Scientific Notation

Scientific notation provides us with the only reasonable means of dealing with a broad range of magnitudes, such as we encounter in astronomy. The "trick" of using special units is useful only so long as everything we are dealing with is of the same order of magnitude (i.e. solar mass units are useful as long as we are dealing only with star-sized objects) and as long as we are not trying to perform physical calculations (fundamental physical constants are usually provided is SI units, so ultimately, it is usually necessary to work in those units for calculations).

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.

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

103=10x10x10=1000.

Note that 10n can be written as a 1 followed by n zeros.

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:

10-n = 1/10n.

Thus
10-3 = 1/103 = 1/1000 = 0.001.

Note that, counting the zero before the decimal point, there are n zeroes before the 1 in the standard representation of 10-n.

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
101 10 10 ten
100 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

Hopefully you get the picture.....

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

10n x 10m = 10(n+m)
10n / 10m = 10(n-m)

So, for example, 103x105=108 and 1033/1027=106. (If you don't believe it, try writing it out in long form and doing the math. A few examples should convince you that the exponential notation is a great improvement.)

Scientific Notation

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

X = a x 10n

where the exponential n is an integer, and the coefficient a is a real number satisfying
1 ≤ |a| < 10

(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:

(a x 10n) x (b x 10m) = ab x 10(n+m)
(a x 10n) / (b x 10m) = a/b x 10(n-m)

If the resulting product ab is greater than 10, simply factor out the 10 (move the decimal one place to the left) and multiply it in with the other powers of 10 (increase the exponent by 1). Similarly, if the quotient a/b is less than 1, use one of the factors of 10 (decrease the exponent by 1) to multiply the coefficient (move the decimal one place to the right). A few examples should clarify things:
(2.2 x 103) x (3.5 x 10-7) = 7.7 x 10-4
(-4 x 107) x (6.02 x 1023) = 24.08 x 1030 = 2.408 x 1031
(8.4 x 10-5) / (2 x 104) = 4.2 x 10-9
(1.989 x 1033) / (5.976 x 1027) = 0.3328 x 106 = 3.328 x 105

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:

(a x 10n) +/- (b x 10m) = (a x 10n) +/- (b x 10m-n x 10n) = (a +/- (b x 10m-n)) x 10n

Again, a couple examples may be helpful:
(2.2 x 107) + (3.5 x 105) = (2.2 x 107) + (3.5 x 10-2 x 107) = (2.2 + 0.035) x 107 = 2.235 x 107
(6.02 x 1023) - (1.5 x 1022) = (6.02 x 1023) - (1.5 x 10-1 x 1023) = (6.02 - 0.15) x 1023 = 5.87 x 1023
(1.989 x 1033) + (5.976 x 1027) = (1.989 x 1033) + (5.976 x 10-6 x 1033) = 1.989005976 x 1033

You may notice that I usually factor out the larger power of 10. This is simpler in the sense that your answer will almost always be of the same order of magnitude as the larger number. You may, however, be more comfortable factoring out the smaller power of 10. This usually requires an extra step to get the result back into standard form:
(1.989 x 1033) + (5.976 x 1027) = (1.989 x 106 x 1027) + (5.976 x 1027) = 1,989,005.976 x 1027 = 1.989005976 x 1033

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 1033, which is not significantly different from the "exact" answer (which is not exact at all - the numbers, in this case the masses of the sun and the earth, have been rounded off to 3 significant digits to begin with).

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 between regular and scientific notation

I'll denote a number in regular (fixed) notation as X, and the same number in scientific notation is a x 10n.

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.

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