Scientific notation and significant figures
The mass of the Sun is roughly \(1,989,100,000,000,000,000,000,000,000,000 \text{ kg}\). The charge of an electron is around \(0.0000000000000000001602 \text {C}\). These, and many more, real-world quantities are either very large or very small numbers that have an unwieldy number of digits when it comes to either writing or performing calculations with them. For this reason, another way of expressing these numbers is required.
Significant figures (or significant digits) are the digits in a number that carry meaningful information about its precision. It is essential in scientific fields to correctly identify and maintain the correct number of significant figures.
These concepts help ensure that measurements and calculations are represented accurately and consistently.
Use this page to revise the following concepts within scientific notation and significant figures:
- Expressing numbers in scientific notation
- Identifying significant figures: rules
- Scientific notation
- Significant figures
Expressing numbers in scientific notation
Scientific notation is a way of expressing very large or very small numbers in a compact form. It is written as the product of a number between 1 and 10, and a power of 10. This method is especially useful in science and engineering to conveniently handle extreme values.
Worked Example
- The number \(300,000,000\) can be written in scientific notation as \(3.0\times10^{8}\)
- The number \(0.000042\) can be written as \(4.2\times10^{-5}\)
Identifying significant figures: rules
The rules for determining the number of significant figures are:
1. Non-zero digits are always significant.
2. Any zeros between significant digits are significant.
3. Leading zeros (zeros before non-zero digits) are not significant.
4. Trailing zeros in a decimal number are significant.
Worked Example
- \(123.45\) has five significant figures.
- \(0.0076\) has two significant figures (7 and 6).
- \(100.0\) has four significant figures.
Significant figures can also affect how a number is written in scientific notation .
Worked Example
\(0.000760\) written in scientific notation is \(7.60\times10^{-4}\)
However, \(0.00076\) written in scientific notation is \(7.6\times10^{-4}\)
Scientific Notation
In scientific notation, numbers are written in the form \(a\times\ 10^{n}\), where \(a\) is a decimal number between \(1\) and \(10\) and \(n\) is an integer (positive or negative).
- A negative exponent indicates how many factors of ten smaller than \(a\) the value is.
- A positive exponent indicates how many factors of ten larger than \(a\) the value is.
- An index of zero indicates that the value is \(a\) because \(10^{0}=1\)
Worked Example
Express \(63,300\) in scientific notation.
Place the decimal point after the first non-zero number, so \(a=6.33\).
Determine how many places the decimal place needed to move from the original value to become \(6.33\). In this case it is \(4\). As the number is greater than \(10\), the exponent will be a positive \(4\).
Hence \(63,300\) is written in scientific notation as \(6.33\times10^{4}\).
Worked Example
Express \(0.00405\) in scientific notation.
Place the decimal point after the first non-zero number, so \(a=4.05\).
Determine how many places the decimal place needed to move from the original value to become \(4.05\). In this case it is \(3\). As the number is less than \(10\), the exponent will be a negative \(3\).
Hence \(0.00405\) is written in scientific notation as \(4.05\times10^{-3}\).
Check your understanding
View
Significant Figures
Significant figures represent the digits that convey how precise a measurement is.
Consider measuring the height of a person. One might use a measuring tape to the nearest centimetre. They may measure the height to be \(165\text{ cm}\), which has three digits to tell us the precision of our measurement.
The number of digits indicating the level of precision of the measurement is the number of significant figures.
Here there are three digits and three significant figures.
This measurement could also have been done in kilometres, where the height would be \(0.00165\). This is still measuring to the nearest centimetre; the same precision.
As the precision has not changed, the number of significant figures does not change. The leading zeros do not affect the number of significant figures.
By converting the height to millimetres to get \(1650 \text{ mm}\) does not necessarily add to the precision. It is unclear whether this is accurate to the nearest millimetre. As such, the number of significant figures is \(3\) or \(4\).
However, if measured again to the nearest millimetre, and found that it were exactly \(1650 \text{ mm}\), there would be more precision, and so \(4\) significant figures.
It does not matter if it is now converted back to kilometres for \(0.001650 \text{ km}\). It is still measured to the nearest millimetre, hence the final zero. So \(0.001650 \text{ km}\) has \(4\) significant figures.
| Rule | Example |
|---|---|
| All non-zero digits are significant. | In the number \(12.34\), all digits are non-zero, and therefore significant. |
| Any zeroes between non-zero digits are always significant. | In the number \(1001.2\), both zeroes are significant because they appear between non-zero digits. This number has five significant figures. |
| Zeros that appear after the last non-zero digit in a decimal number are significant because they indicate additional precision. | Both \(.222\) and \(.200\) have three significant figures, as there are three digits following the decimal point. |
Zeros at the end of a number without a decimal point are not significant unless explicitly indicated by additional context or notation. | The number \(1000\) may have \(1\), \(2\), \(3\) or \(4\) significant figures depending on the context. |
| Leading zeroes – zeroes before the first non-zero number – are not significant. | The numbers \(0145\) and \(0.00145\) both have three significant figures. The zeroes before the \(1\) are not significant. |
Worked Example
A value has been rounded to give \(54100\).
What is the smallest number of significant figures that \(54100\) could have?
Locate the first and last nonzero digit. Count all places from first to last.
The first nonzero digit is \(5\) and the last is \(1\). There are \(3\) digits from \(5\) to \(1\).
\(54100\) could have \(3\) significant figures.