Binary: how 0 and 1 carry all information

A text, a photo, a song, a bank transfer: inside a computer or a phone, all of it is stored and carried in the same form, a string of 0s and 1s. Using only two symbols looks poor. Yet it is the very reason digital electronics is reliable, fast and cheap.

This article gives an overview in six steps:

1. The basics: what a bit and a binary number are.
2. The history: who had the idea, and how several separate ideas came together.
3. Boolean algebra: how we reason with true and false.
4. Logic gates: how that reasoning becomes a component.
5. The flip-flop: how a circuit learns to hold on to a piece of data.
6. Electronics: how a 0 and a 1 actually live inside a chip.

It is written for two audiences: beginners, who will find every term defined where it first appears, and specialists, who will find deeper dives in the "Going further" boxes. The "Bridge to understanding" boxes fill in notions that are often assumed rather than really understood. Parts 3, 5 and 6 are deliberately overviews: each will be covered in detail in a dedicated article in the same series.

01The basics: bit, byte, binary system

• Bit (short for binary digit): the smallest quantity of information, which can only be 0 or 1. A question answered with yes or no is worth one bit.
• Binary system (or base 2): a way of writing numbers with only two digits. Our usual system, decimal (base 10), uses ten.
• Byte: a group of 8 bits.

The principle: in decimal, each position is worth ten times the previous one (units, tens, hundreds…). In binary, each position is worth twice the previous one: 1, 2, 4, 8, 16, 32…

Worked example: writing 13 in binary

We break 13 down into powers of two: 13 = 8 + 4 + 0 + 1. We write a 1 for each value used and a 0 for each value not used:

So 13 is written 1101 in binary. Going the other way, 1101 = 8 + 4 + 0 + 1 = 13.

How many values with n bits?

Each extra bit doubles the number of combinations. With 1 bit: 2 values. With 2 bits: 4 (00, 01, 10, 11). With 3 bits: 8. With 8 bits (one byte): 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256 values, that is the numbers from 0 to 255 (11111111 in binary).

02A history of binary: ideas that took centuries to meet

Binary was not invented in one go to build computers. Independent ideas, born for other reasons, came together in the 20th century: a way of writing numbers (Harriot, Caramuel, Leibniz), a way of calculating with true and false (Boole), a way of building all this with switches (Nakashima, Shannon) and electronic memory (Eccles and Jordan).

Writing numbers with two digits: Harriot, Caramuel, Leibniz

Unpublished manuscripts by the Englishman Thomas Harriot (1560-1621) show that he was already using the digits 0 and 1 around 1600, without seeing any practical use for them. In 1670 the Spaniard Juan Caramuel published a work studying base two, probably the first European publication on the subject. But it is Leibniz who sets out the system in a complete form.

Leibniz is regarded as the first mathematician to set out the binary system in a complete form. A manuscript dated 15 March 1679 already describes a mechanical binary calculator using balls to represent digits, which was never built. In 1703, after a correspondence with the Jesuit Joachim Bouvet, a missionary in China, who recognised Leibniz's binary numbers in the hexagrams of the Yi Jing (the "Book of Changes"), Leibniz sent the Royal Academy of Sciences in Paris his "Explication de l'arithmétique binaire" ("Explanation of binary arithmetic"), printed in 1705. For him, binary was first of all a tool for mathematical and philosophical thought, not a practical calculating tool.

George Boole: an algebra for logic (1854)

In 1854 Boole published "An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities". His central idea: logical reasoning (true/false, and, or, not) can be written with symbols and follow calculation rules. This is Boolean algebra, the subject of section 3. Boole was not thinking about machines: for decades his work remained a topic in logic and mathematics.

William Eccles and Frank Jordan: the electronic flip-flop (1918)

On 21 June 1918, Eccles and Jordan filed a British patent for a circuit with two vacuum tubes (glass bulbs that control the flow of current) that has two stable states, and described it in 1919 in the journal The Electrician. They were thinking of telegraphy and telephony, not computing. The name "flip-flop" caught on afterwards. The circuit would be used in the Colossus code-breaking computer (1943, or 1944 depending on the source), then in the American ENIAC, and would become the basic building block of electronic memory. We come back to it in section 5.

