Many values in science are rather large or incredibly small. It is inconvenient to write these values as standard numbers. For instance, one mole is equal to 602,200,000,000,000,000,000,000 particles and the diameter of an atom is about 0.0000000001 meters. Such values are better represented in scientific notation.
Scientific notation uses exponents to denote large and small values. Each exponent represents a placeholder, or the number of places a decimal needs to be moved to make a standard number. These values can be used in math problems to find solutions.
Example: 5.23 x 108
The value must be a number from 1 to 9 (such as the 5), followed by x 10 to an exponent (such as the 8).
This example means that 5.23 would need its decimal moved 8 times to the right to turn it into a standard number, making it 532,000,000.
If the exponent is negative, move the decimal point to the left.
Example: 8.43 x 10-6
This would become 0.00000843
You can only add and subtract values in scientific notation with the same order of magnitude. In other words, the same power of 10. This is because of the scale of the numbers.
Example: You can add 8.5 x 103 and 1.2 x 103 to get 9.7 x 103. However, you cannot directly add 3.4 x 102 and 6.0 x 105
To add or subtract, you must first convert all numbers to standard numbers, then perform the operation, then convert back to scientific notation.
Example: 3.4 x 102 and 6.0 x 105 → 340 + 600000 = 600340 → 6.0 x 105.
In this case, the 3 x 102 value is insignificant
compared to 6.0 x 105. Why do we drop the 0340? Because of significant figures.
Numbers in scientific notation can be easily multiplied by calculator. First, let's look at how to do so by hand.
Example: 4.2 x 106 x 9.82 x 103
To solve by hand, multiply the two base number as usual: 4.2 x 9.82 = 41.244
Next, add the exponents together: 106 x 103 = 109
Put these two values together: 41.244 x 109
This is not proper scientific notation because the number before the decimal is 41, and it must be a number from 1 to 9. To get this, we need to move the decimal one more space to the left, which means we also need to increase the exponent by 1.
Answer: 4.1244 x 1010
Division works the same basic way, except you subtract the exponents.
5.2 x 107
2.1 x 103
Divide the base values: 5.2 / 2.1 = 2.476
Divide the exponents by subtracting them: 107 / 103 = 104
Put these values together to get: 2.476 x 104
In this example, the number before the decimal is 2 so no other conversion is needed.
When using a calculator, you need to use the correct buttons.
When inputting an exponent, find and use the EE button.
To input 4 x 107, type in 4 EE 7
To input a negative exponent, you need to know how your calculator works. You must use the negative (-) sign, not the subtraction button.
On a graphing calculator, you typically enter the value as shown: use 4 EE -3 to input 4 x 10-3.
Other calculators may require you to input the negative sign after typing the exponent. 4 EE 3 (-)