Arithmetic with Numbers in Scientific Notation

Very large or very small numbers are sometimes difficult to handle. There may be many leading zeros after a decimal point of very small numbers, or there may be many trailing zeros on the end of very large numbers. For example, in chemistry, Avogadro’s number gives the number of molecules in a mole of a substance. This number is 602,000,000,000,000,000,000,000. Rather than writing out all of those zeros, we can write Avogadro’s number as 6.02 x 10²³. This is Avogadro’s number written in scientific notation.

A positive number is in scientific notation when it is written as the product of a number greater than or equal to 1 and less than 10 and an integer power of ten.

A negative number is in scientific notation when it is written as the product of a number less than or equal to -1 and greater than -10 and a power of ten.

For more review of exponents, review the tutorial Properties of Exponents.

Example 1

Write 2456 in scientific notation.

Solution 1

The decimal in 2456 is presently after the 6 2456.

Using the digits 2456 in that order, we would need 2.456 times some power of 10 in order to be in scientific notation, because we are trying to write 2456 as a product of a number between 1 and 10, multiplied by an integer power of 10. To go from 2456 to 2.456, we had to divide by 1000 (or 10³), so to offset this division, we need to multiply by 10³.

Consequently, 2456 = 2.456 x 10³ in scientific notation.

Example 2

Write 1.234 x 10^-4 as a decimal.

Solution 2

10^-4 = . In order to multiply 1.234 by , we must move the decimal point 4 places to the left .0001234

Note that 10^-4 is less than one, so the final number will be smaller than the original.

Example 3

(5.234 x 10⁵)·(2.45 x 10^-3)

Solution 3

Properties of multiplication include associative and commutative. The multiplication here is equal to:
5.234 x 10⁵ x 2.45 x 10^-3

= (5.234 x 2.45) x (10⁵ x 10^-3)

= 12.8233 x 10²
(add exponents to multiply powers of 10)

= 1.28233 x 10³

Note that 12.8233 is larger than 10, so the decimal point had to go one more space to the left, adding one to the exponent on 10.

Example 4

(2.65 x 10⁴) / (5.92 x 10^-8)

Solution 4

It might be easier to express this division problem as a fractional expression:
            (2.65 x 10⁴) / (5.92 x 10^-8)    =
                                                            =

                                                            ≈ .448 x 10¹²
                                                            (subtract exponents to divide powers of 10)

                                                            ≈ 4.48 x 10¹¹

Note that .448 is less than 1, so the decimal point had to go one more space to the right, subtracting one from the exponent on 10.

Notice the change from = to . When we divided 2.65 by 5.92, we only kept 3 significant digits in the result. Often, the numbers we use in such situations are measurements, and they have only limited accuracy. It wouldn’t make sense to do arithmetic with these numbers and gain accuracy!