Nakashima and Shannon: linking Boolean algebra to electricity (1934-1938)

In the 1930s, several engineers realised, independently, that Boolean algebra describes relay circuits: Nakashima in Japan, Victor Shestakov in the USSR, and then Shannon. In his 1937 master's thesis, published in 1938, "A Symbolic Analysis of Relay and Switching Circuits", Shannon shows that it can be used to simplify telephone exchange circuits, and conversely that relays can solve Boolean algebra problems. His work, more general, overshadowed the other two and founded the modern design of digital circuits.

In 1948, in "A Mathematical Theory of Communication", published in the Bell System Technical Journal, Shannon founded information theory. He notes there that a flip-flop, having two stable states, can store one bit of information. The threads of the story are now joined.

03Boolean algebra: reasoning with true and false

3.1 The idea: turning a question into a variable

A Boolean variable is a quantity that can only be 0 (false) or 1 (true). Whenever a question can be answered with yes or no, we can write it as a variable:

• A = "the door is open": A is 1 if yes, 0 otherwise.
• B = "it is raining": B is 1 if it is raining, 0 otherwise.

Boolean algebra is the set of rules for combining these variables. It rests on three basic operations, which we already know from everyday language.

3.2 The three basic operations

NOT (negation, written ¬A or A with a bar): flips the value. "It is not raining" is true when "it is raining" is false.

AND (conjunction, written A·B): is 1 only if both are 1. "I go out if I have my keys AND my wallet": you need both.

OR (disjunction, written A+B): is 1 if at least one of the two is 1. "I take an umbrella if it rains OR if it is windy": either is enough (and both together too).

A truth table lists every possible combination of inputs and the matching output:

04Logic gates: from algebra to component

A logic gate is an electronic circuit that performs a Boolean operation: it takes one or more bits as input and produces one bit as output. Boolean algebra provides the rules; gates build them in hardware.

There is a gate for each operation in section 3 (NOT, AND, OR) and a few very common combinations: NAND ("NOT-AND"), NOR ("NOT-OR") and XOR ("exclusive OR", true when the two inputs are different). By assembling them, we build everything a computer can calculate.

Example: to add two bits, the sum is an XOR of the two and the carry is an AND. Two gates are therefore enough to write 1 + 1 = 10 in binary (sum 0, carry 1). Shannon already presented a base-two adder at the end of his 1937 thesis.

05The flip-flop: giving circuits a memory

A computer has to remember: the result of a calculation so it can use it in the next step, the key you have just pressed, the value of a counter. It needs a component whose output stays after the command has gone, like a bedside lamp with a toggle switch: you press once, it lights up and stays lit, whereas a push-button only lights it while you press.

This component is called a flip-flop (or latch). It has two stable states: once placed in one state, it stays there even when the command has gone. It therefore stores one bit. Picture a ball resting in a landscape with two hollows: in either one it stays still, and it takes a push to move it from one to the other.

The principle relies on feedback: the output of each gate is wired to the input of the other, so each gate "listens" to the other and confirms it. This is exactly the Eccles and Jordan circuit, built today with transistors.

That changes everything. Putting 8 flip-flops side by side stores one byte: this is a register, a basic building block of a processor. A circuit that can remember "where it is" can follow a series of steps, that is, run a program. A clock (a regular signal that sets the rhythm of circuits) decides at which instant each flip-flop records its data.

06Binary in electronics: from voltage to 0 and 1

6.1 A bit is a voltage

In a circuit, a bit is not an abstract symbol: it is a range of voltage, measured in volts (V). By convention, a low voltage stands for 0 and a high voltage stands for 1. The component that does the switching is the transistor, a switch controlled by an electrical signal: it lets current through (the "closed" state) or blocks it (the "open" state).

Successive technologies have changed the component, never the principle: electromechanical relays, vacuum tubes (as in the Eccles and Jordan circuit), then transistors, now packed by the billion onto a single chip.

