How to Check Your Arithmetic, Part 1
Learn how to quickly and easily check your answers to arithmetic problems without using a calculator.
Jason Marshall, PhD
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How to Check Your Arithmetic, Part 1
Do you ever solve a simple arithmetic problem and then wonder if you got the answer wrong? Perhaps this happens to you during tests or maybe it’s while you’re performing the highly-endangered act of balancing a checkbook. No matter the circumstances, rest assured that there are ways for you to rid yourself of those gnawing pangs of dread and ensure that your work is error free. And, as we’ll see in the next several articles, you don’t have to use a bulky calculator to do it.
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How to Check the Sign of Simple Multiplication Problems
The first anxiety relieving tip is designed to help you check that your answers to multiplication problems have the correct sign. As you might guess, this tip is based upon the by-now-familiar sayings about multiplying positive and negative numbers that we talked about in the last article:
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A positive times a positive makes a positive.
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A negative times a negative makes a positive.
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A positive times a negative makes a negative.
When multiplying pairs of numbers, your results had best not disagree with these trusty slogans. So 3×3 must be a positive number, –3×3 must be a negative number, and –3x–3 must be a positive number.
How to Check the Sign of More Complex Multiplication Problems
But that’s not all these sayings are saying. Not only do they tell us what the sign of the answer must be when two numbers are multiplied together, they also tell us what the sign of the answer must be when any number of numbers are multiplied together! Here’s the quick and dirty tip:
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If the list of numbers you’re multiplying contains an odd number of negative numbers, the result must be a negative number.
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If the list of numbers you’re multiplying contains an even number of negative numbers, the result must be a positive number.
So 2x–4x5x3 must result in a negative number since there is an odd number of negative numbers in the problem. And 2x–4x5x–3 must result in a positive number since there is an even number of negative numbers in the problem.
Try this example: –1x–4x10x–37×9=13,320…is that the right answer?
Well, there are three negative numbers here: –1, –4, and –37. And since this is an odd number of negative values, we can immediately tell that the result has to be a negative number. So the answer 13,320 can’t be right. Although, as you can check, it’s very close—the real answer is –13,320.
Why does this method work? Well, it goes back to those old sayings that a negative times a negative (which, you’ll notice, is an even number of negative numbers) makes a positive and a positive times a negative (which, you’ll notice, is an odd number of negative numbers) makes a negative. All we’re doing with these longer problems is applying the basic rules for pairs of numbers several times.
How to Check the Sign of Division Problems
If you think about it, you’ll see that the trick we just talked about for figuring out the sign of the answer to multiplication problems works for division problems too. And not only that, it works for problems that contain both multiplication and division—which means that whenever you’re multiplying, dividing, or doing some combination of these two operations with a list of numbers, you can check if the result must be positive or negative simply by counting if there is an even or an odd number of negative numbers.
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If there is an odd number of negative numbers, the result must be a negative number.
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If there is an even number of negative numbers, the result must be a positive number.
For example, we can immediately tell that the answer to the problem –40 / 10 * –37 * 22 must be a positive number (without doing any multiplication or division at all) since there is an even number of negative numbers. If the last number, 22, had instead been –22, then three of the four numbers in the problem would have been negative, and the answer to the problem would have been negative too.
Okay, we’ll leave it right there for today. Be on the lookout for next week’s article in which we’ll take a look at another method for checking your arithmetic that uses something called the “parity” of the result. And in two weeks we’ll talk about the rather exciting (no joke!) and powerful method called “casting out nines” that will really put your arithmetic checking capabilities into overdrive.
Number of the Week
Before we finish up, it’s time for this week’s featured number from my post on QDT’s blog The Quick and Dirty. This week’s number is a conversion that lots of astronomers (myself included) use all the time when making calculations…and it’s something that everybody should have in their number-crunching bag-o-tricks. The trick is to know that if you multiply the world-famous number pi (about 3.14) by the number 10 million, the result is pretty close (within about half a percent) to the total number of seconds you age each and every trip you take around the Sun.
In other words, pi x 10 million is the approximate number of seconds in a year. Of course, the length of a year in units of seconds doesn’t really have anything to do with the ratio of the circumference of a circle to its diameter (which is where pi comes from)…this is all just a very happy accident. The great thing about memory tricks like this is that even though you might not need to use them all the time, they’re easy to remember—so they’re right there for you if (or rather when) you need them!
Wrap Up
Remember to become a fan of the Math Dude on Facebook where you’ll find a new featured number or math puzzle posted each and every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.comcreate new email.
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
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About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.