One of the more mysterious concepts in computer science is binary code. Even though it is a fundamental building block concerning how computers function, programmers today can get away without a deep understanding of how it works. By the end of this lesson, you'll have that advantage!
For this lesson you'll need to have Python 2.7 installed. If you're on Mac or Linux, this should already be preinstalled on your system. If you're on Windows, you may need to download it.
Let's jump right into some code. Create a new Python file called binary_example1.py. Type this into it:
num = 1
print num
for i in range(11):
num = num << 1
print numRun it, and you'll see that this is what you get:
$ python binary_example1.py
1
2
4
8
16
32
64
128
256
512
1024
2048I bet you're wondering what that << thing does... We’ll get into that in a little while.
Does anybody recognize those numbers from anywhere?
Now, hop over to this game called 2048. Play it for a little while. The objective is to use the arrow keys to move the numbers and combine (add) the matching ones together. It can be addicting, so don't get too carried away! ;)
After you've played it for a bit, you may start to notice a similarity between the numbers in the game and the Python code we just wrote.
Why are the numbers in the game the same ones that came out of our Python program?
I will give you a hint... Instead of thinking about the game as 2 + 2, 4 + 4, 8 + 8, etc., think about it as 2 * 2, 2 * 4, 2 * 8, and so on.
By now you may be thinking, "I thought we were supposed to be learning about binary code! I thought that was all made up of 1s and 0s!" If you are thinking that, you're exactly right! The difference is between how those numbers look to you and how they look to the computer.
What, then, is binary code?
Binary code is what you probably already think it is: sequences of 1s and 0s. It might look like 10011001 or 10101010. But since we are humans, that code doesn't really mean anything to us! Also, why were we playing with 2s, 4s, and 8s when binary is only made up of 0 and 1?
Let's focus on answering a couple of key questions:
- Why is binary code made up of
1s and0s? - What does the binary code mean?
The answer to this question goes back to how the very first computers where built. The most important piece of the answer is a little device called a transistor.
A transistor is a switch, except transistors also have the ability to amplify the signal that is going into them. Because a transistor is a switch, it can either be OFF or ON (Just like a light switch at home). Because a transistor is an amplifier, many of them can be connected together in a circuit without the signal dying. You can put however many transistors you want into a circuit and the signal will still make it to the finish line.
It's not too much of a jump then to see how 0 is like OFF, and how 1 is like ON (You might have also seen other paired values in programming like NO/YES or false/true). This means that a transistor has the ability to store data, but it can only store a value of 0 or 1. That's actually okay, because a computer designer could use as many transistors in a computer as he wants.
So that is the basics of how transistors came to be used in computer hardware. Having a bunch of 0s and 1s around may not seem to have an obvious use. We are about to look at how the ability to store multiple 0s and 1s is incredibly powerful.
Let's look at binary_example1.py again.
num = 1
print num
for i in range(11):
num = num << 1
print numSo, about that << thing. << is actually an operator much like +, -, * or /. However, while +-*/ are arithmetic operators, << is a bitwise operator. It is called left shift, and it operates on the underlying binary representation of the number to the left of <<. What this implies is that for every number (1, 2, 3, 4, ...), there is a binary code equivalent.
How do we know what that binary code is? Well, 0 and 1 are freebies, because they are already part of the binary number system (which is made up of 0s and 1s). However, binary code is usually represented in groups of 8. So, the binary representation of 0 is 00000000, and the binary representation of 1 is 00000001.
A bit is an individual 0 or 1 in the binary code. A group of 8 bits is called an octet. A byte stores one octet, or 8 bits.
The left shift operator (<<) takes the binary version of the number to the left of the operator and shifts all of the bits to the left. The number of places to shift is determined by the number to the right of the << operator. So, num << 1 just means "shift num's bits 1 place to the left".
So throughout our program, the variable num is backed by these values:
00000001 # num = 1
00000010 # num = num << 1
00000100 # num = num << 1
00001000 # num = num << 1
00010000 # num = num << 1
00100000 # num = num << 1
01000000 # num = num << 1
10000000 # num = num << 1See how the 1 bit shifts to the left with every cycle of the for loop? We can map these octets to our program output to see how they match up.
00000001 # 1
00000010 # 2
00000100 # 4
00001000 # 8
00010000 # 16
00100000 # 32
01000000 # 64
10000000 # 128
...What about 256, 512, 1024 and 2048? Python knows that it needs another byte to store those larger numbers, so we get another octet to left shift into.
