Lesson 4 Section 3
That number is written in scientific notation. There is one digit to the left of the decimal point -- 2 -- and it is not 0.
In general, a number written in scientific notation will be multiplied by 10 raised to an "exponent."
2.345 × 103 2.345 × 10−3
Without going into the details of what the exponents 3 and −3 actually mean, we can state the following :
Briefly, the exponent indicates the number of 0's
Therefore, if a number is written in scientific notation, then to express it as a standard number, we can state the following rule:
If the exponent is positive, move the decimal point right as many places as indicated by the exponent.
If the exponent is negative, move the decimal point left as many places as indicated by the exponent.
Example 1. Each number is written in scientific notation. What standard number is it?
a) 5.42 × 103 = 5,420. Move the decimal point three places right.
b) 5.42 × 10−3 = .00542 Move the decimal point three places left.
Example 2. Write each number in scientific notation.
An alternative explanation is based on the following:
A number does not change if we divide it and then multiply the quotient by the same number, or if we multiply and then divide by the same number.
And so to go from 123.4 to 1.234 we divided by 100. Therefore, for the original value not to change -- for the equal sign to mean equals -- we must multiply by 100, which is 10².
123.4 = 1.234 × 10²
We both divided and multiplied by 100.
Problem 1. Each of these is written in scientific notation. What standard number is it?
Problem 2. Write each of the following in scientific notation.
The metric system
Here are metric prefixes that are in common use:
Thus km means kilometers, which is 103 or 1000 meters.
μg means micrograms, which is one-millionth of a gram.
The 1st order or principle unit of length is the meter. The 2nd order unit (which is not in common use) is 10 times more and would have the prefix deka: 1 dekameter. The 3rd order unit (again not in common use) would have the prefix hecto, meaning 100 times. And the 4th order unit of length is the kilometer, which means 1000 meters.
As for the decimal units, the 1st order decimal unit is one-tenth of a meter, and is a called a decimeter ("dessi-meter") -- the prefix deci signifies the tenth part; but that measure of length is not in common use. The prefix centi signifies the hundredth part. The prefix milli signifies the thousandth part.
Relations between units of the same kind
The rule for expressing meters as kilometers, for example, or grams as centigrams, is as follows:
To convert to a different metric unit of the same kind, shift the decimal
Example 1. 4235 m = ? km
Solution. Kilometers lie three orders to the left of meters. Therefore, simply separate three decimal digits in that same direction, from right to left:
4235 m = 4.235 km.
(Effectively, we have divided by 1000, for 1 kilometer is 1000 times 1 meter.)
Example 2. 5.2 g = ? cg
Solution. Centigrams lie two orders to the right of grams. Therefore, shift the decimal point two places right.
5.2 g = 520 cg
(Effectively, we have multiplied by 100, for 1 gram is 100 times 1 centigram.)
Example 3. 478 dL = ? kL
Solution. Kiloliters are four orders to the left of deciliters. Therefore, separate four decimal digits.
478 dL = .0478 kL
a) If you change the whole number unit of 1st order to the unit of 4th
Three places left.
b) Change each of the following to kilometers.
a) If you change the whole number unit of 1st order to the decimal unit
Two places right.
b) Change each of the following to centimeters.
Problem 5. Convert the following.
Continue on to the next Lesson.
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