6.2 Why only two levels? The noise margin

Noise is the set of stray electrical disturbances added to a signal (interference, supply variations, signals from neighbouring wires). With only two well-separated levels, a small disturbance doesn't change the interpretation. This is the deep reason for the binary choice in electronics: robustness.

So that two circuits understand each other, each circuit family sets four voltages:

• VOL (output voltage in the low state, Output Low): the maximum a transmitted 0 can reach.
• VOH (Output High): the minimum a transmitted 1 guarantees.
• VIL (Input Low): the maximum an input can reach and still be read as 0.
• VIH (Input High): the minimum an input needs to be read as 1.

The noise margin is the safety gap between what an output guarantees and what an input requires. Take the TTL (Transistor-Transistor Logic) family, a historic family of logic circuits powered at 5 V and well documented. The values below come from the Texas Instruments datasheet for the SN7400 (a NAND gate in the standard TTL series):

Between 0.8 V and 2.0 V the level is undetermined: the input guarantees no reading. These values are worst-case guarantees: the same datasheet gives more favourable typical values (for example a typical high output of 3.4 V).

Worked example: a gate outputs a 0 at 0.4 V (worst case). A 0.3 V disturbance gives 0.7 V, below 0.8 V: the 0 is read correctly. Conversely, a 1 output at 2.4 V with a 0.5 V disturbance falls to 1.9 V, below 2.0 V: the reading is no longer guaranteed. The guaranteed noise margin here is 0.4 V on each side (0.8 − 0.4 and 2.4 − 2.0).

07Limits and common misconceptions

• "A computer only understands 0 and 1" is a simplification. It handles voltages and switches; 0 and 1 are just our way of writing them down.
• Binary isn't magic. A single bit means nothing on its own: you need a convention (encoding) to know whether it is a number, a letter or a colour.
• Noise isn't eliminated, it is contained. Beyond the noise margin, errors appear, hence the error-detection and error-correction codes in memories and transmissions.
• Undetermined states exist. The forbidden zone between VIL and VIH, flip-flops that can hesitate between 0 and 1: the real world is never perfectly binary.
• The history is more nuanced than the shortcuts. Leibniz was not the first to write numbers in binary (Harriot, Caramuel); it was not he but Bouvet who linked his system to the Chinese hexagrams; and Shannon was not the only one to link Boolean algebra to circuits (Nakashima, Shestakov).

08Conclusion

Binary brings together three things that fit into one another: a number system (from Harriot and Caramuel to Leibniz), a logic (Boole) and a physical implementation (Nakashima, Shannon, Eccles and Jordan, then the transistor). Gates calculate, flip-flops remember, voltages carry it all, and the noise margin makes the whole thing reliable.

What stands out is the gap in time: two centuries separate Leibniz's idea from its use, and none of the pioneers mentioned was aiming for the computer. Understanding binary is therefore understanding how abstract ideas, born elsewhere, become machines when another field (here, electricity) gives them something to run on.

The series continues with three articles: Boolean algebra and learning to calculate, flip-flops, and binary in electronics.

09Glossary

• Bit: a binary digit, 0 or 1.
• Boolean algebra: an algebra in which variables are only 0 or 1, with the operations AND, OR, NOT.
• Byte: a group of 8 bits (256 values).
• Clock: a regular signal that sets the rhythm of sequential circuits.
• Encoding: a convention that links a meaning (letter, colour…) to a number.
• Feedback: sending a circuit's output back to its input.
• Flip-flop: a circuit with two stable states that stores one bit.
• Logic gate: a circuit that performs a Boolean operation.
• NAND, NOR: "NOT-AND" and "NOT-OR" gates, the inverses of AND and OR.
• Noise: stray electrical disturbances added to the signal.
• Noise margin: the guaranteed voltage gap between an output level and the matching input threshold.
• Register: a group of flip-flops storing several bits.
• Transistor: an electrically controlled switch.
• Truth table: a table of outputs for every possible input.
• TTL: a historic family of logic circuits; also the name of a set of voltage levels.
• XOR: exclusive OR, equal to 1 if the two inputs differ.