00000000 00000001 # 1
00000000 00000010 # 2
00000000 00000100 # 4
00000000 00001000 # 8
00000000 00010000 # 16
00000000 00100000 # 32
00000000 01000000 # 64
00000000 10000000 # 128
00000001 00000000 # 256
00000010 00000000 # 512
00000100 00000000 # 1024
00001000 00000000 # 2048Now we need to understand why the position of the 1 bit determines which number it represents.
Let's make a new Python file and call it binary_example2.py. It's going to be a lot like our first example, but we are going to replace the left shift with another expression.
num = 1
print num
for i in range(11):
num = num * 2
print numWhat do you think this program outputs? Here's a hint: remember how after we played the 2048 game I said to think about the numbers as 2 * 2, 2 * 4, 2 * 8, and so on? Lo and behold!
$ python binary_example2.py
1
2
4
8
16
32
64
128
256
512
1024
2048By using a mathematical device called a change of base, binary values can be used to express more complex values than 0 and 1.
Our natural counting system is more precisely called the decimal system, and it has a base of 10. Bases work off of the position of the digits in the number. The number of potential digits in a base is equal to the value of the base, so base 10 has 10 possible digits it can express numbers with. The digits always include 0. If you count how many numbers are between 0 and 9, you'll find there are 10, so [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] is the digit set for base 10.
To figure out the decimal value of a number in a different base, the value of each digit is multiplied by the base to the power of the digit's position (The position of the digit is 0 at the far right, then increases to the left). Then all those are added together. "To the power of" just means multiplying a number by itself that many times. This is called an exponent.
Let's explain the decimal number 123. We are going to convert it from base 10 to base 10, so the number won't actually change, but breaking it down will help explain the concept:
- 3 is in the 0th positon, 2 is in the 1st position, 1 is in the 2nd position.
- The value of the 0th digit is
3, and the base is 10, so its decimal value is3 * 10 ^ 0, or3 * 1. Any number to the 0th power is1.3 * 1equals3 - The value of the 1st digit is
2, so its decimal value is2 * 10 ^ 1, or2 * 10, which equals20. - The value of the 2nd digit is
1, so its decimal value is1 * 10 ^ 2, or1 * 10 * 10, which equals100. - Now we add them all together:
100
20
+ 3
----
123
The binary system is base 2, which contains only 2 digits including 0. That only leaves room for one more, the digit 1. These are our transistors!
Let's look at an example binary octet, 00101010, and convert it to decimal:
- The value of the 0th digit is
0, so its decimal value is0since any number times 0 equals 0. - The value of the 1st digit is
1, so its decimal value is1 * 2 ^ 1, or1 * 2, which equals2. - The value of the 2nd digit is
0 - The value of the 3rd digit is
1, so its decimal value is1 * 2 ^ 3, or1 * 2 * 2 * 2, which equals8. - The value of the 4th digit is
0 - The value of the 5th digit is
1, so its decimal value is1 * 2 ^ 5, or1 * 2 * 2 * 2 * 2 * 2, which equals32. - The value of the 6th and 7th digits are
0 - Now we add them all together, and since we are adding we can ignore the
0s:
32
8
+ 2
---
42 (the meaning of life, the universe and everything!)
So now we know how we can use the binary number system to express large numbers with just 1s and 0s. Well, even though numbers are useful and all, humans think in terms of words. Raw numbers don't leave us with a way to express readable text in a way that the computer can store.
Computer scientists were trying to solve this exact problem about 50 years ago. Computers needed to support the uppercase alphabet (26 characters), the lowercase alphabet (26 characters), numerals (10 characters), plus punctuation and special characters. Computers didn't have a lot of storage space at the time, so the scientists needed to figure out a way to fit a way to express those characters in the smallest amount of space per character.
It turned out that 8 bits could store 256 different combinations (Binary 11111111 is equal to the decimal number 255), and so eventually 8 bits became the compromise between space and flexibility for storing a single character.
So letters on a computer are just an illusion! At least an illusion as far as the computer is concerned.
Here's our last example, binary_example3.py. I am using the ord() and chr() functions to find the number value for a character, messing with that
number, then turning it back into a character.
letter_a = "A"
letters = []
print "UPPER:"
for i in range(26):
idx = ord(letter_a)
new_letter = chr(idx + i)
letters.append(new_letter)
print new_letter
print "\nlower:"
for letter in letters:
idx = ord(letter)
shifted = idx + 32 # Why does adding 32 make it lowercase?
new_letter = chr(shifted)
print new_letterRun the program and see what it does. What is so significant about adding 32? Well, there is a standard mapping of numbers to characters called ASCII. Check out the table and see those numbers for yourself.
Now you have a grasp of how computers handle binary code, and how binary code relates to computer data we humans can understand. We have just covered the basics here, and there are other different ways that different types of data are represented in binary. If you really want to melt your brain, check out how floating point numbers work!