Scientific notation or standard form is a compact way of writing very big or very small numbers. As the name suggests, scientific notation is frequently used in science. 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 scientific notation, numbers are written in the form $a\times10^n$a×10n, where $a$a is a decimal number between $1$1 and $10$10 and $n$n is an integer (positive or negative).
- A negative exponent indicates how many factors of ten smaller than $a$a the value is.
- A positive exponent 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 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 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 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 exponent laws we can manipulate calculations that are relatively straightforward or estimate the size of answers for more complex calculations.
Example 3
Use exponent 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
Express the following number as a basic numeral:
$2\times10^7$2×107
Question 2
Express the following number in scientific notation:
$84245000$84245000Question 3
Express the following number as a decimal number:
$3.62\times10^{-4}$3.62×10−4