Multiplication and Division

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Speaking realistically, the problems discussed below can all be done on a calculator. However, you need to know how to enter values into the calculator, read your calculator screen, and round off to the proper number of significant figures. Your calculator will not do these things for you.

In addition, it is more than likely your instructor has good estimation skills and can look at problems set up in scientific notation and come very close mentally to the correct answer. The ChemTeam feels this is an excellent skill to have. Sometimes a student gives an answer in the ChemTeam's classroom and it does not feel right. This is because the ChemTeam does have these estimation skills. You should too.

How do you go about solving the above problem? How many significant figures will be in the answer? The issue of significant figures is discussed elsewhere, so if you do not know what they are right now, that's OK. Later on, when you are more well-versed in significant figures, you might want to review these words. By the way, the answer to the above problem to the correct number of significant figures is 2.92 x 10^{6}. The calculator answer (2917200) contains too many significant figures.

Lastly, these instructions are aimed at calculators using algebraic logic. Those that use Reverse Polish Notation, such as most Hewlett-Packard calculators, will not be covered below.

When multiplying numbers in scientific notation, enter one number into the calculator and multiply it by the other. The calculator should give you the number in proper scientific notation. You will need to read your calculator properly as well as round off the number to the correct number of significant figures. Here is a sample problem:

You should consult your calculator manual for information on how to enter numbers in scientific notation into the calculator. The right way involves the use of a key usually marked "EXP" or "EE." A usual wrong way involves using the times key, where the student presses times then 10 __then__ presses the "EXP" key.

Typically, this is announced by the student confidently saying, "Uh, <substitute name of teacher here>, your value is off by a factor of 10, the correct exponent should be 7, not 6." Depending on your teacher's mood and personality, the response will range from a nice explanation of why you are wrong to an insulting one. Double- and triple- check your work before announcing confidently to the teacher that they are wrong.

The calculator gives 1.3878 x 10¯^{3} for the above problem. Rounded to three significant figures, the correct answer is 1.39 x 10¯^{3}. If you were to write all the displayed digits down for your answer, you would be wrong.

When you are dividing using the calculator, there is an additional factor to be aware of. Division is not commutative like multiplication, so the order in which numbers are entered into the calculator for division is important. Also, it is important to realize that while there are two phrases which can be used for problems, "divided by" and "divided into," the calculator uses only the first.

If the phrase is "divided by" as in this problem: 2.4 x 10¯^{4} divided by 3.4 x 10^{5}, then you enter the numbers into the calculator in the order written. However, if this is the problem: 2.4 x 10¯^{4} divided into 3.4 x 10^{5}, then you reverse the written order when entering the numbers into the calculator.

Sometimes the calculator will give the answer in the usual way (e.g., 0.00084). Often, a calculator will have a way to convert between the usual way (technically called floating point notation) and scientific notation. Consult the manual that came with your calculator for more information. If it does not have this capacity, you will have to do it manually by counting decimal places.

The biggest error ChemTeam students make is in reading the calculator screen and then writing the answer down on paper. No matter how hard he tries, it seems the ChemTeam coach cannot eradicate this error. For example, let us suppose the answer to some problem is 2.35 x 10¯^{5}. Here is how many calculators represent this number on their screen:

What happens is that the student writes what he or she sees on the calculator screen and never thinks about what is being written. The problem is that 2.35¯^{5} is NOT the same numerical value as 2.35 x 10¯^{5}. What has happened is that the student is ignoring the context the numbers are being shown in.

2.35¯^{5} is simply the calculator's shorthand way to show 2.35 x 10¯^{5}. In the context of the calculator screen, 2.35¯^{5} is understood as shorthand for scientific notation. On a piece of paper, 2.35¯^{5} is something completely different.

The calculator screen is not sacred.

This is how the mental estimation referred to above is done.

Multiply the decimal portions and add the exponential portions. (Remember that adding exponents is how to multiply them.) Here is a sample problem:Here is the rearranged problem:

This can be done since multiplication is commutative.

The exponent is easy, since 6 plus negative 10 is negative 4.

3.05 x 4.55 is easy too. You know it will be a bit bigger than 12, so you estimate 13.

You now have 13 x 10¯^{4}.

So the teacher says out loud, "Well it's about 1.3 or 1.4 times ten to the negative three."

And you, the ChemTeam kid, are astonished. It's like magic! Nah, just a good sense of numbers and the rules which govern them.

Suppose it was division, rather than multiplication. Using the previous sample problem numbers set up as division:Here is the rearranged problem:

The exponent is easy, since 6 minus negative 10 is positive 16. Remember that to divide exponents, you subtract them.

3.05 ÷ 4.55 is easy too. You know it will be a bit less than 0.75, so you estimate 0.7.

You now have 0.7 x 10^{16}.

So the teacher says out loud, "Well it's about 7 times ten to the fifteen."

And once again you, the ChemTeam kid, are astonished. It's like magic! Nah, just a good sense of numbers and the rules which govern them.

Practice Problems

1) (2.68 x 10¯^{5}) x (4.40 x 10¯^{8})

2) (2.95 x 10^{7}) ÷ (6.28 x 10^{15})

3) (8.41 x 10^{6}) x (5.02 x 10^{12})

4) (9.21 x 10¯^{4}) ÷ (7.60 x 10^{5})

Reminders

1) Double check each calculator entry for correctness. Once you begin entering the new number, the old one is removed from the display. Often students seem to automatically assume the answer on the display is correct.

2) Try for reasonable approximations mentally. For example, in (3) above 8.41 x 5.02 is around 40 and 10^{6} x 10^{12} is 10^{18}. So the approximate answer is around 40 x 10^{18} or 4 x 10^{19}. Try and predict (3) as a division, not multiplication.

3) Remember, if you struggle with entering numbers in scientific notation into your calculator, check your calculator manual or get some live help.