Binary is counting in base-2 (see textbook section Binary). Hexadecimal is counting in base-16. Decimal is counting in base-10. We use the subscript like X₂ to indicate the number is written in base-2, or X₁₀ for a base-10 number.

The symbols we use to count by powers are all values less than base:

• Base 2 uses two symbols (0,1) to count.
• Base 10 uses ten symbols (0,1,2,3,4,5,6,7,8,9) to count.
• Base 16 uses symbols sixteen symbols: 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f.

Base 10 Example

1534₁₀ = 1 x 10³ + 5 x 10 ² + 3 x 10 ¹ + 4 x10 ⁰ = 1000 + 500+ 30+ 4

Base 2 Example

1101₂ = 1x 2³ + 1x 2² + 0x 2¹+ 1x 2⁰ = 8 + 4+ 1 = 13₁₀

Base 16 Examples

1534₁₆ = 1 x 16 ³ + 5 x 16² + 3 x16 ¹ + 4x 16⁰= 5428₁₀ 

beef₁₆ = b x 16 ³ + e x 16² + e x16 ¹ + fx 16⁰=  11 x 16 ³ + 14 x 16² + 14 x16 ¹ + 15 x 16⁰= 48879₁₀

8 bits is a byte

A single hex digit can represent any four-digit binary value, or 4 bits: 

0000₂ = 0₁₆, 0001₂ = 1₁₆, …, 1110₂ = e₁₆, 1111₂ = f₁₆

Since a byte is eight bits, it needs two hex digits:

1f₁₆ is the byte 00011111₂

This makes hex a convenient way to write colors, like we discuss when we talk about RGB values.

Pixels Represented in Hexadecimal

A pixel may be specified by the RGB Color Model (Red, Green and Blue). An 8-bit (one byte) number represents the amount of each color, where 0 means nothing, and 255 means all the intensity the monitor can provide. We have 256 possible values for red, green and blue. We usually write these colors in hex or as a sequence of decimal numbers. See more about pixels in the textbook section Colors.

Example

(42, 46, 163) is the (R,G,B) color also written as #2a2ea3 where 42₁₀ = 2a₁₆, 46₁₀ = 2e₁₆, and 163₁₀ = a3₁₆

You need not write a subscript for RGB colors since it’s clear in context which base we’re talking about!

Exercises

1. Since we like to combine things in multiples of bits, we also use base-8 in computing; it’s called octal! Base-32 and Base-64 are even used, but more rarely — search RFC 4648 if you’re interested.
2. How do the hex codes vary as you try different combinations in the RGB color picker?
3. What are the pros and cons of using one base over another?
4. Consider a base-64 number, which uses this set {A,B,…,Z,a,b,…,z,0,1,2,3,4,5,6,7,8,9,-,_} (26 upper and lower case letters, 0-9, a dash and underscore to get 64 symbols). The characters in the above set represent the numbers 0 through 63, in the same order—thus C represents 2, c represents 28, 3 represents 55, and _ represents 63. According to this schema, find the natural number whose base-64 expansion is d9WgXcQ, and find the base-64 expansion of the natural number 715984.