Scientific notation. Metric system conversions -- A complete course in arithmetic

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A R I T H M E T I C

Lesson 3 Section 3

Scientific Notation
Metric System Conversions

11.	When is a number written in "scientific notation"?

	A number is written in scientific notation when there is one non-zero digit to the left of the decimal point.

2.345

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 :

A positive exponent means to multiply by a power of 10. A negative exponent means to divide.

Briefly, the exponent indicates the number of 0's
in the power of 10. To go further into the meaning of an exponent, see Lessons 13 and 21 of Skill in Algebra.

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 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.

a)	123.4 = 1.234 × 10²

	The scientific notation begins 1.234. To get back to 123.4, we have to move the point 2 places right. We have to multiply by 10 with exponent +2.

b)	380,000 = 3.8 × 10⁵

	The scientific notation begins 3.8. To get back to 380,000, we have to move the point 5 places right. We have to multiply by 10 with exponent +5.

c)	.0012 = 1.2 × 10⁻³

	The scientific notation begins 1.2. To get back to .0012, we have to move the point 3 places left. We have to multiply by 10 with exponent −3.

Problem 1. Each of these is written in scientific notation. What normal number is it?

a)	2.401 × 10³ = 2,401	b)	1.006 × 10² = 100.6

c)	7.01 × 10⁵ = 701,000	d)	3.248 × 10 = 32.48

e)	1.86 × 10⁻⁴ = 0.000186	f)	4.05 × 10⁻¹ = 0.405

g)	6.0 × 10⁻⁶ = 0.000006	h)	3.1 × 10⁻² = 0.031

Problem 2. Write each of the following in scientific notation.

a)	1,234 = 1.234 × 10³	b)	12.34 = 1.234 × 10

c)	123.4 = 1.234 × 10²	d)	2,400,000 = 2.4 × 10⁶

e)	0.000081 = 8.1 × 10⁻⁵	f)	0.00395 = 3.95 × 10⁻³

g)	0.00000000628 = 6.28 × 10⁻⁹	h)	0.107 = 1.07 × 10⁻¹

12.	How do we convert units in the metric system?

125 cm = ? m

	Apply the rule given below.

WHOLE UNITS				DECIMAL UNITS
4th order	3rd order	2nd order	1st order	1st order	2nd order	3rd order
1000	100	10	1	.1	.01	.001
kilometer km	(hecto)	(deka)	meter m	(deci)	centimeter cm	millimeter mm
kilogram kg			gram gm		centigram cg	milligram mg
kiloliter kL			liter L	deciliter dL	centiliter cL	milliliter mL

In what is called the metric system, the basic units of measurement are the meter, for measuring length; the gram, for measuring weight; and the liter, for measuring volume. The system would be more vividly called the decimal system, because in it we count by 10's, 100's, 1000's, and so on.

Everything the student will require is shown in the table above. Thus 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, shift the decimal point
the same number of places, and in the same direction, left or right,
as the distance between the orders.

Example 1. 4235 m = ? km

Solution. Kilometers lie three orders to the left of meters. Therefore, simply separate three decimal places 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 gm = ? cg

Solution. Centigrams lie two orders to the right of grams. Therefore, shift the decimal point two places right.

5.2 gm = 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 places.

478 dL = .0478 kL

Problem 3.

a) If you change the whole number unit of 1st order to the unit of 4th a) order, how will you shift the decimal point?

b) Change each of the following to kilometers.

3285 m = 3.285 km		296 m = .296 km

85 m = .085 km		6.25 m = .00625 km

Problem 4.

a) If you change the whole number unit of 1st order to the decimal unit
a) of 2nd order, how will you shift the decimal point?

b) Change each of the following to centimeters.

1.658 m = 165.8 cm		2.36 m = 236 cm

3.8 m = 380 cm		15 m = 1500 cm

Problem 5. Convert the following.

a)	586 cm = 5.86 m	b)	4.28 m = 4280 mm

c)	38.9 mL = 3.89 cL	d)	1375 L = 1.375 kL

e)	12 gm = 1200 cg	f)	1396 mg = .001396 kg

g)	6 km = 6000 m	h)	6 km = 600,000 cm

i)	123 cg = 1.23 gm	j)	123 cg = 1230 mg

k)	18 mL = .018 L	l)	6.5 kL = 65,000 dL

Continue on to the next Lesson.

Section 1 of this Lesson

Section 2 of this Lesson

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