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hambasics:sections:mathbasics [2025/06/25 20:39] – [Base 2] va7fihambasics:sections:mathbasics [2026/04/01 21:33] (current) – [Conversion] va7fi
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 A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit.
  
-In the previous section, we saw that Mhz means a million Hertz.  Here's a list of the most common ones:+In the previous section, we saw that MHz means a million Hertz.  Here's a list of the most common ones:
  
 <WRAP indent> <WRAP indent>
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 {{ youtube>3Dry9HUSkTo }} {{ youtube>3Dry9HUSkTo }}
 +
  
 ===== Alternative Formulation ===== ===== Alternative Formulation =====
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 ===== dBm ===== ===== dBm =====
-A related measurement is the dBm.  While the decibel (dB) is a ratio between two quantities (saying "20 dB" is the same as saying "100 times more"), the dBm is a ratio between one quantity and 1 mW.  That is, it's a measures of how much stronger (or weaker) the power of something is compared to 1 milliwatt. 
  
-For example, a typical fibre optic light signal to a house for internet has a light level of -16 dBm.  Before we go through the math, recall that 16 dB = 10 dB + 3 dB + 3 dB.  Which means, the ratio is 10 × 2 × 2 = 40.  Let's now see why -16 dBm is equal to a power of 25 μW (microwatts):+A related measurement is the dBm.  While the decibel (dB) is a ratio between two quantities (saying "20 dB" is the same as saying "100 times more"), the dBm is a ratio between one quantity and 1 mW.  That is, it's a measure of how much stronger (or weaker) the power of something is compared to 1 milliwatt. 
 + 
 +For example, a typical fibre optic light signal to a house for internet has a light level of -16 dBm.  Before we go through the math, recall that \$16 \text{ dB} = 10 \text{ dB} + 3 \text{ dB} + 3 \text{ dB}\$.  Which means, the ratio is \$10 \times 2 \times 2 = 40\$.  Let's now see why -16 dBm is equal to a power of 25 μW (microwatts):
 <WRAP centeralign> <WRAP centeralign>
 \$$-16 \text{ dBm} = 1 \text{ mW} \div 10^{1.6}  \approx 1 \text{ mW} \div 40 = 0.025 \text{ mW} = 25~\mu \text{W} \$$ \$$-16 \text{ dBm} = 1 \text{ mW} \div 10^{1.6}  \approx 1 \text{ mW} \div 40 = 0.025 \text{ mW} = 25~\mu \text{W} \$$
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 Notice how the minus sign in front of the dBm means that the power is less (not more) so it divides (not multiply). Notice how the minus sign in front of the dBm means that the power is less (not more) so it divides (not multiply).
 +
  
 ====== Binary Numbers ====== ====== Binary Numbers ======
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 Do you see the pattern?  Here's another way of writing it with leading zeros and keeping track of where we are in base 10: Do you see the pattern?  Here's another way of writing it with leading zeros and keeping track of where we are in base 10:
-<WRAP prewrap center 10em>+<WRAP indent>
 ^Base 2 ^ Base 10 | ^Base 2 ^ Base 10 |
 |0000   | 0       | |0000   | 0       |
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 Converting the other way around is a bit different, but kind of makes sense if you think about it.  Here are the steps to convert \$ 13_{10} \$ back to \$1101_2\$ and I'll explain a bit more after: Converting the other way around is a bit different, but kind of makes sense if you think about it.  Here are the steps to convert \$ 13_{10} \$ back to \$1101_2\$ and I'll explain a bit more after:
  
-<WRAP prewrap center 35em>+<WRAP indent>
 ^ Division by 2 ^In Decimal ^As fraction           ^ Quotient ^ Remainder ^ ^ Division by 2 ^In Decimal ^As fraction           ^ Quotient ^ Remainder ^
 |13÷2            | 6.5       |6 + \$\frac{1}{2}\$  | 6        | 1         | |13÷2            | 6.5       |6 + \$\frac{1}{2}\$  | 6        | 1         |
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 Let's look at a trivial example converting 2037 from base 10 to base 10 (trivial indeed!): Let's look at a trivial example converting 2037 from base 10 to base 10 (trivial indeed!):
  
-<WRAP prewrap center 35em>+<WRAP indent>
 ^ Division by 10 ^In Decimal   ^As fraction             ^ Quotient ^ Remainder  ^ ^ Division by 10 ^In Decimal   ^As fraction             ^ Quotient ^ Remainder  ^
 |2037÷10         |203.7        |203 + \$\frac{7}{10}\$  | 203      | 7          | |2037÷10         |203.7        |203 + \$\frac{7}{10}\$  | 203      | 7          |
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           &= 63003_{10}           &= 63003_{10}
 \end{align*} \end{align*}
 +
 +One reason why hexadecimal is so useful in computer science is because one hexadecimal "digit" can represent four binary "digits".  For example,
 +<WRAP indent>
 +^  Decimal ^  Hexadecimal ^      Binary ^
 +|        1 |            1 |           1 |
 +|        2 |            2 |          10 |
 +|        3 |            3 |          11 |
 +|        4 |            4 |         100 |
 +|  ...                                |||
 +|        7 |            7 |         111 |
 +|        8 |            8 |        1000 |
 +|  ...                                |||
 +^       15 ^            F ^        1111 |
 +|       16 |           10 |      1 0000 |
 +|  ...                                |||
 +|       31 |           1F |      1 1111 |
 +|       32 |           20 |     10 0000 |
 +|  ...                                |||
 +|       63 |           3F |     11 1111 |
 +|       64 |           40 |    100 0000 |
 +|  ...                                |||
 +|      127 |           7F |    111 1111 |
 +|      128 |           80 |   1000 0000 |
 +|  ...                                |||
 +^      255 ^           FF ^        1111 1111 |
 +|      256 |          100 |      1 0000 0000 |
 +|  ...                                     |||
 +^     4095 ^          FFF ^   1111 1111 1111 |
 +|     4096 |         1000 |  1 0000 0000 0000 |
 +
 +</WRAP>
  
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hambasics/sections/mathbasics.1750909198.txt.gz · Last modified: by va7fi