Hexadecimal

Objectives

• To be able to explain why hexadecimal is often used in computer science.
• To understand how hexadecimal can be used to represent whole numbers.
• To be able to convert a hexadecimal number into binary, and vice versa
• To be able to convert a binary number into hexadecimal, and vice versa
• To be able to convert a hexadecimal number into decimal, and vice versa
• To be able to convert a decimal number into hexadecimal, and vice versa

Everything that takes place in the computer depends on two-state switches. Each switch can be either on or off which can more easily be represented as either a \(0\) or a \(1\) (or, also, False or True).

For convenience, we also need to know the hexadecimal number system, Base-16, as a shorthand for representing binary.

Warning

The computer only deals with the binary representation, the 0s and the 1s. Obviously, as this represents the state of a particular transistor. Hexadecimal is a shorthand for us as it helps make sense of long binary strings.

Given a 32-bit binary string such as \(01110010101000001111001011100101\). If we had to handle this number of bits it would be easy to make mistakes. To make our lives a little easier we can represent this binary string as a hexadecimal number: \(72A0F2E5_{16}\).

Base-16

It looks a little weird, letters representing numbers but we borrow the first six letters from the alphabet to represent \(10\) through to \(15\). Looking at the following table we can see how Base-16 numbers map directly to 4 binary digits:

Base-10	Base-2	Base-16
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

It is easier to use hexadecimal rather than denary to represent binary numbers as it is also based on powers of 2. It is easy to group the binary digits into groups of four (a nibble) and use the table above to convert into the hexadecimal equivalent.

It saves a lot of space. It takes 8 binary digits to represent a number between \(0\) and \(255\), but only ywo hexadecimal digits for the same range, \(00\) and \(FF\).

Returning to our previous 32-bit number: \(01110010101000001111001011100101\). To arrive at the hexadecimal equivalent:

• split the binary string into groups of four:

\(0111 0010 1010 0000 1111 0010 1110 0101\)

• Use the table above to look up the hexadecimal:

\(0111\)	\(0010\)	\(1010\)	\(0000\)	\(1111\)	\(0010\)	\(1110\)	\(0101\)
\(7\)	\(2\)	\(A\)	\(0\)	\(F\)	\(2\)	\(E\)	\(5\)

Uses of hexadecimal

Repeating the point above:

Warning

The computer only deals with the binary representation, the 0s and the 1s. Obviously, as this represents the state of a particular transistor. Hexadecimal is a shorthand for us as it helps make sense of long binary strings.

We use hexadecimal in computer science for a number of reasons:

• Concise Representation: Hexadecimal allows for a more compact and concise representation of binary data. Each hexadecimal digit represents 4 binary digits (bits), making it easier to read and write large binary numbers.

• Memory Addresses: Memory addresses in computer systems are often expressed in hexadecimal. This is because each digit in hexadecimal corresponds to 4 bits, making it a natural choice when dealing with memory locations in a system where memory is often addressed at the byte level.

• Color Representation: Hexadecimal is commonly used in representing colors in computing. In HTML, for example, colors are often specified using hexadecimal notation (e.g., #RRGGBB, where RR, GG, and BB represent the red, green, and blue components, respectively).

• Binary Conversion: When working with binary data, it's common to convert between binary and hexadecimal for readability. Hexadecimal is more concise than binary and is easier for humans to read and write.

Using hexadecimal is easier for programmers to work with than binary, and it can be more readable than decimal for certain tasks.

Decimal to Hexadecimal conversion

There are two methods that be used here.

Method 1 - Repeated division by 16

As we say when converting a decimal number to binary we can divide the value by \(2\) (the base) and note the remainder. We can do the same here but using the number \(16\):

For example, to convert \(245_{10}\):

• Divide \(245\) by \(16 \rightarrow 15\) remainder \(5\)
• Find the hexadecimal equivalents ot both \(15\) and \(5\) and concatenate them together \(\rightarrow F5\)

Method 2 - Via binary

Alternatively, convert the decimal number into binary (using one of the previous algorithms) and then convert the binary number into hexadecimal.

For example, to convert \(245_{10}\):

• Convert \(245_{10}\) to binary \(\rightarrow 11110101_2\)
• Divide the binary string into groups of 4 bits: \(1111\) and \(0101\)
• Use the look up table to get the equivalent hexadecimal digits, thus \(F5\).

