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Digital Logic Fundamentals

Lessons

Senior 4 · ICT

Digital Logic Fundamentals

5 lessons

Lesson 1 of 5

Number Systems

25 min

A clock that counts in twos

On the front of the railway station in St. Gallen, Switzerland, a giant clock shows the time with nothing but lamps that are either on or off. There are no digits at all. Anyone who knows the binary number system can read it; everyone else sees a pattern of blue shapes.

Computers work the same way. Inside a processor, billions of transistors act as switches that are either on or off, and a wire is at either a high or a low voltage. Two states are easy to make, fast to switch and hard to confuse, so every number, letter, colour and sound in a computer is stored as a pattern of 0s and 1s. To understand the logic circuits in the rest of this unit, you first need to count the way they do.

Definition

Number system (base, radix): a way of writing numbers with a fixed set of digits, in which each position is worth a power of the base. The base is written as a subscript: $101_{2}$ , $57_{8}$ , $57_{10}$ , $2\mathrm{D}_{16}$ .

System	Base	Digits
Binary	2	0, 1
Octal	8	0 to 7
Decimal	10	0 to 9
Hexadecimal (hex)	16	0 to 9, then A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

Bit (binary digit): one 0 or 1. Nibble: 4 bits. Byte: 8 bits.

Place value in any base

In decimal, $157 = 1 \times 10^2 + 5 \times 10^1 + 7 \times 10^0$ . Every base works the same way: a number with digits $d_{n-1} \dots d_1 d_0$ in base $b$ has the value

$N = d_{n-1} \times b^{n-1} + \dots + d_1 \times b^1 + d_0 \times b^0$

In binary each place is worth a power of 2, so the place values, from the right, are 1, 2, 4, 8, 16, 32, 64, 128. A binary number is the sum of the place values where there is a 1.

So $10011101_2 = 128 + 16 + 8 + 4 + 1 = 157_{10}$ . With $n$ bits there are $2^{n}$ different patterns, so $n$ bits can store the whole numbers from $0$ to $2^{n} - 1$ : one byte holds 0 to 255.

Decimal to binary: divide by 2

Divide the number by 2 again and again, writing down each remainder, until the quotient is 0. The remainders read from the last to the first (bottom to top) are the binary digits.

Division	Quotient	Remainder
157 ÷ 2	78	1 (least significant bit)
78 ÷ 2	39	0
39 ÷ 2	19	1
19 ÷ 2	9	1
9 ÷ 2	4	1
4 ÷ 2	2	0
2 ÷ 2	1	0
1 ÷ 2	0	1 (most significant bit)

Reading upwards: $157_{10} = 10011101_2$ , which matches the place-value diagram. The same method with 8 or 16 in place of 2 converts decimal to octal or hexadecimal.

Binary to octal and hexadecimal: group the bits

Because $8 = 2^{3}$ and $16 = 2^{4}$ , every octal digit stands for exactly 3 bits and every hex digit for exactly 4 bits. Split the binary number into groups from the right, add zeros on the left of the last group if needed, and convert each group.

Check: $235_8 = 2 \times 64 + 3 \times 8 + 5 = 157$ and $9\mathrm{D}_{16} = 9 \times 16 + 13 = 157$ . To go back from hex or octal to binary, write each digit as 4 (or 3) bits. Hex is how programmers write binary compactly: a byte is always exactly two hex digits, from 00 to FF. The web colour #1E90FF is three bytes: red $1\mathrm{E}_{16} = 30$ , green $90_{16} = 144$ , blue $\mathrm{FF}_{16} = 255$ .

Crash Course Computer Science #4: Representing Numbers and Letters with Binary. Watch how place value works in base 2, why 8 bits make a byte, and how letters are stored as numbers.

Worked example 1: read the station clock

In the photo, the hours row has 5 lamps, the minutes and seconds rows have 6. Writing lit as 1 and dark as 0, from left to right:

Hours: 0 1 0 0 1 → $8 + 1 = 9$
Minutes: 0 1 1 0 0 1 → $16 + 8 + 1 = 25$
Seconds: 1 0 1 1 1 0 → $32 + 8 + 4 + 2 = 46$

The time is 9:25:46. Why 5 lamps for hours? Hours run from 0 to 23, and 5 bits reach $2^{5} - 1 = 31$ , while 4 bits reach only 15. Minutes and seconds run to 59, which needs 6 bits (up to 63).

Worked example 2: hexadecimal to decimal and binary

Convert $1\mathrm{F}4_{16}$ to decimal and to binary.

$1\mathrm{F}4_{16} = 1 \times 16^2 + 15 \times 16^1 + 4 \times 16^0 = 256 + 240 + 4 = 500_{10}$

Binary: each hex digit becomes 4 bits: 1 → 0001, F → 1111, 4 → 0100, so $1\mathrm{F}4_{16} = 0001\,1111\,0100_2 = 111110100_2$ .

