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IT: device to device communication
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2.5 Working with scientific notation using the Windows calculator

Most electronic calculators will enable you to perform calculations on numbers expressed in scientific notation. This section will take you through an exercise using the Windows calculator to perform the following calculation:

This image shows the following equation: open bracket, three times ten to the power of four, close bracket, times, open bracket two times ten to the power of three, close bracket.
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This image shows the following equation: open bracket, three times ten to the power of four, close bracket, times, open bracket two times ten to the power of three, close bracket.

Notice how we have placed the two terms in brackets. Often this is done to ensure that each step of a calculation is done in the right order. Here it isn't strictly necessary to include brackets since, when multiplying together a number of terms, the result is the same regardless of order. However, brackets do help to tidy things up and show which terms belong together.

Start the Windows calculator running on your computer (go to the Start menu and select Programs > Accessories > Calculator) then follow each step shown in the table below. (The > symbol represents the small black triangle shown on the right of a menu item, which indicates a sub-menu.)

Step Action Calculator display
1Make sure the Windows calculator is in Scientific mode by selecting Scientific from the View menu.0.
2Enter 3.3.
3Click the Exp button which you will find in the left half of the calculator keyboard. ('Exp' stands for 'exponent'.) This tells the calculator that the next number you enter will be a power of 10 and that you are working in scientific notation. The 'e' now showing on the display indicates that the number you have entered is displayed in scientific notation.3.e + 0
4Enter 4. The display is now showing the equivalent of 3×104.You have now finished entering the term in the first set of brackets.3.e + 4
5Click the multiply (*) button. This tells the calculator that you want to multiply the number showing in the display by some other number. The display now changes to show the number you have entered in its full form (3×104= 30000).30000.
6You will now start to enter the term in the second set of brackets. Enter 2.2.
7Click the Exp button.2.e+0
8Enter 3. The display is now showing the equivalent of 2×103. You have now finished entering the term in the second set of brackets.2.e+3
9Click the equal (=) button. This tells the calculator that you want it to display the result of the calculation. This is shown in its full form.60000000.
10Finally, you can force the calculator to display the result of the calculation in scientific notation. Do this by clicking the F-E button which you will find in the left half of the calculator keyboard. ('F-E' stands for 'fixed to exponent') The display is now showing the result in the 'shorthand' form which you can interpret as 6×107. 6.e+7

The result of the above exercise shows that (3×104)×(2×103)=6×107. Simple calculations like these can, in fact, be carried out quite easily without the need for a calculator, as we will explain below.

In calculations where terms are multiplied, the order of the terms isn't important and will not affect the result, so:

This image shows the following equation: open bracket, three times ten to the power of four, close bracket, times, open bracket, two times ten to the power of three, close bracket, equals three times two times ten to the power of four times ten to the power of thirty-one.
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This image shows the following equation: open bracket, three times ten to the power of four, close bracket, times, open bracket, two times ten to the power of three, close bracket, equals three times two times ten to the power of four times ten to the power of thirty-one.

Writing this in full would give:

This image shows the following equation: three times two times, open bracket, ten times ten, times ten, times ten, close bracket, times, open bracket, ten times ten, times ten, close bracket.
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This image shows the following equation: three times two times, open bracket, ten times ten, times ten, times ten, close bracket, times, open bracket, ten times ten, times ten, close bracket.

Since:

This image shows the following equation: ten times ten, times ten, times ten, times ten, times ten, times ten equals ten to the power of seven.
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This image shows the following equation: ten times ten, times ten, times ten, times ten, times ten, times ten equals ten to the power of seven.

we hope you can see that:

This image shows the following equation: ten to the power of four times ten to the power of three equals ten to the power of, open bracket, four plus three, close bracket, equals ten to the power of seven.
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This image shows the following equation: ten to the power of four times ten to the power of three equals ten to the power of, open bracket, four plus three, close bracket, equals ten to the power of seven.

and therefore:

This image shows the following equation: open bracket, three times ten to the power of four, close bracket, times, open bracket, two times ten to the power of three, close bracket, equals three times two, times ten to the power of, open bracket, four plus three, close bracket, equals six times ten to the power of seven.
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This image shows the following equation: open bracket, three times ten to the power of four, close bracket, times, open bracket, two times ten to the power of three, close bracket, equals three times two, times ten to the power of, open bracket, four plus three, close bracket, equals six times ten to the power of seven.

So when multiplying together two or more terms expressed in scientific notation, a shortcut to the result is to add the powers.

Sometimes the calculation will require a little more manipulation in order to express the result in scientific notation. For example:

This image shows the following equation, worked out over five lines: First line: open bracket, eight times ten to the power of six, close bracket, times, open bracket, three point five times ten to the power of three, close bracket. Second line: equals eight times three point five times ten to the power of, open bracket, six plus three, close bracket. Third line: equals twenty-eight times ten to the power of nine. Fourth line: two point eight times ten to the power of one, times ten to the power of nine. Fifth line: two point eight times ten to the power of ten.
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This image shows the following equation, worked out over five lines: First line: open bracket, eight times ten to the power of six, close bracket, times, open bracket, three point five times ten to the power of three, close bracket. Second line: equals eight times three point five times ten to the power of, open bracket, six plus three, close bracket. Third line: equals twenty-eight times ten to the power of nine. Fourth line: two point eight times ten to the power of one, times ten to the power of nine. Fifth line: two point eight times ten to the power of ten.

Similar principles can be used when dividing terms expressed in scientific notation. A shortcut to the result is to subtract the powers.

To demonstrate we'll evaluate (3×104) divided by (2×103):

This image shows the following fraction: three times ten to the power of four over two times ten to the power of three. The 10 in the denominator is crossed out. There is a gap, then the following fraction: three times ten, times ten, times ten, times ten over two times ten, times ten, times ten. Both numerator and denominator are followed by the equals sign, linking to the third fraction: three times ten over two. This is followed by the equation: one point five times ten to the power of one.
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This image shows the following fraction: three times ten to the power of four over two times ten to the power of three. The 10 in the denominator is crossed out. There is a gap, then the following fraction: three times ten, times ten, times ten, times ten over two times ten, times ten, times ten. Both numerator and denominator are followed by the equals sign, linking to the third fraction: three times ten over two. This is followed by the equation: one point five times ten to the power of one.

We hope you can see from this that:

This image shows the fraction: three times ten to the power of four over two times ten to the power of three. This is followed by the equation: three over two times ten to the power of open bracket four minus three, close bracket. The three in the four minus three is crossed out. Following this is the equation: one point five times ten to the power of one.
Show description|Hide description

This image shows the fraction: three times ten to the power of four over two times ten to the power of three. This is followed by the equation: three over two times ten to the power of open bracket four minus three, close bracket. The three in the four minus three is crossed out. Following this is the equation: one point five times ten to the power of one.

Activity 9: self-assessment

  1. Use the 'short-cut' method to evaluate the following, write down your answers and then check your results using the Windows calculator.

    1. (5×102)×(7×103)

    2. This image shows the second part of activity eleven, a. It shows the fraction four times ten to the power of five, over two times ten to the power of two.

  2. Use the Windows calculator to evaluate the following, write down your answers correct to three significant figures.

    1. (8.55×104)×(5.04×106)

    2. (5.24×102)×(7.53×103)

Answer

Answers

  1. 3.5×106

  2. 2×103

  3. 4.31×1011

  4. 3.95×106

If any of your answers differ from ours, it is probably because you entered an incorrect value or clicked on the wrong key. If this happens, try the calculation again.