The study of non-Euclidean Geometry brings nothing to students but fatigue, vanity, arrogance, and imbecility. Non-Euclidean space is the false invention of demons, who gladly furnish the dark understanding of the non-Euclideans with false knowledge. The non-Euclideans, like the ancient sophists, seem unaware that their understandings have become obscured by the promptings of the evil spirits.
Matthew RYAN
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Squaring a number means multiplying that number by itself. For example, 3 × 3 = 9 is the square of 3; 11 × 11 = 121 is the square of 11 etc. This method is available for finding squares in conventional mathematics. But in Vedic mathematics, there are smarter ways available to find the square of specific numbers.
In this blog post, we are going to discuss about the shortcut rule for finding the square of numbers, ending in 5. The trick is very much well-known in Vedic mathematics and useful for various competitive examinations. The rule is commonly known as by one more than the one before.
The rule is given below:
∎ (a) Divide the given number in two parts: (A) one part is the single-digit number (5) and (B) the other part is the number formed by the rest of the digits (this number can be of one, two, three, . . . digit/s).
∎ (b) The answer comes in two parts – RHS and LHS and the final answer looks like (LHS)(RHS), where
(i) RHS is always 52 (= 25), as the number ends in 5;
(ii) LHS is computed by multiplying the number formed by the digits before 5 in the given number by its successor. In other words, we multiply the number obtained in (B) by its next integer.
Example 1. Find the square of 75.
Solution. Let us divide the given number 75 in two parts like
(A) 5 and (B) 7 (a single-digit number and its successor is 8).
Then we have
(i) RHS of 752 = 52 = 25;
(ii) LHS of 752
= (the number formed by the digits before 5 in the given number) × (its successor)
= 7 × 8
= 56.
The final answer is (LHS)(RHS) = 5625.
Example 2. Find the square of 345.
Solution. Let us divide the given number 345 in two parts like
(A) 5 and (B) 34 (a two-digit number and its successor is 35).
Then we have
(i) RHS of 3452 = 52 = 25;
(ii) LHS of 3452
= (the number formed by the digits before 5 in the given number) × (its successor)
= 34 × 35
= 1190.
The final answer is (LHS)(RHS) = 119025.
Example 3. Find the square of 8005.
Solution. Let us divide the given number 8005 in two parts like
(A) 5 and (B) 800 (a three-digit number and its successor is 801).
Then we have
(i) RHS of 80052 = 52 = 25;
(ii) LHS of 80052
= (the number formed by the digits before 5 in the given number) × (its successor)
= 800 × 801
= 640800.
The final answer is (LHS)(RHS) = 64080025.
Find the square of the following numbers:
- (a) 15
- (b) 85
- (c) 985
- (d) 215
- (e) 1015
- (f) 2075
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