Binary

Starting with Base 10

Usually with numbers, we write them in “base 10” which means that there are 10 unique digit values:


0, 1, 2, 3, 4, 5, 6, 7, 8, 9

If you use a counter with three places, it starts at 000 and increments the right-most place by one each time: 001, 002, 003, until we reach 009. After that, it adds one to the next place and resets the right-most place back to zero: 010. Then it continues again 011, 012, … 019, 020, 021 … 099, 100, 101, … up to 999.

If you increment one more time, most counters would “wrap” back to 000 - called “overflow”

Other Bases

How would other bases work? If we had a counter that worked differently and moved to the next place sooner or later, then it would be counting in a different base. For example, counting some duration of time would go:


0 hours,  0 minutes,  0 seconds
0 hours,  0 minutes,  1 second
...
0 hours,  0 minutes, 59 seconds
0 hours,  1 minute,   0 seconds
0 hours,  1 minute,   1 second
...
0 hours, 59 minutes, 59 seconds
1 hour,   0 minutes,  0 seconds
1 hour,   0 minute,   1 second

In this example the seconds and minutes have 60 unique “digit” values before the next place is incremented, so it’s almost like it’s “base 60”.

Base 2

Now let’s go the other direction: what if we only had 2 unique digit values? Let’s choose 0 and 1. Setting up a counter with three places again: 000, then start counting: 001. Wait, looks like we already need to move onto the next place and reset the right-most digit: 010. And if we continuing counting: 011, 100, 101, 110, 111. Now we’ve gotten as far as we can with three places - that didn’t take very long. In fact with three places, we can only increment the counter 7 times! But on the other hand, it’s very easy to count since each digit is only a 0 or 1. Here is an example counter with 4 places:

Take a look at the patterns that come out! The right-most place flip flops between 0 and 1. In fact, if the digit is 0 we’ve incremented the counter an even number of times. If the digit is 1, we’ve done it an odd number of times. Also notice how the next place flip flops half as frequently, then the next place half as frequently again, then the left-most place only when from a 0 to 1 and not back to a 0.

This base-2 system is called binary and computers that interally use binary numbers are called “digital” computers. Counting with binary takes a lot more places than in base-10 which makes it harder for us to write down binary numbers. Most phones or laptops work with binary numbers that have at least 32 binary places: 0000 0000 0000 0000 0000 0000 0000 0000 (spaces added for clarity).

When computer processors are designed, one choice that is made is what the number of places they can handle for a number at time, and a computer processor designed for numbers of 32 binary places is called a “32-bit” processor.

Hexadecimal

Since it’s so many places, we also typically work with another base: base-16 also known as “hexadecimal” (hex: 6 + dec: 10). If you have a binary counter with 4 places, you can represent 16 unique values: 0 through 15. This means it’s very easy to convert a number written in binary to a number written in hexadecimal because each group of 4 binary digits becomes 1 hexadecimal digit which means a lot fewer digits. And since 16 is close to 10 it’s also easier for us humans! For example, here is the number “11433” in a counter with 16 binary places:


0010 1100 1010 1001

If we look at each group of four binary digits and treat them as a separate counter with 4 binary places, then these would be the counts they correspond to:


0010 ->  "2"
1100 -> "12"
1010 -> "10"
1001 ->  "9"

Those numbers greater than 9 are inconvenient because they take two places right now in decimal, so in hexadecimal the notation that is used include not only the common 0-9, but also add a 6 more digits: a, b, c, d, e, f. These represent “10”, “11”, “12”, “13”, “14”, and “15” but can be written with one letter instead of two. This means now we can write the number with 4 letters in hexadecimal:


2ca9

Much easier!

Notation

If a number is written as 1010, usually it’s meant to be read in base-10, but it could be in base-2, or base-16 so some notation is used to indicate the base:

TK Also subscript to indicate

Common notation for binary is 0b or b:


0b1010 or 1010b

Common notation for hexadecimal is 0x or h:


0x1010 or 1010h

The reason the prefix 0b or 0x is because if it was b1010 or x1010 those could technically be variable names so the prefix starts with 0 to signify it is a number

TODO

Odometer
Mechanical visualization
Binary mechanical visualization