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 10n, where n is
the (integer) number of 10's used as factors. Thus
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:
Here is a table with some powers of 10:
|Power of 10||Factorized||Number||"Translation"|
|109||10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10||1,000,000,000||1 billion|
|106||10 x 10 x 10 x 10 x 10 x 10||1,000,000||1 million|
|105||10 x 10 x 10 x 10 x 10||100,000||1 hundred thousand|
|104||10 x 10 x 10 x 10||10,000||ten thousand|
|103||10 x 10 x 10||1,000||1 thousand|
|102||10 x 10||100||1 hundred|
Multiplying (or dividing) powers of 10 is easy: you just add (or
subtract) the exponents.
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
Multiplying (dividing) numbers in scientific notation simply involves
multiplying (dividing) the two coefficientss and separately multiplying
(dividing) the powers of 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
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 105. 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 105 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, n is the logarithm of |X|, rounded down to the nearest integer. a is then X/10n.
In practical terms, if |X| is greater than 1, n is one less than the total number of digits to the left of the decimal point. If |X| is less than one, n is the negative of the number of zeros (including the one to the left of the decimal point) before the first non-zero digit. a can 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, multiply a by 10n to get X.
In practical terms, remove the decimal point from a. If n is positive, reposition the decimal point so that the total number of digits to the left of the decimal point is n+1 (pad with zeros as needed). If n is negative, add |n| zeros to the left of the first (leftmost) digit of a, and place the decimal point between the first and second zero.