10.1.6 Small numbers and Scientific Notation

In the preceding section you have seen how to write numbers that are less than 1 using powers of ten. So you can now write numbers less than 1 in scientific notation. For example,

equation sequence 0.003 equals three divided by 100 equals three divided by 10 multiplication 10 equals three divided by 10 squared equals three multiplication 10 super two minus

An alternative way to write a number less than 1 in scientific notation is first to write it down as a number between 1 and up to but not including 10; in this case it is 3. Then divide this number by 10 repeatedly until you reach the number required. Each time you divide by 10, the power of 10 reduces by 1. So here you have to divide 3 by 10 twice to get 0.03, so 0.03 equals three prefix multiplication of 10 super two minus.

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 four minus equals 5.72 divided by 10 super four equals 5.72 divided by 10 comma 000 equals 5.72 division 10 comma 000 equals 0.000 57

Activity Symbol Activity: Understanding Small Numbers in Scientific Notation

Write down the following numbers as decimals.

(a) three multiplication 10 super four minus

Solution Symbol

Answer

(a) equation sequence three multiplication 10 super four minus equals three divided by 10 super four equals three divided by 10 multiplication 10 multiplication 10 multiplication 10 equals three divided by 10 comma 000 equals 0.0003

(b) 7.5 prefix multiplication of 10 super five minus

Solution Symbol

Answer

(b) equation sequence 7.5 multiplication 10 super five minus equals 7.5 divided by 10 super five equals 7.5 divided by 10 multiplication 10 multiplication 10 multiplication 10 multiplication 10 equals 0.000 075

(c) 1.46 prefix multiplication of 10 super three minus

Solution Symbol

Answer

(c) equation sequence 1.46 multiplication 10 super three minus equals 1.46 divided by 10 cubed equals 1.46 divided by 10 multiplication 10 multiplication 10 equals 0.00146

Notepad iconActivity: Writing Small Numbers in Scientific Notation

Write down the following numbers in scientific notation.

(a) 0.000007

Solution Symbol

Answer

(a) 0.000 equation sequence 007 equals seven divided by one comma 000 comma 000 equals seven divided by 10 super six equals seven multiplication 10 super six minus

(b) 0.0742

Solution Symbol

Answer

(b) equation sequence 0.0742 equals 7.42 divided by 100 equals 7.42 divided by 10 squared equals 7.42 multiplication 10 super two minus

(c) 0.0000000098

Solution Symbol

Answer

(c) 0.000 000 009 equation sequence eight equals 9.8 divided by one comma 000 comma 000 comma 000 equals 9.8 divided by 10 super nine equals 9.8 multiplication 10 super nine minus

Before we move on, think about how scientific notation fits in with what you already know about decimals. Earlier, you saw that when you multiplied a number by 10, the decimal point in the number moved one place to the right, and when you divided a number by 10, the decimal point moved one place to the left. So, extending this idea, if you multiply a number by 10 cubed, the decimal place will move three places to the right to make the number bigger. For example, 4.5 prefix multiplication of 10 super three equals four comma 500. Similarly, if you multiply a number by 10 super four minus, which is the same as dividing by 10 super four, the decimal point will move four places to the left to make the number smaller. For example, two multiplication 10 super four minus equals 0.0002.

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

10.1.5 Negative Exponents

10.1.7 Calculator Exploration: Scientific Notation