Chapter 4: Metric Prefixes & Powers of Ten Gigafun with nanoeffort Presented by: James, VE3BUX

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

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 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 9 7 1 1 2 3 5 8 6 4 7 2 1 4 5

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 . 9 0.71 7 1 0.123 2 3 0.5864 5 8 6 4

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

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!

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 300000000m s-1 300000000m s-1 x10 300000000m s-1 8 7 6 5 4 3 2 1

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 / 30000000Hz 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?

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

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

Reciprocal Values What can we say about a value such as: 1 . 101 1 . 101 What about making it: 100 .

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

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

Metric Prefixes Prefix Symbol Scientific Notation Decimal Common Word tera T  1012 1000000000000 trillion giga G  109 1000000000 billion mega M  106 1000000 million kilo k 103 1000 thousand 100 1 one milli m 10-3 0.001 thousandth micro μ 10-6 0.000001 millionth nano n 10-9 0.000000001 billionth pico p 10-12 0.000000000001 trillionth

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?

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

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

“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

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

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

Questions?