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or result we obtain when a number is squared (raised to power of 2) is called a square number.
2×2 = 4, 4/3×4/3 =16/9 etc.
➤ When we multiply any integer with itself, we always get a whole square number called perfect square. So, a perfect squire is a square of an integer.
(-2)×(-2) = 4,
(-1)×(-1) = 1,
1×1 = 1,
2×2 = 4,
3×3 = 9, etc
➤ For now, we consider Squares of only positive integers i.e, natural numbers,
1×1 = 1,
2×2 = 4,
3×3 = 9, etc
Properties of Square numbers
| Number | Square | Number | Square |
| 1 2 3 4 5 6 7 8 9 10 | 1 4 9 16 25 36 49 64 81 100 | 11 12 13 14 15 16 17 18 19 20 | 121 144 169 196 225 256 289 324 361 400 |
➤ All perfect square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. None of these end with 2, 3, 7 or 8 at unit’s place.
↪ When a square number ends in 1, the number whose square it is, will have either 1 or 9 in unit’s place.
12 = 1, 112 = 121
92 = 81, 192 = 361
↪ When a square number ends in 4, the number whose square it is, will have either 2 or 8 in unit’s place.
22 = 4, 122 = 144
82 = 64, 182 = 324
↪ When a square number ends in 9, the number whose square it is, will have either 3 or 7 in unit’s place.
32 = 9, 132 = 169
72 = 49, 192 = 289
↪ When a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit’s place.
42 = 16, 142 = 196
62 = 36, 162 = 256
↪ When a square number ends in 5, the number whose square it is, will have 5 in unit’s place.
52 = 25, 152 = 225
↪ When a square number ends in 0, the number whose square it is, will have 0 in unit’s place.
102 = 100, 202 = 400
↪ Square numbers can only have even number of zeros at the end which must be double the number of zeros of the number whose square it is.
102 =100, 202 = 400, 302 =900, 1002 = 10000
↪ The squares of even numbers are even numbers and squares of odd numbers are odd numbers.
82 =64, 162 = 256,
72 =49, 152 = 225,
Finding the Square of a Number
→ Multiply the number with itself22 = 2×2 =4
72 = 7×7 =49
→ Using algebraic identity -
Write the number as sum or difference of two appropriate numbers and then use one of these identities
(a+b)2 = a2+b2+2ab, or
(a−b)2= a2+b2−2ab .
262 = (20+6)2 = 202+62+2×20×6
= 400 + 36+240 = 676
or,
262 = (30-4)2 = 302+42-2×30×4
= 900 + 16-240 = 916-240 = 676
Patterns in Squares
➤ Adding Triangular numbersTriangular numbers – Numbers whose dot patterns can be arranged as triangles.
| 1 (⦁) | 3 (⦁⦁⦁) | 6 (⦁⦁⦁⦁⦁⦁) | 10 (⦁⦁⦁⦁⦁⦁⦁⦁⦁⦁) |
15
(⦁⦁⦁⦁⦁⦁⦁⦁⦁⦁⦁⦁⦁⦁⦁) |
| ⦁ | ⦁ ⦁⦁ | ⦁ ⦁⦁ ⦁⦁⦁ | ⦁ ⦁⦁ ⦁⦁⦁ ⦁⦁⦁⦁ | ⦁ ⦁⦁ ⦁⦁⦁ ⦁⦁⦁⦁ ⦁⦁⦁⦁⦁ |
1+3 = 4
| ⦁ + | ⦁ ⦁⦁ | = | ⦁⦁ ⦁⦁ |
| ⦁ ⦁⦁ + | ⦁ ⦁⦁ ⦁⦁⦁ | = | ⦁⦁⦁ ⦁⦁⦁ ⦁⦁⦁ |
| ⦁ ⦁⦁ + ⦁⦁⦁ | ⦁ ⦁⦁ ⦁⦁⦁ ⦁⦁⦁⦁ | = | ⦁⦁⦁⦁ ⦁⦁⦁⦁ ⦁⦁⦁⦁ ⦁⦁⦁⦁ |
Square Roots
↪ Square root of a given number is the value which when multiplied by itself gives that number.↪ Finding the number having the known square is known as finding the square root, it is the inverse operation of squaring.
4 = 2×2, Square Root of 4 is 2.
25 = 5×5, Square Root of 25 is 5.
↪ Every perfect square has two integral square roots, one is positive and other is its negative,
4 = 2×2, also 4 = (-2)×(-2), so square roots of 4 are −2 and 2.
9 = 3×3, also 4 = (-3)×(-3), so square roots of 9 are −3 and 3.
Negative square root of a number is denoted by the symbol -√.
Positive square root of a number is denoted by the symbol √.
e.g., −√4 = −2 & √4 = 2
↪ For now, we consider only positive square roots.
| Statement | Inference | Statement | Inference |
| 12 = 1 | √1 = 1 | 62 = 36 | √36 = 6 |
| 22 = 4 | √4 = 2 | 72 = 49 | √49 = 7 |
| 32 = 9 | √9 = 3 | 82 = 64 | √64 = 8 |
| 42 = 16 | √16 = 4 | 92 = 81 | √81 = 9 |
| 52 = 25 | √25 = 5 | 102 = 100 | √100 = 10 |
Finding Square root
Prime Factorisation method
→ Write the given number as product of its prime factors→ Pair the prime factors.
→ Express the paired prime factors in exponential form of power of 2
→ Take the power of 2 as common
→ Put √ before the number in LHS and remove the power in RHS to get the required value.
e.g. 144 = 2×2×2×2×3×3
⇒ 144 = 2×2×2×2×3×3
⇒ 144 = 22 × 22 × 32
⇒ 144 = (2 × 2 × 3)2
⇒ √144 = 2×2×3 = 12
∴ √144 =12
Division method
➧ For large numbers we use long division method.➧ If a perfect square is of n-digits, then its square root will have n/2 digits for even n or (n +1)/2 digits for odd n.
➧ Steps in long division method
| Step 1⇾ Make pair of digits in given number starting with digit at one’s place. Put bar on each pair. If the number of digits of the given number is odd, then place the bar on this remaining digit also. Step 2⇾ Find the divisor, such that its square is either equal or less than the number under first bar from the left. Here divisor and first digit of quotient will be same. Now do the division and get the remainder. |
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| Step 3⇾ Now bring down the next digits under second bar and place them to the right of the remainder. This will become new dividend. Step 4⇾ Starting digit(s) of new divisor is equal to twice of first digit of quotient (calculated in step 2) Step 5⇾ Now the digit at one’s place of new divisor is equal to next digit of quotient. And it is to be selected in such a way that, when new divisor is multiplied by this next digit of quotient, the product will be equal or less than the new dividend (as calculated in step 3) Step 6⇾ Divide and get the new remainder. This process is to be repeated till all numbers are used under the bars. |
Square root of a Decimal
Division method
⇾ Put a decimal point in the quotient when the decimal part in the number is taken for the division.
| Negative Exponents | |