Many students are daunted by this necessary brush with mathematics, but it
is not as difficult as it may seem. The basic idea is to
*simplify* large and small numbers by
allowing us to deal only with numbers between 1 and 10 and with powers of
10.

To get to the most basic level, what is meant by a power of 10? Simply put,
it is a number that can be arrived at by multiplying 10's together. Such
numbers can be written in the form 10^{n}, where *n* is
the (integer) number of 10's used as factors. Thus

Note that 10

That works as long as *n* is positive, what if *n* is negative?
A number raised to a negative power is simply 1 divided by the number raised
to the positive power:

Thus

Note that, counting the zero before the decimal point, there are

Here is a table with some powers of 10:

Power of 10 | Factorized | Number | "Translation" |
---|---|---|---|

10^{9} |
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 | 1,000,000,000 | 1 billion |

10^{6} |
10 x 10 x 10 x 10 x 10 x 10 | 1,000,000 | 1 million |

10^{5} |
10 x 10 x 10 x 10 x 10 | 100,000 | 1 hundred thousand |

10^{4} |
10 x 10 x 10 x 10 | 10,000 | ten thousand |

10^{3} |
10 x 10 x 10 | 1,000 | 1 thousand |

10^{2} |
10 x 10 | 100 | 1 hundred |

10^{1} |
10 | 10 | ten |

10^{0} |
1 | one | |

10^{-1} |
1/10 | 0.1 | 1 tenth |

10^{-2} |
1/(10x10) | 0.01 | 1 hundredth |

10^{-3} |
1/(10x10x10) | 0.001 | 1 thousandth |

10^{-4} |
1/(10x10x10x10) | 0.0001 | 1 ten-thousandth |

10^{-5} |
1/(10x10x10x10x10) | 0.00001 | 1 hundred-thousandth |

Multiplying (or dividing) powers of 10 is easy: you just add (or
subtract) the exponents.

10

So, for example, 10

Any positive number can be expressed as the product of power of 10 with
a number between 1 and 10. Negative numbers can be expressed by negating
such a product. Thus, for any non-zero number *X*, we can write

where the exponential

(Zero cannot be expressed by a product satisfying these conditions, and remains simply 0).

Multiplying (dividing) numbers in scientific notation simply involves
multiplying (dividing) the two coefficientss and separately multiplying
(dividing) the powers of 10:

(

If the resulting product

(-4 x 10

(8.4 x 10

(1.989 x 10

If adding or subtracting numbers in scientific notation, it is necessary to
ensure that the powers of ten are the same. This may require factoring some
of the powers of 10 out of one of the exponentials and into the coefficient.
This will (temporarily) create a number that is not in standard form (the
coefficient will not be between 1 and 10). The process can be written as
follows:

Again, a couple examples may be helpful:

(6.02 x 10

(1.989 x 10

You may notice that I usually factor out the larger power of 10. This is

Notice that if the exponentials are very different, the smaller one can be treated as effectively zero. As a rule-of-thumb, if the difference in the exponents (6 in the case above) is bigger than the number of decimals in the coefficient of the larger number (3 in the case above), you can ignore the smaller number in additions and subtractions. The result of the previous example would thus be 1.989 x 10

*Note regarding scientific notation on calculators
*

Most calculators and computers cannot display exponentials, and are
therefore unable to explicitly show scientific notation in the correct
form. *(Actually, cannot is too strong a word, since any modern computer
is capable of displaying an exponential, but for historic reasons and for
compatibility with "plain text" output, you will find the they generally
don't use the exponential format for scientific notation.)*
The usual format used for printing scientific notation is to replace
"x 10" with "E" or "e", so your calculator may display
1.3 E05 instead of 1.3 x 10^{5}.
Similarly, when inputting a number in scientific
notation into a calculator, there is usually a button labeled "EE" or "E"
that indicates the exponent (it is pushed at the point where "x 10" appears
in the number). So, if you wanted to enter
1.3 x 10^{5} into
your calculator, you would press
1_._3_EE_5
(where I have used the underscore to separate individual key presses).

Converting from fixed to scientific notation:

Algebraically,nis the logarithm of |X|, rounded down to the nearest integer.ais thenX/10^{n}.In practical terms, if |

X| is greater than 1,nis one less than the total number of digits to the left of the decimal point. If |X| is less than one,nis the negative of the number of zeros (including the one to the left of the decimal point) before the first non-zero digit.acan be obtained by moving the decimal point so that there is only one non-zero digit to the left of it.

Converting from scientific to fixed notation:

Algebraically, multiplyaby 10^{n}to getX.In practical terms, remove the decimal point from

a. Ifnis positive, reposition the decimal point so that the total number of digits to the left of the decimal point isn+1 (pad with zeros as needed). Ifnis negative, add |n| zeros to the left of the first (leftmost) digit ofa, and place the decimal point between the first and second zero.