Why Digital, and Why Binary

In an analogue system, the quantity of interest is the voltage: 2.4 V might mean 24 °C. Between any two voltages there is another one, so a single wire theoretically carries unlimited precision. That sounds like a strictly better deal than two levels, and on a perfect wire it would be.

Real wires pick up interference from every motor, radio and neighbouring track. If 2.4 V leaves and 2.29 V arrives, the received value is a perfectly legal analogue value — indistinguishable from a genuine 2.29 V. No circuit downstream can detect the error, let alone repair it.

Worse, the errors accumulate. Each stage adds its own, so a signal through ten stages is measurably wrong and through a thousand is meaningless. This is why a tenth-generation tape copy is unlistenable.

Stop using the exact voltage. Declare that anything above a threshold means 1 and anything below another means 0, and deliberately leave the range between them unused.

Send 3.3 V, receive 2.9 V after the same interference — still comfortably logic 1. The information was never in the precise voltage, so corrupting the voltage did not corrupt the information. That unused middle range is the noise margin, and it is the entire product.

A gate does not pass its input along. It decides which band the input is in, and then drives its own output to a full, clean level. The noise is discarded rather than carried forward.

That is why a signal can pass through billions of transistors and arrive exactly as correct as it started. Errors cannot accumulate, because every stage throws the accumulated error away. Analogue circuits have no equivalent, and this single asymmetry is why digital won.

Noise margin

How much interference a signal can absorb and still be read correctly. NM_high = V_OH − V_IH and NM_low = V_IL − V_OL: the gap between what a driver guarantees to output and what a receiver promises to accept.

V_OH / V_OL

The output levels a gate guarantees to drive — deliberately further from the middle than the input thresholds require.

V_IH / V_IL

The input levels a gate guarantees to interpret correctly. The deliberate gap between these and the output levels is what creates the margin.

More levels would carry more data per wire — four levels is two bits, ten is about 3.3. But squeezing more levels into the same 0–5 V narrows every gap, and margin is the product being bought.

• Widest margins. Two levels leaves the largest possible gap, which is what tolerates temperature, ageing and interference.
• Matches the hardware. A transistor is a switch — naturally either conducting or not. No comparator is needed to tell two states apart.
• Boolean algebra already exists. A complete, well-understood mathematics for two-valued logic was available a century before anyone needed it.
• Cheap and reproducible. Only two states must be manufactured reliably, across billions of devices.

Multi-level signalling is not fantasy — flash memory stores several bits per cell exactly this way, and pays for it in error-correction machinery and reduced write endurance.

One wire distinguishes two values: one bit. Put n wires side by side and you can distinguish 2ⁿ patterns — 8 wires give 256.

The hardware attaches no meaning to a pattern. Deciding that 10000001 is 129, or −127, or the letter Á, or an instruction, is a convention chosen by whoever designed the system. Choosing those conventions is the whole of the next module.