODMAS Rule: It is used for the ordering of mathematical operations in a mathematical situation. In any mathematical situation, the first thing to be considered is Brackets followed by Division, Multiplication, Addition and Subtraction in that order.

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Number System

  • Integers: Numbers having no decimal point in it
    • Positive Integers - 1, 2, 3, …
    • Zero
    • Negative Integers: -1, -2, -3, …
  • Decimals: A decimal number is a number with a decimal point in it, like these: 1.5, 3.21, 4.173, 5.1, etc.
  • Natural numbers: These are the numbers (1, 2, 3, etc.) that are used for counting. In other words, all positive integers are natural numbers. Based on divisibility, there could be two types of natural numbers: Prime and Composite.
    • Prime numbers - A natural number larger than unity is a prime number if it does not have other divisors except for itself and unity. (Note: Unity (i.e. 1) is not a prime number.)
    • Composite Numbers: It is a natural number that has at least one divisor different from unity and itself.
      Every composite number n can be factored into its prime factors. (This is sometimes called the canonical form of a number.)
  • Whole Numbers: The set of numbers that includes all natural numbers and the number zero are called whole numbers. Whole numbers are also called non-negative integers.
  • Real Numbers: All numbers that can be represented on the number line are called real numbers.
  • Rational Numbers: A rational number is defined as a number of the form a/b where a and b are integers and b ⧣ 0.
  • Irrational numbers: Fractions, that are non-terminating, non-periodic fractions, are irrational numbers. Some examples of irrational numbers are √2, √3, √5, e, pi, etc.
  • Co-prime numbers are any two numbers which have an HCF of 1, i.e. when two numbers have no common prime factor apart from the number 1, they are called co-prime or relatively prime to each other. Eg. (5, 6), (29,30)

Operations on Numbers

1. Exponents and Powers

  • Exponents, or powers, are an important part of maths as they are necessary to indicate that a number is multiplied by itself for a given number of times.

  • When a number is multiplied by itself it gives the ‘square of the number’. Thus,

  • n * n = n^2 (e g. 3 * 3 = 3^2)

  • If the same number is multiplied by itself twice we get the cube of the number. Thus, n * n * n= n^3 (e g. 3 * 3 * 3= 3^3)
    n * n * n * n= n^4 and so on.

  • With respect to powers of numbers, there are 5 basic rules which you should know: For any number ‘n’ the following rules would apply:

    • Rule 1: n^a × n^b = n^(a+b). Thus, 4^3 × 4^5 = 4^(3+5) = 4^8
    • Rule 2: n^a / n^b = n^(a-b). Thus, 3^9 / 3^4 = 3^5.
    • Rule 3: n^ (a^b)= n^(a×b). Thus, 3^ (2^4 )= 3^8.
    • Rule 4: n^ (-a) = 1/n^a . Thus, 3^ (-4)=1/3^(4)
    • Rule 5: n^0=1. Thus, 5^0=1.

These are also popularly known as the rules of indices.

2. Properties

  • Commutative property of addition: a + b = b + a.
  • Associative property of addition: (a + b) + c = a + (b + c).
  • Commutative property of multiplication: ab = ba.
  • Associative property of multiplication: (ab) * c= a(b*c).
  • Distributive property of multiplication with respect to addition: (a + b) c = ac + bc.
  • Subtraction and division are defined as the inverse operations to addition and multiplication respectively.

LCM and HCF

LCM (Least Common Multiple)

  • Smallest natural number 'n' which is divisible by n1 & n2
    How to find LCM?
  • Find Standard forms of n1, n2
  • Write down all prime factors
  • Raise each prime factor to highest of the powers
  • The product will be LCM

Eg. Find the LCM of 150, 210.

  • Step 1: Writing down the standard form of numbers
    • 150 = 5 * 5 * 3 * 2
    • 210 = 5 * 2 * 7 * 3
  • Step 2: Write down all the prime factors: that appear at least once in any of the numbers: 5, 3, 2, 7.
  • Step 3: Raise each of the prime factors to their highest available power (considering each to the numbers).
    • The LCM= 2^1 * 3^1 * 5^2 * 7= 2 * 3 * 25 * 7 = 1050

Highest Common Factor (HCF)

  • For n1, n2: If n1 and n2 are exactly divisible by the same number x, then x is a common divisor (CD) of n1, n2. Highest of all such CDs of n1 and n2 is HCF.

How to find HCF?

  • Find Standard forms of n1, n2
  • Write down all prime factors
  • Find Common factors in both n1, n2.
  • Raise each prime factor to lesser of the powers
  • The product will be HCF

E.g., Find the HCF of 150, 210, 375.

  • Step 1: Writing down the standard form of numbers
    • 150 = 5 * 5 * 3 * 2
    • 210 = 5 * 2 * 7 * 3
  • Step 2: Writing Prime factors common to all the numbers: 5, 2, 3 (Note: we didn’t consider 7 as it is only a common factor of 210 and not of 150!)
  • Step 3: Raising each prime factor to lesser of the power, i.e.- 2^1, 5^1 and 3^1
  • Step 4: Hence, the HCF will be 2 * 5 * 3 = 30

Important Rule:
HCF (n1, n2) * LCM (n1, n2) = n1*n2
i.e., The product of the HCF and the LCM equals the product of the numbers.

Rule for finding out HCF and LCM of fractions
HCF of two or more fractions is given by:

• LCM of two or more fractions is given by:

The Remainder Theorem

  • Remainder of a division or expression can be expressed as positive and negative numbers, though negative numbers cannot be remainders. But we can use them, to find out the correct remainder easily.
  • Consider following expression,
  • 107/9, here remainder is positive remainder is +8, and negative remainder is 8-9 = -1.
  • This method helps in finding the reminders for large expressions, refer to the example given below.
    (1753 * 1749 * 83 * 171 / 17)
    Remainders for each number in the numerator,
    (+2 * -2 * -2 * 1)/17 = 8/17
    Hence, remainder is 7.

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