F1.03: Example 6

Example 6. Scientific notation.   Multiply 8,000,000 by 60,000.

Solution: When you do this on your calculator, you’ll get a strange-looking answer with a E in it. You must learn to interpret that answer. That’s the way calculators give scientific notation.

In some algebra classes you learned scientific notation as a shorter way of writing some very large or very small numbers.

[latex]4,120,000=4.12\times{{10}^{6}}[/latex]   and   [latex]0.000089=8.9\times{{10}^{-5}}[/latex]

When using a calculator or spreadsheet we might easily obtain a number that is too big for the display and must be expressed in scientific notation. But calculator displays usually use shorthand for this.   On most calculators and spreadsheets, we’ll have 3.12 E 05 or 6.7 E–10. These mean

3.12 E 05 = [latex]3.12\,\times{{10}^{5}}=312,000[/latex]

6.7 E–10 = [latex]6.7\times{{10}^{-10}}=0.00000000067[/latex]

On your calculator, multiply 8,000,000 by 60,000. What do you get? How would you write it in scientific notation? How would you write it in regular notation? (Answer: 480,000,000,000,= [latex]4.8\,\times{{10}^{11}}[/latex])

Review:   Additional review of scientific notation is available from the course website.

Going further: Scientific notation is also used to convey the precision of measured values clearly and concisely. We will discuss that in later Topics in this course.

Discussion. When and how much should you round the results of a calculator computation?

Calculators keep more accuracy in calculations than we will probably want to do in our hand calculations.   Typically, that is about 12 decimal places for the inexpensive scientific calculators. When computing, it is tempting for students to use the calculator to do each individual operation and then write down that result correct to about three decimal places and then do the next individual operation. This is not considered good practice because if it is done for several steps, then quite a bit of accuracy can be lost.

Good practice in using a calculator is to do all the calculations in the problem by keeping the intermediate results in the calculator and only round at the end to report the final answer. That enables us to keep as much accuracy in our result as our original data had and not to introduce inaccuracy as a result of our computation.

However, when formulas are particularly long or complicated, you may need to write down intermediate results in order to better understand the techniques. While that is acceptable to help you make progress, it is important for you to understand that you are losing accuracy.   If there is anything important depending on your computed result, you should learn to keep all the computed values in the calculator to produce the final answer.

Discussion. Checking your work

When learning to use the keys on a calculator, use problems that you can easily do by hand or even mentally, so that you can easily check to see whether the calculator has given the correct answer.   In fact, using such problems to experiment is one of the main ways people learn to do new things on their calculators. After all, what percentage of time do most of us spend reading the calculator manual? In fact, how many of us can even remember where we put the calculator manual?

Notice that most of the examples so far used simple numbers so that it would be easy for you to do the calculation by hand and check your calculator work.   That is always how you should approach learning to use a key on your calculator that you have never used before.

When using the calculator to do messy problems that we wouldn’t want to do by hand, it is also important to do some checking. (It is very easy to punch in the wrong numbers, decimals, or operations.)   In such cases, you should estimate what the answer will be and then be sure that your final answer on the calculator is reasonably close to your estimate. If it is not, you’ll need to determine whether your estimate was wrong or the something went wrong with the calculation in the calculator.

Example 7: A sofa is priced at $887 in the furniture store. The sales tax is 8.25%.

  1. Estimate the total amount you’ll have to pay with both the cost and the tax.
  2. Use your calculator to compute the total amount you’ll have to pay.
  3. Is your calculator answer close to your estimate?

Solution.

  1. Since $887 is not too far from to $1000, let’s estimate the cost of the sofa as $1000. 8.25% is pretty close to 8%, so let’s compute 8% of $1000.Since 8% of $100 is $8, then 8% of $1000 must be ten times that, so it’s about $80.   Thus, sales tax amount is pretty close to $80.So how much is the whole bill? It’s about $80 more than the cost of the sofa itself. Here we could continue to use the estimate of the sofa as $1000, but it’s really closer to $900.

    It’s pretty easy to add $900 and $80, so we estimate the total is about $980.

  2. Using a calculator, we compute 887 + 0.0825*887 = 960.18.
  3. Yes, the calculator answer is pretty close to my estimate, so it seems likely that it is correct.