Scientific Notation: What is it?
Astronomers deal with quantities ranging from the truly microcosmic to the macrocosmic. It is very inconvenient to always have to write out the age of the universe as 15,000,000,000 years or the distance to the Sun as 149,600,000,000 meters. To save effort, powers-of-ten notation is used. For example, 10 = 101; the exponent tells you how many times to multiply by 10. As another example, 10-2 = 1/100; in this case the exponent is negative, so it tells you how many times to divide by 10. The only trick is to remember that 100 = 1. Using powers-of-ten notation, the age of the universe is 1.5 x 1010 years and the distance to the Sun is 1.496 x 1011 meters.
If the number is between 1 and 10, so that it would be multiplied by 100 (=1), then it is not necessary to write the power of 10. For example, the number 4.56 already is in scientific notation (it is not necessary to write it as 4.56 x 100, but you may write it this way if you wish).
If the number is a power of 10, then it is not necessary to write that it is multiplied by 1. For example, the number 100 can be written in scientific notation either as 102 or as 1 x 102. (Note, however, that the latter form should be used when entering numbers on a calculator.)
The use of scientific notation has several advantages, even for use outside of the sciences:
Because it is so easy to multiply powers of ten in your head (by adding the exponents), scientific notation makes it easy to do "in your head" estimates of answers.
Use of scientific notation makes it easier to keep track of significant figures; that is, does your answer really need all of those digits that pop up on your calculator?
Converting from "Normal" to Scientific Notation
Place the decimal point after the first non-zero digit, and count the number of places the decimal point has moved. If the decimal place has moved to the left then multiply by a positive power of 10; to the right will result in a negative power of 10.
Example: To write 3040 in scientific notation we must move the decimal point 3 places to the left, so it becomes 3.04 x 103.
Example: To write 0.00012 in scientific notation we must move the decimal point 4 places to the right: 1.2 x 10-4.
Converting from Scientific Notation to "Normal"
If the power of 10 is positive, then move the decimal point to the right; if it is negative, then move it to the left.
Example: Convert 4.01 x 102. We move the decimal point two places to the right making 401.
Example: Convert 5.7 x 10-3. We move the decimal point three places to the left making 0.0057.
Working with Scientific Notation: Addition, Multiplication, ...
Addition and Subtraction
When adding or subtracting numbers in scientific notation, their powers of 10 must be equal. If the powers are not equal, then you must first convert the numbers so that they all have the same power of 10.
Example: (6.7 x 109) + (4.2 x 109) = (6.7 + 4.2) x 109 = 10.9 x 109 = 1.09 x 1010. (Note that the last step is necessary in order to put the answer in scientific notation.)
Example: (4 x 108) - (3 x 106) = (4 x 108) - (0.03 x 108) = (4 - 0.03) x 108 = 3.97 x 108.
Multiplication and Division
It is very easy to multiply or divide just by rearranging so that the powers of 10 are multiplied together.
Example: (6 x 102) x (4 x 10-5) = (6 x 4) x (102 x 10-5) = 24 x 102-5 = 24 x 10-3 = 2.4 x 10-2. (Note that the last step is necessary in order to put the answer in scientific notation.)
Approximations with Scientific Notation
Because working with powers of 10 is so simple, use of scientific notation makes it easy to estimate approximate answers. This is especially important when using a calculator since, by doing mental calculations, you can verify whether your answers are reasonable. To make approximations, simply round the numbers in scientific notation to the nearest integer, then do the operations in your head.
Example: Estimate 5,795 x 326. In scientific notation the problem becomes (5.795 x 103) x (3.26 x 102). Rounding each to the nearest integer makes the approximation (6 x 103) x (3 x 102), which is 18 x 105, or 1.8 x 106 (the exact answer is 1.88917 x 106).
Example: Estimate (5 x 1015) + (2.1 x 109). Rounding to the nearest integer this becomes (5 x 1015) + (2 x 109). We see immediately that the second number is nearly 1015/109, or one million, times smaller than the first. Thus, it can be ignored in the addition problem and our approximate answer is 5 x 1015. (The exact answer is 5.0000021 x 1015).
Numbers should be given only to the accuracy that they are known with certainty, or to the extent that they are important to the topic at hand. For example, your doctor may say that you weigh 130 pounds, when in fact at that instant you might weigh 130.16479 pounds. The discrepancy is unimportant and will change anyway as soon as a blood sample has been drawn.
If numbers are given to the greatest accuracy that they are known, then the result of a multiplication or division with those numbers can't be determined any better than to the number of digits in the least accurate number.
Example: Find the circumference of a circle measured to have a
radius of 5.23 cm using the formula:
C = (2 x p x R). Since the value of p stored in your calculator is probably 3.141592654, the numerical solution will be
(2 x 3.141592654 x 5.23 cm) = 32.86105916 = 3.286105916 x 101 cm.
If you simply write down all 10 digits as your answer, you are implying that you know, with absolute certainty, the circle's circumference to an accuracy of one part in 10 billion! That would mean that your measurement of the radius was in error by no more than 0.000000001 cm; that is, its true value was at least 5.229999999 cm, but no more than 5.230000001 cm (otherwise, your calculator would have shown a different number for the circumference).
In reality, since your measurement of the radius was known to only three decimal places, the circle's circumference is also known to only (at best) three decimal places as well: you round the fourth digit and give the result as 32.9 cm or 3.29 x 101 cm. It may not look as impressive, but it's an honest representation of what you know about the figure.
Since the value of "2" was used in the formula, you may wonder why we're allowed to give the answer to three decimal places rather than just one: 3 x 101 cm. The reason is because the number "2" is exact - it expresses the fact that a diameter is exactly twice the radius of a circle - no uncertainty about it at all. Without any exaggeration, we could have represented the number as 2.0000000000000000000, but merely used the shorthand "2" for simplicity - so we really didn't violate the rule of using the least accurately-known number.