5.4: Decimals- Scientific Notation
Learning Outcomes
- Convert from scientific notation to decimal notation and back.
In this section, we will look at how to read scientific notation. A very common error that statistics students make is not noticing that the calculator is giving an answer in scientific notation.
Scientific Notation
When a calculator presents a number in scientific notation, we must pay attention to what this represents. The standard way of writing a number in scientific notation is writing the number as a product of a number greater than or equal 1 but less than 10 followed by a power of 10. For example:
\[ 602,000,000,000,000,000,000,000 = 6.02 \times 10^{23} \nonumber\]
The main purpose of scientific notation is to allow us to write very large numbers or very small numbers close to 0 without having to use so many digits. Most calculators and computers use a different notation for scientific notation, most likely because the superscript is difficult to render on a screen. For example, with a calculator:
\[0.00000032 = 3.2E-7 \nonumber\]
Notice that to arrive at 3.2, the decimal needed to be moved 7 places to the right. The "E" represents "times 10 raised to the power of."
Example \(\PageIndex{1}\)
A calculator displays:
\[2.0541E6 \nonumber\]
Write this number in decimal form.
Solution
Notice that the number following E is 6. This means move the decimal over 6 places to the right. The first 4 moves is natural, but for the last 2 moves, there are no numbers to move the decimal place past. We can always add extra zeros after the last number to the right of the decimal place:
\[2.0541E6 = 2.054100E6 \nonumber\]
Now we can move the decimal place to the right 6 places to get
\[2.0541E6 = 2.054100E6 = 2,054,100 \nonumber \]
Example \(\PageIndex{2}\)
If you use a calculator or computer to find the probability of flipping a coin 27 times and getting all heads, then it will display:
\[ 7.45E−9 \nonumber\]
Write this number in decimal form.
Solution
Many students will forget to look for the "E" and just write that the probability is 7.45, but probabilities can never be bigger than 1. You can not have a 745% chance of it occurring. Notice that the number following E is −9. Since the power is negative, this means move the decimal to the left, and in particular 9 places to the left. There is only one digit to the left of the decimal place, so we need to insert 8 zeros:
\[ 7.45E−9 = 000000007.45E−9 \nonumber\]
Now we can move the decimal place to the right 9 places to the left to get
\[ 7.45E−9 = 000000007.45E−9 = 0.00000000745 \nonumber\]
This is a very small probability and essentially rounds to 0.