Binary to hexadecimal conversion

• Split the binary string into groups of four bits
• Find the corresponding hex value for each group of four

For example: \(01101100_2\)

• Split into groups of 4: \(0110_2\) and \(1100_2\)
• Using the look-up table: \(0110_2\) is \(6_{16}\), \(1100_2\) is \(C_{16}\)
• Thus: \(01101100_2\) is \(6C_{16}\)

Hexadecimal to binary conversion

• Take each hex digit
• Use the look up table to find the corresponding binary nibble
• Concatenate the results together

For example: \(5D_{16}\)

• Use the look up table to find the binary equivalent for both \(5_{16}\) and \(D_{16}\)
• \(5_{16}\) is \(0101_2\), and \(D_{16}\) is \(1101_2\)
• Thus, \(5D_{16}\) is \(0101 1101_2\)

Hexadecimal to decimal

The simplest approach here is to convert the hexadecimal number into binary and then convert the binary number into decimal.

For example: \(B3_{16}\)

• Look up the binary equivalent for \(B_{16} \rightarrow 1011_2\)
• Look up the binary equivalent for \(3_{16} \rightarrow 0011_2\)
• Concatenate the two together: \(10110011_2\)
• Convert to decimal: \(((128 * 1) + (32 * 1) + (16 * 1) + (2 * 1) + ( 1 * 1)) \rightarrow 179\)

Questions

1. Why is hexadecimal often used in computer science?

It is easier for humans to read compared to binary.

It is the native number system of computers.

It is more compact than decimal for representing large binary numbers.

It is the default number system used in arithmetic operations.

Hexadecimal is often used because it provides a more compact representation of binary-coded values.

Submit

2. How can hexadecimal be used to represent whole numbers?

By using binary-coded decimal.

By grouping binary digits into sets of four and converting to hexadecimal.

By using a direct conversion from decimal to binary.

By performing arithmetic operations in base-10.

Hexadecimal can represent whole numbers by grouping binary digits into sets of four and converting each group to a hexadecimal digit.

Submit

3. What is the hexadecimal representation of the decimal number $156_{10}$?

9C_{16}

A6_{16}

B4_{16}

C2_{16}

The decimal number $156_{10}$ converts to $9C_{16}$ in hexadecimal.

Submit

4. Convert the hexadecimal number $2A7_{16}$ to decimal.

639

679

723

745

The hexadecimal number $2A7_{16}$ converts to $639$ in decimal.

Submit

5. Why is hexadecimal commonly used in memory addresses in computer systems?

It matches the binary structure of memory directly.

It reduces the number of digits compared to binary and is easier to read.

It is simpler to convert from decimal addresses.

It automatically adjusts for different address sizes.

Hexadecimal is used in memory addresses because it reduces the number of digits compared to binary and is easier for humans to read.

Submit

6. What is the hexadecimal representation of the binary number $110110101011_{2}$?

1B5_{16}

1D6_{16}

1E5_{16}

1E6_{16}

The binary number $110110101011_{2}$ converts to $1B5_{16}$ in hexadecimal.

Submit

7. How does hexadecimal provide a more compact representation for binary-coded values?

By reducing the number of digits needed to represent the same value.

By using fewer bits per digit.

By allowing for direct conversion from decimal.

By simplifying arithmetic operations.

Hexadecimal provides a more compact representation by reducing the number of digits needed to represent the same binary value.

Submit

8. Convert the hexadecimal number $4F_{16}$ to binary.

01001111_{2}

00101111_{2}

10011111_{2}

11001111_{2}

The hexadecimal number $4F_{16}$ converts to $01001111_{2}$ in binary.

Submit

9. If a memory address is expressed as $1A3F_{16}$ in hexadecimal, what is the equivalent decimal value?

6719

6928

6767

6912

The hexadecimal address $1A3F_{16}$ converts to $6719$ in decimal.

Submit

10. Describe the relationship between hexadecimal and binary, and how each hexadecimal digit corresponds to a set of binary digits.

Each hexadecimal digit corresponds to a 3-bit binary group.

Each hexadecimal digit corresponds to a 2-bit binary group.

Each hexadecimal digit corresponds to a 4-bit binary group.

Each hexadecimal digit corresponds to an 8-bit binary group.

Each hexadecimal digit corresponds to a 4-bit binary group, making it a compact representation of binary numbers.

Submit