7.03 Scientific notation

Worked examples

Example 1

Express $63300$63300 in scientific notation.

Think: We need to express the first part of scientific notation as a number between $1$1 and $10$10 and then work out the index of $10$10 required to to make the number equivalent. 

Do: 

To find the first part of our scientific notation we place the decimal point after the first non-zero number, so$a=6.33$a=6.33.

To find the power of ten, ask how many factors of ten bigger is $63300$63300 than $6.33$6.33?

$63300$63300 is $10000$10000 or $10^4$104 times bigger than $6.33$6.33. (You can also see this by counting how many places to the right the decimal point has shifted). So in scientific notation, we would write this as $6.33\times10^4$6.33×104.

Example 2

Express $0.00405$0.00405 in scientific notation.

Think: We need to express the first part of scientific notation as a number between $1$1 and $10$10 and then work out the index of $10$10 required to to make the number equivalent. 

Do: 

To find the first part of our scientific notation we place the decimal point after the first non-zero number, so$a=4.05$a=4.05.

To find the power of ten, ask how many factors of ten smaller is $0.00405$0.00405 than $4.05$4.05?

$0.00405$0.00405 is $1000$1000 or $10^3$103 times smaller than $4.05$4.05. (You can also see this by counting how many places to the left the decimal point has shifted or the number of zeros including the one before the decimal point). So in scientific notation, we would write this as $4.05\times10^{-3}$4.05×10−3.

We will often use our calculator to evaluate expressions with scientific notation. However, knowing our index laws we can manipulate calculations that are relatively straightforward or estimate the size of answers for more complex calculations.

Example 3

Use index laws to simplify $2\times10^6\times6\times10^5$2×106×6×105. Give your answer in scientific notation.

Simplifying first we find:

$2\times10^6\times6\times10^5$2×106×6×105	$=$=	$12\times10^{6+5}$12×106+5
	$=$=	$12\times10^{11}$12×1011

Then we need to adjust our answer to obtain scientific notation, as the first number is larger than ten.

$12$12 can be as expressed as $1.2\times10^1$1.2×101. We will use this to write our answer in scientific notation.

$1.2\times10^1\times10^{11}$1.2×101×1011	$=$=	$1.2\times10^{1+11}$1.2×101+11
	$=$=	$1.2\times10^{12}$1.2×1012

 

Practice questions

Question 1

Question 2

 

Scientific notation and calculators

Calculators will often display numbers in scientific notation but the format may vary between different models. A common variation from showing $2.95\times10^8$2.95×108 is the display $2.95$2.95E$8$8 where the E is for exponent of $10$10. Most calculators will also have a button for entering numbers in scientific notation. This may look like $\times10^x$×10x or like the button EXP circled in blue in the picture. Look carefully at your calculator and ensure you are familiar with the display format and syntax for entering numbers in scientific notation.

 

For example, to write $1.5\times10^9$1.5×109 on this calculator, you would press: 

 

Practice question

Question 3

 

Significant figures

We often round numbers to indicate the precision to which measurements were made and sometimes for convenience. If reporting on a crowd at a concert it may be convenient to state there was a crowd of $60000$60000 fans rather than $59759$59759. If stating the speed of a car to be $76.82$76.82 km/h it appears as though the measurement is accurate to the nearest hundredth of a km/h but what measurements were used in the calculation? Can we justify showing so many figures? Are they all significant? Questions may often indicate the number of significant figures required for the answer and in science it is common practice to give answers to the same number of significant figures as the measurement with the least number of significant figures.

Here is an illustration of this:

Note: In our example of a concert reported as having $60000$60000 attendees, when in fact it had $59759$59759, we cannot tell if the report has been given to one or two significant figures, since both would round to $60000$60000. Often we indicate after an answer to be clear. For example $60000$60000 ($2$2 significant figures) or $60000$60000 ($2$2 s.f.) for short.

 

Worked example

Example 4

Express the following to three significant figures:

(a) $10432$10432

$10432$10432 to $3$3 significant figures $=$= $10400$10400    The zero between the $1$1 and the $4$4 is counted as significant.

(b) $2.4983$2.4983

$2.4983$2.4983 to $3$3 significant figures $=$= $2.50$2.50     Remember to state the trailing zero if necessary.

(c) $6.53126\times10^7$6.53126×107

$6.53126\times10^7$6.53126×107 to $3$3 significant figures $=$= $6.53\times10^7$6.53×107     

 

Practice questions

Question 4

Question 5

Question 6