 Basic Math: Scientific Notation In this section, you will occasionally be asked to answer some questions. Whenever a problem set is given, you should answer the questions on a separate sheet of paper and then verify your answers by clicking on "Answers."

The first thing to learn is how to convert numbers back and forth between scientific notation and ordinary decimal notation. The expression "10n", where n is a whole number, simply means "10 raised to the nth power," or in other words, a number gotten by using 10 as a factor n times:

105 = 10 x 10 x 10 x 10 x 10 = 100,000 (5 zeros)

108 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000 (8 zeros)

Notice that the number of zeros in the ordinary decimal expression is exactly equal to the power to which 10 is raised.

If the number is expressed in words, first write it down as an ordinary decimal number and then convert. Thus, "ten million" becomes 10,000,000. There are seven zeros, so in powers of ten notation ten million is written 107.

A number which is some power of 1/10 can also be expressed easily in scientific notation. By definition,

1/10 = 10-1     ("ten to the minus one power")

More generally, the expression "10-n" (where n is a whole number) means ( 1/10 )n. Thus

10-3 = ( 1 / 10 )3 = 1 / ( 10 x 10 x 10) = 1/1000

10-8 = ( 1 / 10 )8 = 1/100,000,000

Scientific notation was invented to help scientists (and science students!)deal with very large and very small numbers, without getting lost in all the zeros. Now answer the following on a separate sheet of paper and check your answers by clicking on "Answers":

First Problem Set

Express 1-6 in scientific notation, and 7-10 in ordinary notation:

 1. 100 2. 10,000,000 3. 1 / 10,000 4. one million 5. 1 / 10,000,000 6. one ten millionth 7. 103 8. 10-5 9. 109 10. 1 X 10-2

What about numbers that are not exact powers on ten, such as 2000, 0.0003, etc.? Actually, they are only a little more complicated to write down than powers of ten. Take 2000 as an example:

2000 = 2 x 1000 = 2 x 103

As another example, take 0.00003, or "three ten-thousandths":

0.0003 = 3 x 1 / 10,000 = 3 x 10-4

There is a simple procedure for getting a decimal number into the "standard form" for scientific notation:

First, write down the number as the number itself times 100. This can be done because 100 equals one, and any number times one equals that number. The number is now in the standard form:

coefficient x 10 exponent

Second, start moving the decimal point in the coefficient to the right or left. For each place you move the decimal place to the left, add 1 to the exponent. For each place you move it to the right, subtract 1 from the exponent. What you are doing is dividing (or multiplying) the coefficient by 10 each time, while at the same time multiplying (or dividing) the exponent term by 10 each time. Since what you do to the exponent term undoes what you do to the coefficient, the total number does not change.

Some examples will hopefully make it clear:

2000 = 2000 x 100= 200 x 101= 20 x 102= 2 x 103

0.0003 = 0.0003 x 100= 0.003 x 10-1 = 0.03 x 10-2 = 0.3 x 10-3= 3 x 10-4

You should move the decimal point until there is exactly one nonzero digit to the left of the decimal point, as in the last case of each example given. We then say that the number is fully in the standard form. You should always express scientific notation numbers in the standard form. Notice that you don't really have to write down each of the steps above; it is enough to count the number of places to move the decimal point and use that number to add or subtract from the exponent. Some examples:

250,000 = 2.5 x 105       5 places to the left

0.000035 = 3.5 x 10-5     5 places to the right

0.00000001 = 1 x 10-8 = 10-8  8 places to the right

Second Problem Set

Express 1-6 in scientific notation, and 7-10 in ordinry notation:

 1. 342,000,000 2. 0.000923 3. eight million 4. 0.0000064 5. 47,682 6. 0.0249 7. 4 x 107 8. 3.22 x 10-3 9. 8.4 x 1010 10. 6.33 x 10-6

The most difficult kind of calculation that can be done with numbers expressed in scientific notation turns out to be addition or subtraction. Multiplication, division, and raising to powers is actually easier. So, we'll deal with these first.

The rule for multiplying two numbers expressed in scientific notation has three steps:

1. Multiply the coefficients to get the new coefficient.
2. Add the exponents (watch the signs!) to get the new exponent.
3. Get the number into the standard form, if needed.

Examples:

(4 x 103) x (2 x 107) = ( 4 x 2 ) x ( 103 + 7 ) = 8 x 1010

(2 x 10-5) x (2.5 x 108) = ( 2 x 2.5 ) x ( 10-5+ 8 ) = 5 x 103

(3 x 10-7) x (3 x 10-8) = ( 3 x 3 ) x ( 10 -7 + (-8) ) = 9 x 10-15

(4 x 107) x (3 x 105) = ( 4 x 3 ) x ( 10 7 + 5 ) = 12 x 1012= 1.2 x 1013

The steps for division are similar:

1. Divide the coefficients to get the new coefficient
2. Subtract the "bottom" exponent from the "top" one (really watch the signs!) to get the new exponent.
3. Get the number into the standard form, if needed.

