8.06 Standard form
Standard form (or scientific notation) is a compact way of writing very big or very small numbers. For example:
- The sun has a mass of approximately $1.988\times10^{30}$1.988×1030kg which is much easier to write than $1988000000000000000000000000000$1988000000000000000000000000000kg
- The mass of at atom of Uranium (one of the heaviest elements) is only approximately $3.95\times10^{-22}$3.95×10−22g. That is $0.000000000000000000000395$0.000000000000000000000395g.
In standard form, numbers are written in the form $a\times10^b$a×10b, where $a$a is a number between $1$1 and $10$10 and $b$b is an integer (positive or negative).
- A negative index indicates how many factors of ten smaller than $a$a the value is.
- A positive index indicates how many factors of ten larger than $a$a the value is.
- A index of zero indicates that the value is $a$a because $10^0=1$100=1.
Worked examples
Example 1
Express $63300$63300 in standard form.
Think: We need to express the first part of standard form 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 standard form 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 the decimal point has shifted). So in standard form, we would write this as $6.33\times10^4$6.33×104.
Example 2
Express $0.00405$0.00405 in standard form.
Think: We need to express the first part of standard form as a number between $1$1 and $10$10 and then work out the index of $10$10 required to to make the number equivalent.
Do:
To find the first part of our standard form we place the decimal point after the first non-zero number, so$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 the decimal point has shifted or the number of zeros including the one before the decimal point). So in standard form, we would write this as $4.05\times10^{-3}$4.05×10−3.
We will often use our calculator to evaluate expressions with standard form. 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 standard form.
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 standard form, 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 standard form.
| $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
Use index laws to simplify $\frac{4\times10^{-5}}{16\times10^4}$4×10−516×104. Give your answer in scientific notation.
Question 2
If we round to $1$1 significant figure, sound travels at a speed of approximately $0.3$0.3 kilometres per second, while light travels at a speed of approximately $300000$300000 kilometres per second.
Express the speed of sound in kilometres per second in scientific notation.
Express the speed of light in kilometres per second in scientific notation.
How many times faster does light travel than sound?
Standard form and calculators
Calculators will often display numbers in standard form 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 standard form. 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 standard form.
For example, to write $1.5\times10^9$1.5×109 on this calculator, you would press:
