A couple of days ago I blogged about this book: Rapid Math Tricks and Tips: 30 Days to Number Power by Edward H. Julius (Wiley, 1992) ISBN 0-471-57563-1. I’ve had quite a few people saying how fun that last tip was (multiplying by 99). Here’s my favourite trick of all in the first book: how to square any number that ends in 5. (Remember that the symbol (^) means raise to the power, so 5^2 is 5 raised to the power of 2, or 5 x 5.)

Here’s what Edward H. Julius has to say:

Strategy:This trick is one of the oldest in the book, and one of the best! To square a number that ends in 5,

- first
multiply the tens digit by the next whole number.- To that product,
affix the number 25.The number to affix (25) is easy to remember, because 5^2 = 25. Although a calculation such as 7.5 x 750 is technically not a square, it too can be solved using this technique. This trick will also work for numbers with more than two digits.

For example: 15^2:

- Multiply the tens digit (1) by the next whole number (2): 1 x 2 = 2.
- Affix 25 to that answer: 225 (the answer).

Another example, 65^2:

- Multiply 6 x 7 = 42
- Affix 25: 4,225 (the answer)

How about, 450^2:

- Ignore the zero and think 45^2
- Multiply 4 x 5 = 20
- Affix 25: 2025
- For each zero initially disregarded in a squaring problem two must eventually be affixed to obtain the product.
- Affird two zeros to the end: 202,500.

And what about the not-squared problem? This method only works if the two numbers have the same digits, eg 65 x 6.5 or 950 x 9.5 but not 65 x 3.5. So for 65 x 6.5:

- Disregard the decimal point for now
- Multiply 6 x 7 = 42
- Affix 25: 4225
- Applying a test of reasonableness you can see that 65 x 5 would be 325, so x 6.5 must be between 325 and 650 (which is 65 x 10). So re-inserting the decimal point gives the answer: 422.5.

How cool is that! Now, go buy the book! It’s only £6.96 at Amazon UK.