All you need is a simple calculator and Jerry’s magic formula
Issue 22 : Jan/Feb 2002
The mathematicians who read this should be prepared to forgive me. It is my only contribution to “applied mathematics.”
About 100 years ago, I used to eat supper with my calculus teacher before our night-school class. On one occasion I said to him: “Could you give me just one example of how I might use the concept we are studying?” His response was simply “No.” Thus was born my personal idea of applied mathematics, which I distinguished from the other mathematics — stuff that I would study, rarely or never use, and then forget from disuse.
There are times when a worker will want to measure, calculate, or specify something in decimal inches. This happens to me often when I am measuring parts and spaces with a vernier caliper (or even a micrometer) which allows resolution to a thousandth of an inch (0.001 inch). It also happens when I am doing something using fractions, because the vast majority of modern calculators deal with decimals directly, but must convert fractions to decimals to process them.
It is well known that one may easily convert any fraction to its decimal equivalent by simply dividing the numerator (the number above the bar) by the denominator (the number below the bar). For example, the fraction 3⁄4 is converted by entering into the calculator the number 3 divided by the number 4. The answer will appear as 0.75.
The difficulty, of course, is in going the other way, from a decimal to a fraction. This is necessary for me when I move from the calculation stage to the measuring and cutting stage, which often involves laying out something with my trusty machinist’s square. It reads in fractions of an inch. It also happens when I have to use a calibrated square or measuring tape. In these cases, I like to be able to convert the decimal-inch measurement that I’ve calculated into a fraction.
Nearest fraction
You are thinking at this point that most decimals don’t convert evenly to fractions or at least not into the fractions with standardized denominators found on machinist’s squares and tape measures. This is true, and it is also true that if you actually need to make the finished parts to a very high tolerance and fit, you should work with decimal-inch measurements throughout the project. In many cases, however, the nearest fraction of an inch will do nicely at the end of a long calculation. So then, how do we convert a decimal to the nearest fraction? You can carry a chart of decimal-to-fraction conversions with you, but it can also be done with the simplest of four-function calculators, and this method is my contribution to “applied mathematics.”
Let’s convert 0.75 back to a fraction. In your calculator, enter .75 x 4 + 400.” You get back the number 403, which, if read in pairs from right to left, can be interpreted as 3⁄4. You now say this is rigged, and how did I pick those numbers? Well, I wanted the answer in 4ths so I used “times 4 plus 400.” If I’d wanted the answer in 16ths I’d have used “times 16 plus 1600” and gotten 1612, which I would have read as 12⁄16. Then, noting that the denominator was even, I would have keyed in “divided by 2.” This would have yielded 806, to be read as 6⁄8, and again the denominator is even, so I would have divided by 2 again to get back to 403 or 3⁄4.
Even if you have your calculator out by now and have not gone to sleep over this, you are thinking that I still have not dealt with decimals that don’t come out even. Let’s look at one of those. Let’s take the decimal 0.4567 for example. We can get this back to a fraction with almost any dominator we might want, as we saw above.
Admittedly, 128ths are little tricky, but 64ths are easy. So enter “.4567 x 64 + 6400.” Wow, it equals 6429.2288, which we read as a little larger than 29⁄64. In fact, it’s .2288 64ths larger. Now there’s a concept, a fraction with a decimal numerator. That’s OK, however. If you are measuring something in 64ths, you will know that 0.4567 is about a quarter of a 64th larger than 29⁄64, which is good enough in most cases. In another example, the decimal 0.4525 would convert to 6428.96 or 28⁄64 with .96 of a 64th for a remainder. Naturally, you’d round that one up to 29⁄64.
Handle separately
This technique is best limited to numbers smaller than one. Handle the whole units separately. It will work for any denominator you choose, but the common ones you need have two digits, and they are the least confusing.
The general case is (the target decimal, times the desired denominator) plus (the desired denominator, times one hundred). Read the answer in pairs from right to left, where the first pair of digits is the numerator, and the second pair is the denominator. (The pairs themselves are read from left to right in the normal way.)
Now you can throw away your table of fractions to decimals, and convert to any denominator you want with a low-cost, four-function calculator.
Please, don’t ask me how I think this stuff up.
I will not write about mathematics again. I will not write . . .
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