Electronics & Circuits · Study deck

Binary Number Systems

Picture a pressure alarm stored as eight bits.

Voltage Vera is your guide for this deck.

analogdigitalbinary

Voltage Vera, the module guide, in a scene from this chapter.

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!

ADC conversion from a physical quantity through a sensor and analog voltage to a bounded digital code interpreted by software

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!
Each arrow changes the quantity or its representation.

Analog vs Digital Signals: Continuous Ramp vs Discrete Steps

Major section

Binary Number System

Each added bit has two states and doubles the combinations, so four bits produce $2^4=2\times2\times2\times2=16$ possible values.

Eight bits similarly provide 256 values numbered 0 through 255.
This distinction matters for ADC work: $2^n$ counts the available quantization levels, while $2^n-1$ is the largest code.
Because an unsigned code begins at zero, those values are numbered 0 through 15 rather than 1 through 16.

Major section

ADC Resolution vs Range

The Mistake: Assuming that a "12-bit ADC" can measure 12 volts, or that increasing bit depth extends the measurable voltage range.

For a 12 V input, the pin is 12/6.6 = 1.818 V; a calibrated 1.818 V reading gives 12.0 V.
Raw code alone is not a 3.3 V/4095 voltmeter.
They expected accurate readings because "12-bit should handle 12 volts." The ESP32 immediately damaged because its maximum input is 3.3V, not 12V.

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 test whether two computed floating-point sensor values are approximately equal?

ASlow comparisons make exact floating-point equality unsuitable in real time.

BRounding in binary floating point makes exact comparison unreliable.

CFloating-point values cannot be compared by any numeric operator.

DCompiler errors prevent exact equality checks on computed sensor values.

Show answer

Answer: B see answers page

Q2What is the decimal value of binary 1101?

ADecimal 11 (binary 1011)

BDecimal 13 (binary 1101)

CDecimal 15 (binary 1111)

DDecimal 9 (binary 1001)

Show answer

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?

A16,384

B32,768

C131,072

D65,536

Show answer

Answer: D 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?

A2× more precise (2 more bits = double precision)

B12× more precise (12 bits vs 10 bits)

C1.2× more precise (linear relationship)

D4× more precise (2^2 = 4 times more voltage levels)

Show answer

Answer: D 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.

Q5What quantity is being encoded before any binary bit position is interpreted in the 37 example?

ADecimal input: 37

BBit 5 = 1 (value 32)

CBit 2 = 1 (value 4)

DBit 0 = 1 (value 1)

Show answer

Answer: A 37 is the quantity being encoded before any bit position is interpreted.

Print reference

Answers 1 of 2

Answer key.

B · Due to binary representation limitations, operations like 0.1 + 0.2 may not exactly equal 0.3 in floating-point arithmetic. Use threshold comparison instead: if (abs(a - b) < 0.0001). This is critical for IoT sensor thresholds, PID controllers, and any comparison involving floating-point sensor readings to avoid unpredictable behavior.
B · Answer: B) 13 $1101_2 = (1×8) + (1×4) + (0×2) + (1×1) = 8 + 4 + 0 + 1 = 13_{10}$ .

Print reference

Answers 2 of 2

Answer key.

D · 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).
D · Each additional bit doubles the resolution: 10-bit = 1,024 levels, 12-bit = 4,096 levels.
A · 37 is the quantity being encoded before any bit position is interpreted.