Lesson 2 of 4Introduction to Computers

Unit 1 · Lesson 2 of 4

Binary, Decimal and Hexadecimal

Why computers count in twos, why programmers write in sixteens, and how to convert between all three without a calculator.

3 min read11 practice questionsInteractiveAsk AI about this →

Key things to remember

1Binary place values from the right: $1, 2, 4, 8, 16, 32, 64, 128$ — doubling each time.
2An $n$ -bit number can represent $2^{n}$ different values, from $0$ to $2^{n} - 1$ .
3One byte = 8 bits = values 0 to 255 = exactly two hex digits.
4Four bits = one hex digit, always. $16 = 2^{4}$ is the reason hex exists.
5Decimal → binary: divide by 2 repeatedly and read the remainders BOTTOM to TOP.
6Hex digits above 9: A=10, B=11, C=12, D=13, E=14, F=15.

A transistor is a switch. It is either conducting or it is not — on or off, high voltage or low. There is no reliable "a bit on", because component tolerances, temperature and electrical noise would make a ten-level signal unreadable. So computers count in a system with exactly two digits: binary. Everything else follows from that one hardware fact.

Place value is the whole trick

You already understand binary, because you already understand place value. In decimal, 305 means 3 hundreds, 0 tens, 5 ones — each column is worth ten times the one to its right, because there are ten digits. In binary there are two digits, so each column is worth twice the one to its right: 1, 2, 4, 8, 16, 32, 64, 128. Nothing else changes.

Fig 1One byte read three ways. The bits never change — only how we group them.

Binary to decimal

Write the place values above the bits, then add up only the places with a 1 underneath. That is the entire method, and it works for any width.

1011010 1_{2} = 128 + 32 + 16 + 4 + 1 = 18 1_{10}

Add the columns that hold a 1; ignore the rest.

Decimal to binary

Two methods work, and you should know both. The subtraction method is faster once you know the powers of two by heart: take the largest power of 2 that fits, write a 1, subtract it, repeat. The division method is more mechanical and harder to get wrong: divide by 2 over and over, writing down each remainder, then read the remainders bottom to top.

Divide by 2	Quotient	Remainder
181 ÷ 2	90	1
90 ÷ 2	45	0
45 ÷ 2	22	1
22 ÷ 2	11	0
11 ÷ 2	5	1
5 ÷ 2	2	1
2 ÷ 2	1	0
1 ÷ 2	0	1

Read the remainder column from the BOTTOM up: 10110101.

The single most common error

Reading the remainders top-to-bottom instead of bottom-to-top. The last remainder you calculate is the most significant bit. Always check your answer by converting it back — if it does not give the number you started with, you read the column the wrong way.

Why hexadecimal exists

Binary is correct but unreadable — nobody can dictate 11010110 over the phone without a mistake. Hexadecimal (base 16) fixes this because 16 is 2⁴, so exactly four bits make exactly one hex digit, with no arithmetic across the boundary. A byte becomes two characters. A colour becomes six. A memory address becomes eight. That is why every error code, MAC address and colour code you will ever see is written in hex.

Decimal	Binary (4-bit)	Hex
0	0000	0
1	0001	1
2	0010	2
3	0011	3
4	0100	4
5	0101	5
6	0110	6
7	0111	7
8	1000	8
9	1001	9
10	1010	A
11	1011	B
12	1100	C
13	1101	D
14	1110	E
15	1111	F

Memorize this table. Every hex conversion is four lookups, not a calculation.

Never go through decimal

To convert binary to hex, split into nibbles of four bits from the RIGHT and look each one up. 10110101 → 1011 0101 → B5. Converting to decimal first is slower and adds a place to make an arithmetic slip.

Loading interactive…

Flip bits and watch all three representations move together.

Notation

Binary: a subscript 2 ( $101 1_{2}$ ), a 0b prefix (0b1011), or the word stated — never assume.
Hexadecimal: a subscript 16, a 0x prefix (0xB5), or a # in colour codes (#FF8800).
Decimal: usually nothing, or a subscript 10 when the context is ambiguous.
101 with no marking is genuinely ambiguous — it is 101 in decimal, 5 in binary, and 257 in hex.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Convert

1100111 0_{2}

to decimal.

Example 2

Convert

17 3_{10}

to 8-bit binary using the division method.

Example 3

Convert

0 x 3F

to binary and to decimal.

Example 4

How many different values can a 12-bit number represent, and what is the largest one?

Where marks get lost

Reading division-method remainders top-to-bottom instead of bottom-to-top, which reverses the whole answer.
Splitting a binary number into nibbles from the LEFT. Always group in fours starting from the right, padding the left with zeros.
Writing $2^{8} = 256$ and then claiming the largest 8-bit value is 256. It is 255, because counting starts at 0.
Treating hex letters as variables. In $0 x B_{5}$ , B is the digit 11, not an unknown.
Converting binary to hex by going through decimal — it works but wastes time and invites arithmetic errors.

Flashcards 9

Why do computers use binary?

Binary place values (8 bits)

How many values in n bits?

Largest 8-bit unsigned value

+ 5 more

Practice

Convert

1001011 0_{2}

to decimal.

Convert

0 x C_{4}

to decimal.

Convert

9 2_{10}

to 8-bit binary.

Convert

1110101 0_{2}

to hexadecimal (two digits, no prefix).

How many distinct values can 6 bits represent?

Why is hexadecimal preferred over decimal for writing memory addresses?

What is the largest value that can be stored in 10 bits?

Order these values from smallest to largest.

1 $0 x 2A$
2 $0 x FF$
3 $6 0_{10}$
4 $0000111 1_{2}$

Match each 4-bit group to its hex digit.

1010

1100

0111

1111

A colour is written #FF8800. What is the decimal value of the green channel (the middle two hex digits, 88)?

A student converts

4 5_{10}

by dividing by 2 and writes the remainders as 1, 0, 1, 1, 0, 1 reading downwards. What is the correct binary answer?

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