In the previous example you should have noticed that the answer is presented in what is called scientific notation.
Scientific notation…
…is a way to express very small or very large numbers
…is most often used in "scientific" calculations where the analysis must be very precise
…consists of two parts: A Number and a Power of 10. Ex: 1.22 x 103
For a number to be in correct scientific notation only one digit may be to the left of the decimal. So,
\begin{align} 1.22 & \times 10^3 \text{ is correct} \\ 12.2 & \times 10^2 \text{ is not} \end{align}
How to convert non-exponential numbers to exponential numbers:
Example 1
$$ 234,999 $$
This is a large number and the implied decimal point is at the end of the number.
$$ 234,999. $$
To convert this to an exponential number we need to move the decimal to the left until only one digit resides in front of the decimal point. In this number we move the decimal point 5 times.
$$ 2.34999 \text{ (five numbers)} $$
…and thus the exponent we place on the power of 10 is 5. The resulting exponential number is then:
$$2.34999 \times 10^5 $$
Other examples:
\begin{align} 21 & \to 2.1 \times 10^1 \\ 16600.01 & \to 1.660001 \times 10^4 \\ 455 & \to 4.55 \times 10^2 \end{align}
Small numbers can be converted to exponential notation in much the same way. You simply move the decimal to the right until only one non-zero digit is in front of the decimal point. The exponent then equals the number of digits you had to pass along the way.
Example 2
$$ 0.000556 $$
The first non-zero digit is 5 so the number becomes 5.56 and we had to pass the decimal point by 4 digits to get it to the point where there was only one non-zero digit at the front of the number so the exponent will be -4. The resulting exponential number is then:
$$ 5.56 \times 10^{-4} $$
Other examples
\begin{align} 0.0104 & \to 1.04 \times 10^{-2} \\ 0.0000099800 & \to 9.9800 \times 10^{-6} \\ 0.1234 & \to 1.234 \times 10^{-1} \end{align}
So to summarize, moving the decimal point to the left yields a positive exponent. Moving the decimal point to the right yields a negative exponent.
Another reason we often use scientific notation is to accommodate the need to maintain the appropriate number of significant figures in our calculations.
Significant Figures
There are three rules on determining how many significant figures are in a number:
- Non-zero digits are always significant.
- Any zeros between two significant digits are significant.
- A final zero or trailing zeros in the decimal portion ONLY are significant.
Examples
- 2003 has 4 significant figures
- 00.00300 has 3 significant figures
- 00067000 has 2 significant figures
- 00067000.0 has 6 significant figures
Exact Numbers
Exact numbers, such as the number of people in a room, have an infinite number of significant figures. Exact numbers are counting up how many of something are present, they are not measurements made with instruments. Another example of this are defined numbers, such as
$$ 1 \text{ foot} = 12 \text{ inches} $$
There are exactly 12 inches in one foot. Therefore, if a number is exact, it DOES NOT affect the accuracy of a calculation nor the precision of the expression. Some more examples:
- There are 100 years in a century.
- Interestingly, the speed of light is now a defined quantity. By definition, the value is 299,792,458 meters per second.
In order to present a value in the correct number of significant digits you will often have to round the value off to that number of digits. Below are the rules to follow when doing this:
The application of significant figures rules while completing calculations is important and there are different ways to apply the rules based on the type of calculation being performed.
Significant Figures and Addition or Subtraction
In addition and subtraction the number of significant figures that can be reported are based on the number of digits in the least precise number given. Specifically this means the number of digits after the decimal determine the number of digits that can be expressed in the answer.
Example
Significant Figures and Multiplication or Division
In multiplication and division the number of significant figures is simply determined by the value of lowest digits. This means that if you multiplied or divided three numbers: 2.1, 4.005 and 4.5654, the value 2.1 which has the fewest number of digits would mandate that the answer be given only to two significant figures.