Binary Numbers Explained for Beginners

Counting in any base

Every number system you have ever used works the same way. The digits are worth different amounts depending on where they sit, and the multiplier for each position is a power of the base.

In decimal, the number 4,271 means:

4 x 1000  =  4000     (4 x 10^3)
2 x 100   =   200     (2 x 10^2)
7 x 10    =    70     (7 x 10^1)
1 x 1     =     1     (1 x 10^0)
            ------
              4271

Binary is identical except the base is 2, so the positions are worth 1, 2, 4, 8, 16, 32 and so on. The number 1101 means:

1 x 8  =  8     (1 x 2^3)
1 x 4  =  4     (1 x 2^2)
0 x 2  =  0     (0 x 2^1)
1 x 1  =  1     (1 x 2^0)
         ----
           13

That is the whole idea. There is nothing more to positional notation than “multiply each digit by its place value and add up”. What makes binary feel unfamiliar is only that you have decades of practice with the base-10 version.

Contrast this with a non-positional system such as Roman numerals, where X always means ten regardless of position. That is precisely why Roman numerals are hopeless for arithmetic and why the Hindu-Arabic system displaced them.

Why computers use two digits

The answer is physics, not mathematics. A computer could in principle work in base 10, and some early machines tried — ENIAC was decimal internally.

The problem is reliability. To represent ten digits electronically you need ten distinguishable voltage levels. If your supply is 5 V, each level is separated by about half a volt, and half a volt of electrical noise — from a nearby switching transistor, a fluctuating supply, or heat — turns a 6 into a 7.

With two levels the margin is enormous: anything below 0.8 V is 0 and anything above 2.0 V is 1. A transistor does not need to be a precise analogue device, only a reliable switch. That tolerance is what allows a modern processor to perform billions of operations per second with essentially zero error rate.

The same two-state logic applies at every layer:

• A transistor conducts or it does not.
• A capacitor in DRAM holds charge or it does not.
• A magnetic domain on a hard disk points one way or the other.
• A pit on an optical disc reflects or scatters.
• A fibre-optic pulse is present or absent.

Claude Shannon proved in 1948 that any information whatsoever — text, images, sound, video — can be represented as a sequence of binary digits without loss. The bit became the universal unit of information, and the practical advantages of two-state electronics did the rest.

Bits, bytes and why eight

A single bit holds one of two values. Group bits together and the number of possible combinations doubles with each one:

Eight bits became the standard byte with IBM’s System/360 in 1964. It was not inevitable — earlier machines used 6, 7, 9 and 12-bit bytes — but eight won because it comfortably held one character, it was a power of two (making address arithmetic cheap), and it divided neatly into two 4-bit halves that each map to a single hexadecimal digit.

That last property is why hexadecimal is used everywhere binary data is displayed: one byte is always exactly two hex digits, from 00 to FF. See the conversion chart for every value.

Converting by hand

Binary to decimal: add the place values

Write the place values above the digits and add up the ones with a 1 beneath them.

128  64  32  16   8   4   2   1
  1   0   1   1   0   1   0   1

128 + 32 + 16 + 4 + 1 = 181

Decimal to binary, method 1: subtract the largest power

Find the largest power of two that fits, subtract it, repeat. Faster than division once you know the powers.

181 - 128 = 53   -> bit 128 = 1
 53 -  32 = 21   -> bit  64 = 0, bit 32 = 1
 21 -  16 =  5   -> bit  16 = 1
  5 -   4 =  1   -> bit   8 = 0, bit 4 = 1
  1 -   1 =  0   -> bit   2 = 0, bit 1 = 1

Result: 10110101

Decimal to binary, method 2: divide by two

Divide repeatedly by 2, recording each remainder, then read the remainders bottom to top.

181 / 2 = 90 r 1
 90 / 2 = 45 r 0
 45 / 2 = 22 r 1
 22 / 2 = 11 r 0
 11 / 2 =  5 r 1
  5 / 2 =  2 r 1
  2 / 2 =  1 r 0
  1 / 2 =  0 r 1

Read upwards: 10110101

Binary to hex: split into nibbles

Group the bits in fours from the right and convert each group separately. No arithmetic across groups is needed.

1011 0101
  B    5     ->  0xB5  ->  181

Check any of these with the number system converter, which shows the full working.

Negative numbers and two’s complement

Binary has no minus sign, so a convention is needed. The obvious approach — reserve the top bit as a sign flag — was tried and abandoned, because it gives two representations of zero (+0 and −0) and requires separate circuits for addition and subtraction.

Every modern computer uses two’s complement. To negate a number, invert every bit and add one:

  5 =  00000101
invert  11111010
add 1   11111011  = -5

This has an elegant consequence: subtraction becomes addition. To compute 12 − 5, add 12 and −5:

  00001100   (12)
+ 11111011   (-5)
-----------
 100000111   -> discard the 9th bit -> 00000111 = 7

One adder circuit handles both operations, there is exactly one zero, and comparison still works. The only oddity is the asymmetric range: eight signed bits cover −128 to +127, because zero occupies one of the “positive” slots. Try it in the bitwise calculator at 8-bit width.

How everything becomes binary

Text is a lookup. Each character has a numeric code point defined by Unicode, and that number is stored in binary. The letter A is 65, which is 01000001. Watch it happen in the text to binary converter.

Images are grids of pixels, each pixel three or four numbers. A pure red pixel is red 255, green 0, blue 0 — 11111111 00000000 00000000. A 1920×1080 uncompressed image is 6.2 million bytes, which is why compression exists.

Sound is a wave sampled thousands of times per second. CD audio measures amplitude 44,100 times per second and stores each measurement as a 16-bit number, giving 65,536 possible levels.

Video is images plus sound plus time, which is why compression matters so much: uncompressed 1080p at 30 fps is about 187 MB per second.

Programs are numbers too. Each machine instruction has a numeric opcode; the processor reads them as binary and executes. There is no fundamental distinction between code and data at the hardware level — which is both the foundation of stored-program computing and the reason buffer overflow attacks work.

Practice problems

Work these out, then check them in the converter.

1. Convert 11001 to decimal.
2. Convert 200 to binary.
3. What is 1010 + 0110 in binary?
4. How many bits are needed to represent 1,000 distinct values?
5. What is 0xFF in binary and decimal?
6. Represent −12 in 8-bit two’s complement.
7. What ASCII character is 01001010?

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