4.15. The Hexadecimal System

The hexadecimal system works in base 16. This means it uses the digits 0-9 and the letters a-f.

\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)	\(11\)	\(12\)	\(13\)	\(14\)	\(15\)
\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)	a	b	c	d	e	f Conversion between the hexadecimal system and decimal system works much the same way as conversions between the binary and decimal systems.

4.15.1. Hexadecimal to Decimal

Example

Let’s take the number 6d as an example and fill it into the table below.

\(16^1\) \(16\)	\(16^0\) \(1\)
6	d

The way we interpret the hexadecimal number 6d is

\((\textbf{6}\times16^1) + (\textbf{d}\times16^0) = \textbf{109}\)

or alternatively

\((\textbf{6}\times16) + (\textbf{13}\times1) = \textbf{109}\)

Example

Let’s take the number 2f9a as an example and fill it into the table below.

\(16^3\) \(4096\)	\(16^2\) \(256\)	\(16^1\) \(16\)	\(16^0\) \(1\)
2	f	9	a

The way we interpret the hexadecimal number 2f9a is

\((\textbf{2}\times16^3) + (\textbf{f}\times16^2) + (\textbf{9}\times16^1) + (\textbf{a}\times16^0) = \textbf{12186}\)

or alternatively

\((\textbf{2}\times4096) + (\textbf{15}\times256) + (\textbf{9}\times16) + (\textbf{10}\times1) = \textbf{12186}\)

What you may have noticed is that you can represented much larger numbers with fewer characters in the hexadecimal system compared to the binary system.

4.15.2. Applications: Colours

While computers don’t ‘think’ using the hexadecimal system, programmers often use it because it’s easier for humans to read. 8 bits can be represented using just 2 digits in the hexadecimal system. A common example is colours, which are often represented using the hexadecimal system.

We often represent colours using RGB (red, green, blue) values, where were specify a number between 0 and 255 for the amount of red, green and blue a colour has.

Examples

• Red would be represented as (255, 0, 0), which has maximum red, and no green and no blue

• Blue would be represented as (0, 0, 255), which has no red, no green, but maximum blue

• Purple could be represented as (100, 0, 100), which has no green, but equal amounts of red and blue

• White would be represented as (255, 255, 255), which is maximum red, green and blue

These RGB values are then often represented in hexadecimal with a 6-character number. The first 2 characters indicate the amount of red, the middle 2 characters indicate the amount of green and the last 2 characters indicate the amount of blue.

** Examples**

We would represent red using ff0000

	Red	Green	Blue	Final Representation
Decimal	255	0	0	(255, 0, 0)
Hexadecimal	ff	0	0	ff0000

And we can represent purple using 640064

	Red	Green	Blue	Final Representation
Decimal	100	0	100	(100, 0, 100)
Hexadecimal	64	0	64	640064

Question 1

What is the largest integer you can represent using 2 characters in the hexadecimal system? Give your answer as a decimal number.

Solution

The largest value you can represent using 2 characters in hexadecimal is ff.

\(16^1\) \(16\)	\(16^0\) \(1\)
f	f

ff corresponds to \((15 \times 16) + (15 \times 1)\), which is the decimal number 255. (This is equivalent to the binary number 11111111!)

Question 2

What is the hexadecimal number 7c2 as a decimal number?

Solution

Solution is locked

Question 3

The colour grey can be made using equal amounts of red, green and blue. Which of the following corresponds to a shade of grey? Select all that apply.

1. ababab

2. 388338

3. 555555

4. 2288ee

Solution

Solution is locked

Code Challenge (Extension): Converting Between Hexadecimal and Decimal

Write a module called conversions that can be used to convert from binary to hexadecimal and vice versa. You should be able to import functions from your module into your main script main.py. Your module should contain the following 2 functions.

Hexadecimal to binary specification

• name: hex_to_dec

• parameters: n (str)

• return: n as a decimal number (int)

Binary to hexadecimal specification

• name: dec_to_hex

• parameters: n (int)

• return: n as a hexadecimal number (str)

Example 1

import conversions

print(conversions.bin_to_dec(10101))
print(conversions.bin_to_dec(10011101))

21
157

Example 2

import conversions

print(conversions.dec_to_bin(21))
print(conversions.dec_to_bin(157))

10101
10011101

Example 3

import conversions

for i in range(20):
    print(conversions.dec_to_hex(i))

0
1
2
3
4
5
6
7
8
9
a
b
c
d
e
f
10
11
12
13

Solution

Solution is locked