What is a number?

While seemingly a deeply philosophical question, the way numbers are represented in electronics is different from the decimal system we naturally work with. Understanding and being able to transition between these numbering systems is a useful skill to have.

There are infinitely many ways to represent a number, but the four you will typically work with in software are decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16). Octals are rarely used today, as they come from a time when processors were 8-bit (but the theory is exactly the same)—you typically will work with one of the other three.

The key to understanding different numbering systems is the notion of place value. If you recall from elementary school, you probably learned that for decimals, the rightmost place denotes the value of the ones place, then the next corresponds to the tens place, then the hundreds... What this means is that for each digit, its value depends on its position. If you understand decimal (base 10) and place value, you should be able to understand any of the other numbering systems.

Decimal: Base 10

There are 10 unique digits in decimal (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) which represent an integer value 0-9. In general, what this means is that for a decimal number with digits say

abcdefgh

this is actually

Decimal Digits

Examples

Here's an example (hopefully it's just a review)

Example

The number

9001

can be represented as

Counting in Decimal

Here's how to count from 0 to 32 in decimal.

Binary: Base 2

Binary is the natural way most digital circuits represent and manipulate numbers. There are 2 unique digits in binary (0, 1) which represent an integer value of 0 or 1. The numbering system works similarly to decimal, where the rightmost place represents the smallest unit (1), then the next corresponds to the next-smallest (2), then the one following (4), and so on... In general, what this means is that for a number (in binary) with digits say

pqrstuvw

this is actually

Binary Digits

Examples

Here's a few examples

Example 1

The number

145

can be represented as

or

0b10010001

Example 2

The number

255

can be represented as

or

0b11111111

Counting in Binary

Counting in binary is similar to counting in decimal, except there are only 2 digits. Here's how to count from 0 to 32 in binary.

Hexadecimal: Base 16

Hexadecimal—also known as hex or base 16—is a system we can use to represent numerical values in a way that is more compact than binary. Hex, like the decimal system, combines a set of 16 unique digits to create large numbers. 

Hexadecimal Digits

Examples

Here's a few examples

Example 1

The number

9

can be represented by

or

0x9

Example 2

The number

100

can be represented as

or

0x64

Example 3

The number

140

can be represented as

or

0x8C

Example 4

The number

255

can be represented as

or

0xFF

Counting in Hex

Counting in hex is similar to counting in decimal, except there are an additional 6 digits. Here's how to count from 0 to 32 in hex.

Protip

You can think of hex as just binary, but with groups of 4. So, when performing the conversion from binary to hex

1. Pad with any necessary 0s so the number of digits is divisible by 4.
2. Group the binary representation into groups of 4.
3. Convert each group of 4 to the hex digit

Example

Consider the number

159

which has the binary representation

0b10011111

and can be grouped into two groups of 4 binary digits

1001 1111

We know that 

0b1001 = 0x9

and 

0b1111 = 0xF

so the hex representation is

0x9F

Base n

In general, for any numbering system with base b, we can represent a number with n + 1 digits as