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Scientific Notation

Until now, we have been writing numbers in "decimal notation." Sometimes,
especially with large numbers, we need to convert numbers into scientific
notation.

To write a number in scientific notation, we write it as a product of a single
digit and a power of 10. Here are the steps to writing a number in scientific
notation:

- Write the first non-zero digit of the number times a power of ten--see
Exponents
and Negative Exponents.
- Place a decimal point after the single digit number, and put the remaining
digits in the same order after the decimal point. If the number is a whole
number that ends with zeros, drop the zeros.

Thus, to write 527 in scientific notation:

- Write the first digit times a power of ten: 500 = 5×10
^{2}

- Put the remaining digits in order after a decimal point:
5.27×10
^{2}

527 = 5.27×10^{2}
To write 1,108.4 in scientific notation:

- 1, 000 = 1×10
^{3} - 1.1084×10
^{3} (Note that there is no decimal point between 8 and 4)

1, 108.4 = 1.1084×10^{3}
To write 0.0963 in scientific notation:

- 0.09 = 9×10
^{-2}

- 9.63x10^

0.0963 = 9.63×10^{-2}
To write 78,000 in scientific notation:

- 70, 000 = 7×10
^{4}

- 7.8×10
^{4} (Note that 78,000 is a whole number, so we dropped the
zeros)

78, 000 = 7.8×10^{4}
To write 15.200 in scientific notation:

- 10 = 1×10
^{1}

- 1.5200×10
^{1} (Note that 15.200 is a decimal, so we did not drop
the zeros)

15.200 = 1.5200×10^{1}
Observe: the exponent on "10" corresponds to the number of places that the
decimal point has moved--it is positive if the decimal point has moved to the
left and negative if it has moved to the right.

One of the trickiest things about scientific notation is remembering the rules
for zeros: if a number ends in one or more zeros, *do not* include the
zeros if the number is a whole number, but *do* include the zeros if the
number is a decimal. For example, 820 = 8.2×10^{2} in scientific
notation, and 0.820 = 8.20×10^{-1} in scientific notation. Zeros in the middle of a number are treated as normal digits.

Scientific notation makes it easy to compare very large (or very small numbers).
The number with a larger exponent on "10" is always greater. For example,
6.7103×10^{13} is greater than 9.2×10^{7} and
8.3×10^{-5} is greater than 2.3×10^{-11}.