Class 8 Mathematics: Chapter - Square and Square Roots

Quick Revision Notes & Formula Sheet (MP Board)

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1. Basic Definitions (मूल परिभाषाएं)

• Square of a Number (संख्या का वर्ग): When a number is multiplied by itself, the result is called the square of that number.

  • Formula: Square of x = x × x = x^2
  • Example: 5^2 = 5 × 5 = 25

• Perfect Square (पूर्ण वर्ग): A natural number is called a perfect square if it is the square of some natural number.

  • Example: 1, 4, 9, 16, 25, 36... are perfect squares.

• Square Root (वर्गमूल): The square root of a number x is that number which, when multiplied by itself, gives x. It is denoted by the symbol √.

  • Formula: If x^2 = y, then √y = x
  • Example: If 6^2 = 36, then √36 = 6

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2. Important Properties of Square Numbers (वर्ग संख्याओं के गुण)

1. Ending Digits Rule:

   • A perfect square NEVER ends with 2, 3, 7, or 8 at the unit's place.
   • Numbers ending in 0, 1, 4, 5, 6, or 9 MAY be perfect squares.

2. Unit Digit Patterns:

   Unit digit of Number	Unit digit of its Square
   1 or 9	1
   2 or 8	4
   3 or 7	9
   4 or 6	6
   5	5
   0	0

3. Number of Zeros:

   • A square number always contains an EVEN number of zeros at the end.
   • Example: 100 (2 zeros - Perfect Square), 1000 (3 zeros - Not a Perfect Square).

4. Odd and Even Rules:

   • Square of an Even Number is always Even. (4^2 = 16)
   • Square of an Odd Number is always Odd. (7^2 = 49)

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3. Key Concepts & Formulas (महत्वपूर्ण नियम एवं सूत्र)

A. Non-Square Numbers between Two Consecutive Squares

Between the squares of two consecutive numbers n and (n + 1), there are 2n non-square numbers.

• Formula: Total non-square numbers between n^2 and (n+1)^2 = 2n
• Example: Non-square numbers between 4^2 (16) and 5^2 (25) = 2 × 4 = 8 numbers.

B. Sum of Consecutive Odd Numbers

The sum of first n odd natural numbers is equal to n^2.

• Formula: 1 + 3 + 5 + ... + (2n - 1) = n^2
• Example: 1 + 3 + 5 + 7 + 9 = 5^2 = 25 (Sum of first 5 odd numbers).

C. Pythagorean Triplets (पाइथागोरस त्रिक)

A set of three numbers (a, b, c) is called a Pythagorean Triplet if a^2 + b^2 = c^2.

• General Formula: For any natural number m > 1, the triplet is given by: $$\text{Triplet} = (2m,\ m^2 - 1,\ m^2 + 1)$$

  • Smallest member: 2m
  • Other members: m^2 - 1 and m^2 + 1

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4. Methods to Find Square Roots (वर्गमूल ज्ञात करने की विधियाँ)

Method 1: Repeated Subtraction Method (बारंबार घटाने की विधि)

• Subtract consecutive odd numbers (1, 3, 5, 7, 9...) successively from the given number until you get 0.
• The total number of steps performed is the square root.

Method 2: Prime Factorisation Method (अभाज्य गुणनखंडन विधि)

1. Find the prime factors of the given number.
2. Form pairs of identical prime factors.
3. Take one factor from each pair and multiply them together.

• Example for 36: 36 = 2 × 2 × 3 × 3 $\rightarrow$ Pairs: (2 × 2) and (3 × 3) $\rightarrow$ √36 = 2 × 3 = 6

Method 3: Long Division Method (भाग विधि)

• Used for finding square roots of large numbers and decimals.
• Steps:
  1. Group digits in pairs starting from the unit digit (place a bar over each pair).
  2. Find the largest number whose square is less than or equal to the first period/pair.
  3. Divide and bring down the next period.
  4. Double the quotient to form the new trial divisor.

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5. Square Roots of Decimals (दशमलव संख्याओं का वर्गमूल)

• Place bars on the integral part from right to left (standard way).
• Place bars on the decimal part from left to right.
• Proceed with the standard long division method.

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6. Quick Reference Table: Squares from 1 to 20

Number (n)	Square (n^2)	Number (n)	Square (n^2)
1	1	11	121
2	4	12	144
3	9	13	169
4	16	14	196
5	25	15	225
6	36	16	256
7	49	17	289
8	64	18	324
9	81	19	361
10	100	20	400

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💡 Quick Exam Tips:

• If a question asks "How many numbers lie between $12^2$ and $13^2$?", directly use 2n = 2 × 12 = 24.
• If a question asks to find the "Smallest number to multiply/divide to make a number a perfect square", always use the Prime Factorisation Method and find the unpaired factor.
• If a question asks to find the "Smallest number to add/subtract to make a number a perfect square", use the Long Division Method.