PhysicsLAB: Metric Prefixes, Scientific Notation, and Conversions

Common Metric Prefixes
Factor	Name	Symbol
10⁹	giga	G
10⁶	mega	M
10³	kilo	k
10^-1	deci	d
10^-2	centi	c
10^-3	milli	m
10^-6	micro	μ
10^-9	nano	n
10^-12	pico	p

Given below is an expanded diagram of a meter stick showing the relationships between meters, decimeters (10^-1 m), centimeters (10^-2 m), and millimeters (10 ^-3 m).

relationship between meters, dm, cm, and mm

Scientific Notation

The easiest way to convert one unit of measurement to another unit of measure is to initially convert its metric prefix to its associated power of ten while also rewriting the original numerical value in scientific notation. The final answer can then be simplified by just combining exponents.

A number is expressed in scientific notation when it is in the form

a × 10ⁿ where 1≤ a <10 and n ∈ { ... -3, -2, -1, 0, 1, 2, 3 ... )

Before continuing with the next section of lesson in which we convert measurements in one metric prefix to another, let's practice converting between decimal notation and scientific notation.

Refer to the following information for the next five questions.

Express the following numbers in scientific notation.

	362

	11.8

	0.00362

	0.118

	1986

Refer to the following information for the next four questions.

Convert these numbers in scientific notation to decimal form.

	8.71 x 10^-3

	2.56 x 10²

	1.28 x 10⁶

	5.21 x 10^-4

Basic Conversions

Before looking at some examples of converting measurements from one metric prefix to another we need to remember the following rules for working with exponents:

Rules of exponents:

10^x × 10^y = 10^{(x + y)}

10^x ÷ 10^y = 10^{(x - y)}

10^-n = 1 ÷ 10ⁿ

Example #1

628 nm = ? meters
628 nm = 628 × 10^-9 meters [definition of the metrix prefix nano]
628 nm = 6.28 × 10² × 10^-9 meters
628 nm = 6.28 × 10^2+(-9) meters
628 nm = 6.28 × 10^-7 meters

Example #2

628 nm = ? μm
using the answer to Example #1 we can replace 628 nm with 6.28 × 10^-7 meters
6.28 × 10^-7 meters = ? μm
since 1 μm equals 10^-6 meters, we rewrite -7 as -1+(-6)
6.28 × 10^-1+(-6) meters = ? μm
remember that adding exponents means multiplying powers of 10
6.28 × (10^-1 × 10^-6 ) meters = ? μm
(6.28 × 10^-1) × (10^-6 meters) = 6.28 × 10^-1 μm = 0.628 μm

Now you practice two conversions problems.

	1. How many mm are in each of the following measurements: 500 μm, 50 cm, 5 m?

	2. Rank the following from smallest to largest: 500 µm, 50 cm, 50 dm, and 5 mm.

Some Advanced Conversions

Example #3

If a liter equals 1000 cm³, then a cube that is 100 cm on each edge would hold how many liters?

V = l w h = (100 cm)³ = (10² cm)³ = 10⁶ cm³

our question has now become how many liters are in 10⁶ cm³

10⁶ cm³ = ? L
since 1 L = 10³ cm³, we rewite 6 as 3+3
10⁽³⁺³⁾ cm³ = ? L
remember that adding exponents means multiplying powers of 10
(10³× 10³) cm³ = ? L
10³× (10³ cm³) = 10³ L

Example #4

How many m³ does a cube 100 cm on an edge occupy?

V = l w h = (100 cm)³ = (10² cm)³ = 10⁶ cm³
V = l w h = (100 cm)³ = (1 m)³ = 1 m³

Therefore, 10⁶ cm³ = 1 m³

Example #5

How many m³ are present in a cube having a volume of 1000 cm³?

1000 cm³ = 10³ cm³ = ? m³
using the answer to Example #4 we know that 10⁶ m³ = 1 m³
10³ cm³ = 10^(-3+6) cm³ = ? m³
remember that adding exponents means multiplying powers of 10
(10^-3 × 10⁶) cm³ = ? m³
10^-3 × (10⁶ cm³ ) = 10^-3 m³