Succeed with maths – Part 2
Succeed with maths – Part 2

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Succeed with maths – Part 2

2.5 Small numbers and scientific notation

By using negative powers of ten it is possible to write numbers less than one in scientific notation. Let’s take the example of 0.03.

0.03 is three one hundredths, so this can be shown as a fraction, with 100 as the denominator. This is important, as 100 is a multiple of 10, and can therefore be shown using powers of 10 as follows:

equation sequence 0.03 equals three divided by 100 equals three divided by 10 multiplication 10 equals three divided by 10 squared

From the previous section you know that equation left hand side one divided by 10 squared equals right hand side 10 super minus two.

Hence, 0.03 equals three multiplication 10 super negative two

Now 0.03 is shown in scientific notation, since there is a number between 1 and 10, that is then multiplied by a power 10.

This shows you why the process works, but it may not necessarily be the most straightforward way of writing numbers that are less than one in scientific notation. Another way of thinking about this is in two steps, as with large numbers:

  • Write the number as one between 1 and 10.
  • From this, decide on the power of 10 required.

To establish the power of 10 required, work out how many times you would need to divide the number from step 1 by ten to reach the original number. Each time you divide by 10, the negative power of 10 reduces by 1, starting from -1.

Following these steps for 0.03 again:

  1. Write the number as one between 1 and 10.

This gives us 3.

  1. From this, decide on the power of 10 required.

To do this divide by 10 and then by 10 again to return to the original number. Hence, the negative power will be negative two times open negative one minus one close.

To convert a number in scientific notation back into decimal form, write down the negative power of 10 as a fraction and then divide the numerator by the denominator. For example:

equation sequence 5.72 multiplication 10 super negative four equals 5.72 divided by 10 super four equals 5.72 divided by 10 000 equals 5.72 division 10 000 equals 0.00057

Now use what you have learned in this section, as well as your previous knowledge from the week, to complete the following activity and hone your skills.

Activity _unit6.2.5 Activity 6 Understanding and writing small numbers in scientific notation

Timing: Allow approximately 10 minutes

Write down the following numbers as decimals.

  • a.three multiplication 10 super negative four

Discussion

Start by converting the negative power into a positive power, hence showing the number as a fraction.

Answer

  • a.equation sequence three multiplication 10 super negative four equals three divided by 10 super four equals three divided by 10 multiplication 10 multiplication 10 multiplication 10 equals three divided by 10 000 equals 0.0003
  • b.7.5 multiplication 10 super negative five

Answer

  • b.equation sequence 7.5 multiplication 10 super negative five equals 7.5 divided by 10 super five equals 7.5 divided by 10 multiplication prefix multiplication of 10 multiplication 10 multiplication 10 multiplication 10 equals 7.5 divided by 100 000 equals 0.000075
  • c.1.46 multiplication 10 super negative three

Answer

  • c.equation sequence 1.46 multiplication 10 super negative three equals 1.46 divided by 10 cubed equals 1.46 divided by 10 multiplication 10 multiplication 10 equals 1.46 divided by 1000 equals 0.00146

Write down the following numbers in scientific notation:

  • d.0.000007

Answer

  • d.equation sequence 0.000007 equals seven divided by one 000 000 equals seven divided by 10 super six equals seven multiplication 10 super negative six
  • e.0.0742

Answer

  • e.equation sequence 0.0742 equals 7.42 divided by 100 equals 7.42 divided by 10 squared equals 7.42 multiplication 10 super negative two
  • f.0.0000000098

Answer

  • f.equation sequence 0.0000000098 equals 9.8 divided by one 000 000 000 equals 9.8 divided by 10 super nine equals 9.8 multiplication 10 super negative nine

If you want some more practice with scientific notation before you move on to look at how to use it on a calculator, have a go at a scientific notation game [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] .

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