Some examples:

(6 x 105) / (2 x 103) = ( 6 / 2 ) x ( 10 5 - 3 ) = 3 x 102

(9 x 108) / (3 x 10-5) = ( 9 / 3 ) x ( 10 8 - (-5) ) = 3 x ( 10 8 + 5 ) = 3 x 1013

(5 x 103) / (2 x 107) = ( 5 / 2 ) x ( 10 3 - 7 ) = 2.5 x 10-4

(2 x 105) / (4 x 102) = ( 2 / 4 ) x ( 10 5 - 2 ) = 0.5 x 103= 5 x 102

If you are given a number in scientific notation to raise to a power, remember that all this means is that it is used as a factor that many times. Simply write the number down as many times as the power to which it is to be raised, and use the rules for multiplication repeatedly.

Example:

(2 x 105)3 = (2 x 105) x (2 x 105) x (2 x 105)

(2 x 105)3 = ( 2 x 2 x 2 ) x ( 105 x 105 x 105)

(2 x 105)3 = 8 x ( 10 5 + 5 + 5 ) = 8 x 1015

In a situation where you have to raise things to a power and do multiplication or division, always finish raising to the power first, then do the other operation.l

Example:

(2 x 109)3 / (6 x 10-2)2

(2 x 2 x 2) x (109 x 109 x 109) / (6 x 6) x (10-2 x 10-2)

(8 x 10 9 + 9 +9) / (36 x 10 -2 -2)

(8 x 1027) / (36 x 10-4)

(8/36) x (10 27 - (-4))

0.22 x 10 31 = 2.2 x 10 30

Third Problem Set

Calculate the following:

 1. (7 x 106) / (2 x 104) 2. (2 x 107) x (4 x 10-9) 3. (5 x 108) x (5 x 103) 4. (6 x 103)3 / (3 x 106)4 5. (5 x 103) / (2 x 103) 6. (3 x 104)2 7. (4 x 10-6)3 8. (2 x 105) x (6 x 107) / (4 x 108)

Addition and subtraction are a little more involved. There are four basic steps:

1. Find the number whose exponent is algebraically the smallest (remember, negative numbers are algebraically smaller than positive ones, and the "more negative" the number, the smaller it is).
2. If the exponents of the numbers are not the same, change the number with the smaller exponent. Do this by moving the decimal point of the coefficient of that number to the left, and adding one to the exponent of that number, until the two exponents are equal.
3. Add or subtract the coefficients of the two numbers. The result is the coefficient of the result. The exponent is the exponent of the number you did not change.
4. Put the result in standard form, if necessary.

Examples:

a) (3 x 10-6) - (2 x 10-7)

The algebraically smallest exponent is -7, so we change the second term:

2 x 10-7 = 0.2 x 10-6 The exponents are now the same

(3 x 10-6) - (0.2 x 10-6) = ( 3 - 0.2 ) x 10-6 = 2.8 x 10-6

b) (9.39 x 105) + (8 x 103) = (9.39 x 105) + (0.08 x 105) = 9.47 x 105

In situations where addition and subtraction are mixed with multiplication and division, do the multiplication and division first, then do the addition and subtraction. And don't forget that raising things to powers always takes priority over multiplication and division!

Examples:

a) (5 x 106) x (3 x 10-3) + (2.2 x 105)

(5 x 3) x (106 x10-3) + (2.2 x 105)

(15 x 106-3) + (2.2 x 105)

(15 x 103) + (2.2 x 105)

(0.15 x 105) + (2.2 x 105)

2.4 x 105

b) (6.3 x 103) - (4 x 104)3 / (8 x 105)2

(6.3 x 103) - (4 x 4 x 4 x 104 x 104 x 104) / (8 x 8 x 105 x 105)

(6.3 x 103) - (64 x 1012) / (64 x 1010)

(6.3 x 103) - ( (64 / 64) x 10 12 - 10 ) )

(6.3 x 103) - (1 x 102)

(6.3 x 103) - (0.1 x 103)

6.2 x 103

Fourth Problem Set

Calculate the following:

 1. (5.7 x 106) + (3 x 105) 2. (4.2 x 10-8) - (2.3 x 10-8) 3. (3.8 x 105) - (2.1 x 106) 4. (7.43 x 105) + (1.97 x 107) / (2 x 103) 5. (1.35 x 107) + (8 x 105) 6. (6.52 x 103) - (1.41 x 105) x (2.31 x 10-3) 7. (8.52 x 10-9) + (2.16 x 10-9) 8. (4.73 x 104) + (3.16 x 1011) / (7.4 x 103)2 Updated 8/26/99