Posted by admin on February 21, 2010
It is the mark of an instructed mind to rest satisfied with the degree of precision which the nature of the subject permits, and not to seek an exactness where only an approximation of the truth is possible.
Have you ever listened to the news and heard one of the talking heads make a particularly nonsensical assertion? Did you think to yourself, “That is just not believable?” When you have that experience, do you try to prove to yourself that the assertion is far fetched?
With this type of problem, you generally need only a rough estimate to demonstrate that the assertion is feasible or improbable. Many day-to-day problems are of this type. Engineers and scientists use estimates to quickly evaluate these problems. Learning and using simple estimating tricks will greatly improve your ability to critically evaluate events and opportunities in your day-to-day life.
An extremely simple yet useful estimate is to determine before beginning a calculation if the answer is greater than or less than 1. This comes in handy when converting fractions to decimals. Take 8/5 for example. When we convert this to a decimal, should the answer be greater than one or less than one? Since 8 in the numerator is greater than 5 in the denominator, we know the answer should be greater than 1. If we do the division correctly, we should get this:
However, if we mistakenly do this:
We know right away that we’ve made an error because our answer is less than 1.
Another simple but useful estimating trick is to determine if an answer should be positive or negative before beginning. It is said that the single most common error made by engineers using advanced mathematics is making a plus or minus sign error. Take for example, – ½ * ¾ * -2
We see that there are two minus signs so our answer should be positive. If we punch the numbers in our calculator and come out with a negative answer, we know immediately again that we have made an error.
The third simple estimating trick is using an order of magnitude estimate. An order of magnitude simply means a power of ten. When we estimate to an order of magnitude, we ask simply, is it closer to 1 or to 10, to 10 or 100, to 100 or 1,000, and so on. We can also estimate small numbers similarly. Is it closer to 1 or 1/10, to 1/10 or 1/100?
Using order of magnitude estimating, we calculate a “ballpark” answer before beginning a problem. If we get a final answer that is very much different, we know that we have made an error. Take for example, 7543 / 0.4359. Since 7543 is closer to 10 thousand than to 1 thousand, we round it to 10 thousand. Since 0.4359 is closer to 0.1 than to 1, we round it to 0.1. Dividing 10 thousand by 1 tenth, we get 100 thousand. The true answer, 17,204.4276 is within an order of magnitude of 100 thousand. If we made a mistake, say multiplying instead of dividing, we’d get 3,287.9937, which is closer to 1 thousand and is more than an order of magnitude from the correct answer, letting us know that we made a mistake.
Order of magnitude estimates are useful when an exact answer is not required. They can help understand big questions readily. For example, if you were to take off in a space ship just powerful enough to leave earth orbit, would it be feasible to travel to our nearest star, Proxima Centauri?
Taking escape velocity as 11.2 km per second, the distance to Proxima Centauri as 39.7 quadrillion km, and 31.5 million seconds in a year, we would come up with an order of magnitude estimate of 10 quadrillion km divided by 10 km/sec all divided by 10 million seconds per year. Using the back of an envelope, we write 10 quadrillion km / 10 km/second. We cancel one zero from the top and bottom leaving 1 quadrillion seconds. We then divide 1 quadrillion seconds by 10 million seconds per year. We cancel 7 zeros from the top and bottom, leaving 1 with 5 zeros, 100 thousand years, as our final answer. Something will certainly break on our spacecraft before 100 thousand years has passed. So, the answer is that our spacecraft may eventually get there, but it will in all likelihood be broken.
If we do the real math, we find that our answer is not appreciably different:
3.97 × 10^13 km / 11.2 km/sec = 3.54 × 10^12 secs
3.54 × 10^12 secs / 31.5 × 10^6 seconds/ year = 112,528 years
Whether it takes 100 thousand years or 112 thousand years is not going to make a difference. None of us will be around by the time the spacecraft gets there. In this example, the order of magnitude estimate comes very close to the calculated answer. Other times, it will be further off. For our purpose, we don’t care. We know that the answer is between 50 thousand and 500 thousand years. Long enough for us to decide that this endeavor is not feasible.
What’s with the 10^13 stuff, you may ask? That’s called “Scientific Notation.” It’s an easy way to work with very large or very small numbers, numbers with a lot of zeros. It will be the subject of the next article.
In this post, we’ve talked about three easy to use estimating techniques. It is a good practice to use one or more of these when we start to solve a math problem. It is easy to make errors when performing complex calculations. Having a rough guess at the correct answer before beginning the calculation gives us a way to check to see if we have performed the calculation correctly.
In addition to improving our math, these techniques can be used to understand problems that involve large numbers. A calculation using order of magnitude estimates, usually easy enough to be done in your head or on a scrap of paper, can answer many questions of the type, “Is this feasible?” or “Is this plausible?”
Try using these techniques to answer questions in your everyday life. I would love to hear about your experience. Do you find them useful? Please leave a comment to let me know what you think!