0.4.5 The Hexadecimal Number System

Hexadecimal Is Base 16

The hexadecimal number system is a base-16 number system.

It needs sixteen single-digit symbols.

The first ten are familiar:

0 1 2 3 4 5 6 7 8 9

Hexadecimal then uses letters for the next six digit values:

A B C D E F

The values are:

After F, hexadecimal moves to another position:

10₁₆

which represents decimal 16.

Hexadecimal Place Values Grow by Powers of 16

Starting from the right:

1, 16, 256, 4096, ...

These are powers of 16:

16^0 = 1
16^1 = 16
16^2 = 256

This follows the same positional-number-system idea you already used in decimal and binary.

The base changes.

The place-value logic does not.

Example: 2A

Consider:

2A₁₆

The left digit is in the 16s place.

The right digit is in the 1s place.

A represents decimal 10.

Therefore:

2 × 16 = 32
A × 1 = 10

Add them:

32 + 10 = 42

So:

2A₁₆ = 42₁₀

Example: 3F

3F₁₆ has:

3 × 16 = 48
F = 15

Therefore:

48 + 15 = 63

So:

3F₁₆ = 63₁₀

Hex Digits Are Values, Not Letters in a Word

In hexadecimal:

A

means the digit value ten.

F

means the digit value fifteen.

The letters are used because base 16 needs more single-digit symbols than decimal provides.

Treat A through F as hexadecimal digits when they appear in a hexadecimal value.

How Hexadecimal Values Are Marked

Different technical contexts use different ways to show that a value is hexadecimal.

You may see notation such as:

2A₁₆

or, in many programming-related contexts:

0x2A

The notation helps prevent confusion with decimal.

For this module, focus on understanding the number system itself rather than memorizing every notation convention.

Hexadecimal Is More Compact Than Binary

The decimal value 15 is binary:

1111

but hexadecimal:

F

The decimal value 255 is binary:

11111111

but hexadecimal:

FF

Hexadecimal can represent long binary patterns with fewer written symbols.

That makes it useful when people need to read or write values closely connected to binary information.

The Same Quantity Has Several Representations

The quantity fifteen can be written as:

15₁₀
1111₂
F₁₆

The symbols differ because the bases differ.

The represented quantity is the same.

This is the central number-system idea in the module:

value and representation are related, but they are not the same thing.

Hexadecimal Connects Naturally to Four Bits

One hexadecimal digit has sixteen possible values:

0 through F

Four binary bits have sixteen possible patterns:

0000 through 1111

That one-to-one match is extremely useful.

It means a four-bit binary pattern can be represented by exactly one hexadecimal digit.

The next activity develops that connection directly.