Electronics & Circuits · Study deck
Binary Number Systems
Picture a pressure alarm stored as eight bits.
Voltage Vera is your guide for this deck.
After studying this chapter
Learning objectives
You will be able to:
- Convert Binary Numbers: Translate between decimal and binary number systems using place-value decomposition
- Apply Powers of 2: Calculate ADC resolution ranges from bit depth
- Interpret Place Values: Use positional notation in base-2 arithmetic
- Analyse Floating Point: Explain IEEE 754 representation and identify its precision limitations
Major section
Start Simple
The same row can mean a value, several flags, or part of a signed number.
- The hardware contract must say which meaning is correct.
- An analog-to-digital converter is a circuit that turns a changing voltage into a number.
- ADC is its short name.
- This runway does not prove measurement accuracy or every number format.
Major section
Phoebe's Field Notes: How Many Bits Does A Reading Actually Need
The mathematical gist.: To resolve a smallest change \(\Delta\) across span V_{FS}, require \(2^N\ge V_{FS}/\Delta\), hence \(N_{min}=\lceil\log_2(V_{FS}/\Delta)\rceil\).
- Resolving 1% of a 50 mV conditioned span needs 7 bits; leaving that signal on a bare 5 V range needs 14 bits.
Major section
For Kids: Meet the Sensor Squad!
But your feelings are analog - smooth like sliding down a slide." This was a big problem!
- Analog and digital are like the difference between a slide (smooth) and stairs (steps)!
- I can only work with exact step numbers, like climbing stairs.
- The Sensor Squad's translation team was complete!
Major section
For Kids: Meet the Sensor Squad! (continued)
But I can make my stairs have MORE steps - instead of big steps like 20, 21, 22, I can use tiny steps like 20.0, 20.1, 20.2.
- Temperature Terry had a problem. "Max, I can feel temperatures that go smoothly from cold to hot - like 20.1, 20.2, 20.3 degrees.
- He's a translator!" ADC Andy arrived with a big smile. "I can help!
- The more steps I use, the closer I get to Sammy's smooth feelings!" Now Sammy could measure temperatures, Andy would translate them into step-numbers, and Max could process them.
Major section
Getting Started: Analog vs Digital
The analog trace can occupy any intermediate voltage, while the digital trace can occupy only one of the four pictured step levels.
- The left-hand trace has effectively infinite possible values; the right-hand trace assigns each sample to one of $2^n$ fixed levels.
- Everything we want to measure is analog, but computers only understand digital!
Major section
Binary Number System
An $n$-bit word provides $2^n$ combinations, but an unsigned ADC labels them from 0 through $2^n-1$: 4 bits give codes 0–15 and 8 bits give codes 0–255.
- The same doubling rule explains why adding one register bit doubles the representable states.
Deck summary
Key takeaways
The same row can mean a value, several flags, or part of a signed number.
- The mathematical gist.: To resolve a smallest change \(\Delta\) across span V_{FS}, require \(2^N\ge V_{FS}/\Delta\), hence \(N_{min}=\lceil\log_2(V_{FS}/\Delta)\rceil\).
- But your feelings are analog - smooth like sliding down a slide." This was a big problem!
- But I can make my stairs have MORE steps - instead of big steps like 20, 21, 22, I can use tiny steps like 20.0, 20.1, 20.2.
Retrieval practice
Recall check 1 of 3
Voltage Vera says: answer from memory, then check your reasoning.
Q1Why should you avoid using == to compare floating-point numbers in IoT sensor code?
Show answer
Answer: B
Q2What is the decimal value of binary 1101 ?
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Answer: B Answer: B) 13 $$1101_2 = (1×8) + (1×4) + (0×2) + (1×1) = 8 + 4 + 0 + 1 = 13_{10}$$.
Retrieval practice
Recall check 2 of 3
Voltage Vera says: answer from memory, then check your reasoning.
Q3How many different values can a 16-bit ADC represent?
Show answer
Answer: C Answer: C) 65,536 $$Values = 2^{bits} = 2^{16} = 65,536$$ The actual digital output ranges from 0 to 65,535 (one less than total values).
Q4If you increase an ADC from 10-bit to 12-bit, how much more precise is each voltage step?
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Answer: B Each additional bit doubles the resolution: 10-bit = 1,024 levels, 12-bit = 4,096 levels.
Retrieval practice
Recall check 3 of 3
Voltage Vera says: answer from memory, then check your reasoning.
Q5Place each binary-conversion element where it lives so you can prove that 100101 represents decimal 37.
Show answer
Answer: A Separate the decimal input, active bit weights, and final sum so you can verify a binary conversion instead of memorising the bit string.
Print reference
Answers 1 of 2
Answer key.
- B
- B · Answer: B) 13 $$1101_2 = (1×8) + (1×4) + (0×2) + (1×1) = 8 + 4 + 0 + 1 = 13_{10}$$.
- C · Answer: C) 65,536 $$Values = 2^{bits} = 2^{16} = 65,536$$ The actual digital output ranges from 0 to 65,535 (one less than total values).
- B · Each additional bit doubles the resolution: 10-bit = 1,024 levels, 12-bit = 4,096 levels.
Print reference
Answers 2 of 2
Answer key.
- A · Separate the decimal input, active bit weights, and final sum so you can verify a binary conversion instead of memorising the bit string.