Today at work I was going out of my mind. I needed something to think about, something to process. I knew that if I could find a source of random numbers, I could amuse myself adequately. I ended up using the dollar/cents numbers of people's transations (worked well enough for my purposes). At first, I would record a string of them just to get a set of numbers, then when I had downtime, I would try to see how cleverly I could come up with their product: $22.76 turned into 22 and 76. Then I multiplied 22 by 100 (since 76 is about 3/4 of 100, which is pie to multiply). Since I knew I was aiming for quarters (going for 75% of 22*100), I halved my result (1100, then added half of that again: 1650. I added 22 to that, and had my product: 1672. I checked with a calculator (I wasn't always correct - in my mental math, I'd drop a tens place or two every once in a while).

Eventually I became bored and wanted to try something else. I started trying to find the square roots of numbers, accurate to two decimal places. I wasn't as successful in this venture, but I could get within +/- .05 reliably. I also used percents and fractions to get my answers: take 32 for example. I know that 5^2 = 25, and 6^2 = 36. There is a different of 11, and 32 is 7 more than 25, thus - the answer was probably 5 and 7/11-ish. Figuring out the decimal was the hardest part. 11 is almost 10, so it's a little less than 5.7. To get the second decimal place, I'd guess based on how far from 10 the denominator was (which would affect my percent error for the numerator). For this example, I guessed 5.68 - the real answer is 5.656... So I was pretty close!

Then I made a personal discovery.

When doing so many of these square roots, I was comparing many common squares and finding the difference of them, I noticed a pattern... 7,9,11,13,15... the difference of a sequence of squares is the odd-number series! start with a number, and create a sequence of x^2 - (x-1)^2 as x increases. You'll get sequential odd numbers. Fascinating! I realize it's not a real discovery, but I have never been taught this and have not been in a math class for about three years.