Chapter 4:Metric Prefixes & Powers of Ten

Gigafun with nanoeffort

Presented by: James, VE3BUX

2

Base-10: Quick review of “tens”

• We count in base 10 where there are 1s, 10s, 100s, etc .. Columns– We also count in base-24 and base-60 … we are just more

familiar with base-10 for math

• Reconsider the columns in terms of powers of 10 as follows:Column 100’000s 10’000s 1000s 100s 10s 1s

Exponent 105 104 103 102 101 100

# of 0s 5 4 3 2 1 0

3

Counting in Base-10

– 9 = 9 x 100

– 71 = 7 x 101 + 1 x 100

– 123 = 1 x 102 + 2 x 101 + 3 x 100

– 5860 = 5 x 103 + 8 x 102 + 6 x 101 + 4 x 100

– 721645 = 7 x 105 + 2 x 104 + 1 x 103 + 0 x 102 + 4 x 101 + 5 x 100

Column 100’000s 10’000s 1000s 100s 10s 1s

Exponent 105 104 103 102 101 100

9 .

71 .

123 .

5864 .

721045 .

9

7 1

1 2 3

5 8 6 4

7 2 1 0 4 5

4

Base-10: Quick review of “tenths”

• What about “decimal values” ?

– 0.9 = 0 x 100 + 9 x 10-1

– 0.71 = 0 x 100 + 7 x 10-1 + 1 x 10-2

– 0.123 = 0 x 100 + 1 x 10-1 + 2 x 10-2 + 3 x 10-3

– 0.5864 = 0 x 100 + 5 x 10-1 + 8 x 10-2 + 6 x 10-3 + 4 x 10-4

Column 1s 10ths 100ths 1000ths 10’000ths

Exponent 100 10-1 10-2 10-3 10-4

0.9 0 . 9

0.71 0 . 7 1

0.123 0 . 1 2 3

0.5864 0 . 5 8 6 4

5

Scientific / Engineering Notation

• Is there a more effective method of expressing a large (or small) value such as:

– 300 000 000m s-1

• (Speed of light)

– 0.000000000000000000160217657C• The charge (in Coulombs) of an electron

6

Base and Index: A Brief Review

• Any number A which is multiplied by itself “b times” can be expressed in the base-index form:

Ab

• A = base• b = index (or power)

• Eg: 10 x 10 x 10 can be expressed as 103

– Tip: Count the zeros!

7

• Given the following constant (the speed of light in a vacuum): how can we express this in terms of base and index?

Or re-written as: 3 x 108m s-1

• The 3 term preceding the base 10 is the coefficient and is generally what you will perform basic arithmetic on, saving exponent math for the base and index

Base and Index: Example

300000000m s-1300000000m s-1

300000000m s-1

12345678

x10

8

Scientific Notation & Its Uses

• When dealing with large numbers, or converting between bases, it is helpful to use the base-index (scientific notation) form

• Eg:λ = 300000000m s-1 / 30000000Hzλ = 3 x 108m s-1

3 x 107 s-1

λ = 108m / 107

– λ (lambda) is wavelength in m– 1Hz = 1 cycle per second .. so 1 reciprocal second (ie. s-1)

… okay, but how do we solve that?

9

Exponent Math: Mult. & Div.

• When you multiply or divide exponential values,(ie. λ = 108m / 107) from the previous slide

we must observe some special yet simple practices:

• When multiplying, simply add the indices (powers):103 x 104 = 10(3 + 4) = 107

• When dividing, subtract the indices:107 / 102 = 10(7-2) = 105

Take note: This can only be done when the bases are the same. Ie. 102 x 23 ≠ 205

λ = 108m / 107

λ = 10(8-7)m = 101m … or 10m

10

Base and Index: Small numbers

• So we can express very large numbers using the Ab format, how about very small numbers?

• Consider for a moment what a number such as 0.1 means– One tenth– 1/10– 1 . 101

11

Reciprocal Values

• What can we say about a value such as:

1 .101

• What about making it: 100 . 101

12

Exponent Math: Division

100 . 101

• Recall that when we divide exponential values, we subtract them

100 .= 100 – 101 = 10(0-1) = 10-1

101

13

Decimal Values & Scientific Notation

• Since we know 0.1 can be express as 10-1, what about 0.000001 ?

• Again, count the number of times you move the decimal place to the right in order to make 1.0 x 10?

0.000001 = 10-6

14

Metric Prefixes

Prefix Symbol Scientific Notation Decimal Common

Wordtera T 1012 1000000000000 trilliongiga G 109 1000000000 billionmega M 106 1000000 millionkilo k 103 1000 thousand

100 1 onemilli m 10-3 0.001 thousandthmicro μ 10-6 0.000001 millionthnano n 10-9 0.000000001 billionthpico p 10-12 0.000000000001 trillionth

15

Metric Prefixes: Practical Examples

• 3.5MHz = ? M = mega = 106 therefore … 3.5 x 106Hz

• 1.5mA = ?m = milli = 10-3 so … 1.5 x 10-3A

• 3.3kV = ?k = kilo = 103 thus … 3.3 x 103V

• 220μH = ?μ = micro = 10-6 … 2.2 x 10-4H… did I catch you on that one?

16

Engineering Notation

• Scientific notation is nice and all, but it has its ease-of-use limitations in practice

• Engineering notation works in “groups of three” such that the unit value will respect 10n where n is a multiple of 3

• Eg: 220μH from the previous slide was presented in engineering notation– Scientific would have read 2.20x10-4 from the start

17

Engineering Notation

• Values are given in “base” units such as M, k, m, μ, n, p□ (where □ represents an SI unit of measure such as metres or Hz)

– 500μH as opposed to 5.0 x 10-4H– 33kV as opposed to 3.3 x 104V– 0.5nF or 500pF as opposed to 5 x 10-11F• In this example, the 500pF is preferred over 0.5nF

because it avoids using a decimal value

18

Mind your 0s!

• Often when dealing with components, values will be listed on a schematic such that:

“all values of capacitance will be given as μF”

• It may be necessary to become comfortable working in “hybrid units,” eg:– 0.1μV = 100nV– 5000nF = 5μF– 1000μH = 1mH

19

Conversion between prefixes

• Is there a foolproof way to convert between any two prefixes?

• Absolutely! Use known ratios!• 1 MHz = ?? μHz

– 1Mhz = 106Hz and 1μHz = 10-6Hz– Put another way, there are 106Hz in 1MHz and 106μHz per 1Hz

• 1MHz x 106Hz x 106μHz = 1MHz 1Hz

106 x 106μHz = 1012μHz

20

Conversions made simple

• A quicker method of base conversion is to look at the absolute “distance” between two units

• Beware .. you must know something about the “direction” you are converting.– Large to small means +ve exponent (index)– Small to large means –ve exponent

– 1G□ is 10+18n□• (G = 109 & n= 10-9 so |9| + |-9| = 18)• Large unit to smaller, so the index is +ve

– 1n□ is 10-15M□ • (n = 10-9 & M = 106 so |-9| + |6| = 15)• Small unit to larger, so the index is -ve

21

• Questions?