Chapter 4:Metric Prefixes & Powers of Ten

Gigafun with nanoeffort

Presented by: James, VE3BUX

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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

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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 .

97 1

1 2 3

5 8 6 4

7 2 1 0 4 5

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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

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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

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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!

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• 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

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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?

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Exponent Math: Mult. & Div.• When you multiply or divide exponential values,

(ie. λ = 108m / 107) from the previous slidewe 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

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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

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Reciprocal Values

• What can we say about a value such as: 1 .101

• What about making it: 100 . 101

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Exponent Math: Division

100 . 101

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

100 .= 100 – 101 = 10(0-1) = 10-1

101

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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

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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

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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?

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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

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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

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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

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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

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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

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• Questions?