Check: $256 + 128 + 64 + 32 + 16 + 4 = 500$ .

Worked example 3: add in binary

The rules are $0 + 0 = 0$ , $0 + 1 = 1$ , $1 + 1 = 10_{2}$ (write 0, carry 1) and $1 + 1 + 1 = 11_{2}$ (write 1, carry 1). Add $101101_{2} + 011011_{2}$ .

	2⁶	2⁵	2⁴	2³	2²	2¹	2⁰
Carry	1	1	1	1	1	1
First		1	0	1	1	0	1
Second		0	1	1	0	1	1
Sum	1	0	0	1	0	0	0

Result: $1001000_{2}$ . Check in decimal: $45 + 27 = 72 = 64 + 8$ . In lesson 3 you will build a circuit that adds two bits.

python

# Check your conversions (you will learn Python in Unit 4).
print(bin(157), oct(157), hex(157))   # decimal to binary, octal, hex
print(int("10011101", 2))             # binary to decimal
print(int("1F4", 16))                 # hex to decimal
print(bin(0b101101 + 0b011011))       # binary addition

Code Explanation

Press Run. Python writes binary with the prefix 0b, octal with 0o and hex with 0x, so the first line prints 0b10011101 0o235 0x9d. int(text, base) converts the other way. Use it to check your answers to the practice questions, after you have worked them by hand.

Warning

Read the remainders upwards. The first remainder is the least significant bit (the rightmost). Writing them top to bottom reverses the number.

Group from the right. $10011101_{2}$ grouped in threes from the left gives 100 111 01, which is wrong. Always start at the rightmost bit, and pad the leftmost group with zeros.

Tip

Learn two lists by heart. The powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024) and the 16 nibbles (0000 = 0 up to 1111 = F). With them, most conversions take seconds.

✦

Practice

Convert $1101_{2}$ to decimal.
Convert $100_{10}$ to binary by repeated division.
Convert $11010110_{2}$ to octal and to hexadecimal.
Convert $3\mathrm{A}_{16}$ to decimal and to binary.
Convert $157_{8}$ to decimal.
Add $1011_{2} + 0110_{2}$ and check in decimal.
How many different values can 12 bits represent, and what is the largest?
A web page uses the colour #C8C8C8. Find the decimal value of each component and explain why the colour is a grey.

Show answersHide answers

$8 + 4 + 1 = 13$ .
100 → 50 r 0, 25 r 0, 12 r 1, 6 r 0, 3 r 0, 1 r 1, 0 r 1; reading upwards: $1100100_{2}$ . Check: $64 + 32 + 4 = 100$ .
Octal: 011 010 110 → $326_{8}$ . Hex: 1101 0110 → $\mathrm{D}6_{16}$ . (Both equal 214.)
$3 \times 16 + 10 = 58$ ; binary 0011 1010 → $111010_{2}$ .
$1 \times 64 + 5 \times 8 + 7 = 111$ .
$10001_{2}$ ; $11 + 6 = 17 = 16 + 1$ .
$2^{12} = 4096$ values, from 0 to 4095.
$\mathrm{C}8_{16} = 12 \times 16 + 8 = 200$ for red, green and blue alike. When the three components are equal the colour has no tint, so it is a grey (a light one, since 200 is near 255).

Quick check

Key points

Each place in base $b$ is worth a power of $b$ ; a binary number is the sum of the powers of 2 where it has a 1.
Decimal to another base: divide repeatedly and read the remainders upwards.
Binary to octal or hex: group 3 or 4 bits from the right; one hex digit = one nibble, two hex digits = one byte.
$n$ bits give $2^{n}$ values, from 0 to $2^{n} - 1$ . Binary addition carries when a column reaches 2.

Sources and credits

Binary number and Hexadecimal, Wikipedia contributors, Wikipedia, licensed under CC BY-SA 4.0. Adapted by Ganzaa.
Python functions bin(), oct(), hex() and int(): Built-in Functions, Python documentation, Python Software Foundation (linked for reference).
Photo: Binary clock Swiss railway station, BBCLCD, CC BY-SA 4.0, via Wikimedia Commons.
Video: Representing Numbers and Letters with Binary, Crash Course Computer Science #4, CrashCourse, embedded from YouTube.
The diagrams, worked examples, code and practice items were written by Ganzaa.

This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1A clock that counts in twos

2Place value in any base

3Decimal to binary: divide by 2

4Binary to octal and hexadecimal: group the bits

A clock that counts in twos

Place value in any base

Decimal to binary: divide by 2

Binary to octal and hexadecimal: group the bits