1. 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

3. Counting in Base-10 9 = 9 x 100 71 = 7 x 101 + 1 x 100
Column
100’000s
10’000s
1000s
100s
10s 1s
Exponent
105
104
103
102
101
100 9 . 71
123
5864
721045
9 = 9 x 100
71 = 7 x x 100
123 = 1 x x x 100
5860 = 5 x x x x 100
= 7 x x x x x x 100 9 7 1 1 2 3 5 8 6 4 7 2 1 4 5

4. Base-10: Quick review of “tenths”
What about “decimal values” ?
0.9 = 0 x x 10-1
0.71 = 0 x x x 10-2
0.123 = 0 x x x x 10-3
= 0 x x x x x 10-4
Column 1s
10ths
100ths
1000ths
10’000ths
Exponent
100
10-1
10-2
10-3
10-4
0.9 . 9
0.71 7 1
0.123 2 3
0.5864 5 8 6 4

5. Scientific / Engineering Notation
Is there a more effective method of expressing a large (or small) value such as:
m s-1
(Speed of light) C
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. Base and Index: Example
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
m s-1
m s-1
x10
m s-1 8 7 6 5 4 3 2 1

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 = m s-1 / Hz
λ = 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
1 .
101
What about making it:
100 .

12. Exponent Math: Division
100 .
101
Recall that when we divide exponential values, we subtract them
100 .= 100 – 101 = 10(0-1) = 10-1

13. Decimal Values & Scientific Notation
Since we know 0.1 can be express as 10-1, what about ?
Again, count the number of times you move the decimal place to the right in order to make 1.0 x 10?
= 10-6

14. Metric Prefixes Prefix Symbol Scientific Notation Decimal Common Word
tera T
 1012
trillion
giga G
 109
billion
mega M
 106
million
kilo k
103
1000
thousand
100 1
one
milli m
10-3
0.001
thousandth
micro μ
10-6
millionth
nano n
10-9
billionth
pico p
10-12
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. “all values of capacitance will be given as μF”
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 Hz
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?