Unlock the magic of mathematics as we delve into a fascinating trick for squaring numbers ending in 5. Prepare to be amazed by the simplicity and elegance of this mathematical shortcut, a testament to the beauty of numbers and their patterns.
Squaring Numbers that End in 5
You multiply the number that precedes the 5 by the next number higher than it, and then tack on 25 to the end of the product you get.
- For example, to square 25, you multiply the 2 times the next higher number, 3, and get 6; append 25, and you have 625, which is the square of 25.
- For longer numbers, you can do the same thing with 12 by 13 and stick on the 25 to get 15,625 which is square of 125.
- This works for longer numbers such as 125.
For example, consider the following squares: 25, 35, 85, and 75, in columns A, B, C, and D respectively:
First multiplication: 25
- Second multiplication: 125
- Third multiplication: 175
- Fourth multiplication: 625
- Fifth multiplication: 7225
- Sixth multiplication: 6800
- Multiply by even numbers and carry the 2 in the next column
- The last two digits in the product of the first and second multiplication will always end in 25, and the right zero is always a zero
Solution
Since (2)(30)(5) = (10)(30) if we just multiply the 2 times the 5 first, we can derive from step 2 that [(30 + 5)(30, + 5)] = [(30)2 + (10(30) + 25]
- The “25” will end up being the end result of the final squared number
- More generally, what we have when we multiply any two digit number ending in 5 will be 25 (from squaring the 5’s) PLUS the square of the number in the ten’s place, plus 10 more of the numbers in the 10’s place
- We are multiplying the number by one more than what we multiplied it by originally, which is what the ‘trick’ says to do before adding the 25
Why you get the 25 as the last two digits in columns B and D:
First multiplication will always end in 75
- With odd numbers in front of the 5, multiplying the two 5’s together gives you 25
- Second multiplication always ends in 50
- The right zero is automatic, as it is in any multiplication of the ten’s column numeral
- And the second number will always be a 5 because you are multiplying 5 times an odd number in the second column
Full proof
Consider first any two or more digit number, where the last digit is “a” and the number before it is 5.
- The whole number can be represented as (10a + 5), and the square of the number ending in 5 will be the sum of 25 and the product of multiplying “the portion of [10a+5]” with “one more than the portion of “the number before the 5″”
- Notice that (a + 1) is simply one more than a, and since we are multiplying it by 100, which adds two zeroes to the end of the product, the result is 100 + 25.
A Better, More General Explanation
A better, more general explanation
- I can explain the idea intuitively and logically, using some algebra
- It is easier to see than the full algebraic proof, but it is harder to give both an intuitive and logic explanation simultaneously
- The intuitive explanation is more general, and the logic explanation is clearer
The square of the sum of any two numbers x and y is represented by: (x + y)2 which is always equal to (x2 + 2xy + y2).
That is a known algebraic equation that comes from simply multiplying each of the components times each other and adding the products, in the way one does multiplication.
