Bitwise Operations: A Simplified Guide for Beginners

Bitwise operator is one of the most important operators in programming languages. As easy as it is, it may be difficult for those who are learning it for the first time to understand the fun and what seems to be like magic behind it.

I will be explaining what exactly Bitwise is and give you a clear and simple explanation as if you're 10 years old.

Note: I will be using Python to run a few bitwise operations here. But the idea is not to write code but to explain the simple mathematics behind Bitwise operations.

Python needs no introduction, but I think explaining the concept of the programming language is important for absolute beginners.

Aside from printing "Hello world", one of the basic fundamentals we learn for everything programming language is operators.

Just like the English language where you learn the Alphabet A - Z before you can construct words and then go on to become a writer, we also need to understand the basic stuff like operators and data types before jumping into real programming or engineering.

With no further Ado, let's explore the concept behind Bitwise Operators in programming languages.

What you need to know.

You must first of all understand ordinary binary operations, not deeply but the basic high school knowledge you have is okay.

Can you convert 25 (a natural or decimal number) to binary?

It's simple, the basic mathematics will give you 11001.

You should also be able to convert from binary back to decimal.

Bitwise entirely depends on Binary operations.

Recap: Binary operations are carried out in 1s and 0s… Of course, Binary is a number consisting of 1 and 0 only.

1 is otherwise used as ON while 0 is used as OFF, in some contexts, the same 1 and 0 can also be used as input and output.

But I'm going to use the concept of ON and OFF to explain Bitwise Operators.

The Bitwise Operators

Some Bitwise operators in programming languages:

1. AND (&)
2. OR (|)
3. XOR (^)
4. NOT (~)
5. Arithmetic Right SHIFT (>>)
6. Logical Right SHIFT (>>>)
7. Left SHIFT (<<)

Note that when performing a bitwise operation, it is common to represent the binary using a fixed shape of binary which is commonly a byte (8 bits make one byte).

The 25 decimal we converted to binary earlier which gives us 11001 can be made to fit 8 bits shape simply by adding 0s at the back until it's 8-digits.

E.g., 11001 becomes 0001 1001.

Although we can have derivatives like AND Assignment ("&=") which is just a combination of AND bitwise operator and assignment arithmetic operator.

Let's take them one after the other using a set of switches as an example. A set of switches is a binary number, remember we are dealing with bitwise.

For instance, if you're given 15 and you're asked to find it's bitwise, convert it to binary and the result is our switch.

Note: Switch in this context is simply what you know, like that kind of switch you use to ON and OFF your lamp.

1. Bitwise AND Operator ("&")

To perform AND bitwise operators, we need at least 2 switches to work with.

Let’s assume we are asked to find x in:
let x = 6 & 15

6 is a decimal, likewise 15.

Step 1: Convert them to binary so that:

6 = 110
15 = 1111

Step 2: Make each of them the same shape

6 = 0000 0110
15 = 0000 1111

Now we can perform AND bitwise operator on the two switches since they're of the same shape.

The rule for AND bitwise operator is:

• Keep the same element
• Choose OFF over ON

Using that rule, we will have:

6 = 0000 0110
15 = 0000 1111

(6 & 15) = 0000 0110

You can safely remove all 0s that comes before the first 1, so that our final answer is

110.

Since you're given the original number in decimal, you can convert binary 110 to decimal too.

So therefore (6 & 15) is 6.

That's the simple logic behind the operation. Remember, you always have to convert any given number to binary and ensure the two numbers are in the same shape.

You can calculate the same operation using python.

def bitwise_and(x, y):
  return x & y

print(bitwise_and(6, 15))

2. Bitwise OR Operator ("|")

Understanding AND operation has opened our understanding to other operations.

To perform OR operations,

The rule for OR is:

• Keep the same
• Choose ON over OFF

Using the same switches (6, 15). To find (6 | 15), we also do:

conversion and having equal shape

6 = 0000 0110
15 = 0000 1111

result

(6 | 15) = 0000 1111

Finally result for (6 | 15) is 1111, converting to decimal, we have 15.

The python program is:

def bitwise_or(x, y):
  return x | y

print(bitwise_or(6, 15)) 

3. Bitwise XOR Operator ("^")

XOR is otherwise known as eXclusive OR. And it follows the same pattern as the OR operation.

But the difference is you don't keep the same element, but you choose OFF if they are the same.

Here is what I mean using the rule:

• For the same element, choose OFF
• Choose ON over OFF.

To demonstrate that, let's do (6 ^ 15)

Remember:

6 = 0000 0110
15 = 0000 1111

Ignore the same first 0s.

(6 ^ 15) = 1001

In decimal, (6 ^ 15) is 9.

4. Bitwise NOT Operator ("~")

The bitwise NOT Operator is quite different from the first 3 we've explained.

Bitwise NOT Operator doesn't compare, you don't have 2 sets of switches here but one.

The rule is simple:

• If it's a binary, Flip the element, change ON to OFF, and vice versa. Meaning 0 becomes 1 and vice versa.
• If it's a positive decimal, add 1 to it and make it negative
• If it's a negative decimal, make it positive and subtract 1 from it

Let's use the same figures as an example.

(~ 6) becomes -7
(~ 15) becomes - 16.
(~ -6) becomes 5
(~ -15) becomes 14

Here's the binary operation happening behind the scene using the rule.

6 = 0000 0110

If you flip, you'll have 1111 1001

So ~6 in bit level is 1111 1001

Now here is the confusion:

But decimal -7 is not the same as binary `1111 10001`. Yeah, that's the surprising behavior of the NOT Operator.

For the negative or positive sign that's changing, the reason is simply because a number has its own sign, flipping the number also flips the sign.

For example, every natural number has a positive sign which is generally not written. Flipping the number will make the sign change.

That's why ~6 becomes -7 and ~-6 becomes 5

5. Bitwise Right SHIFT (">>")

Right shifting is just exactly as it is called, you simply move a number by the number of times you're given or need it to shift.

But before explaining, here is the rule.

• Divide the given natural number by 2 and round it down.
• If it's a binary, move each digital one by one to the right
• When shifting, you'll lose a digital, replace it with 0

For example:

If you're to right 6 by 1, that is to move it forward by 1 or 6 >> 1

To get your answer in decimal, simply divide by 2 and your answer is 3.

6 >> 2 becomes 1.

explanation:

6 >> 2 is 6 ÷ (2×2) which is 6 ÷ 4

You'll get 1 remainder 2, throw the remainder out of the window.

8 >>> 3 is 8 ÷ (2×3) which is 8/6 = 1

Question: What if you want to right shift 6 in 3 times (6 >> 3)

Your quick understanding is 6 ÷ (2*2*2) which is 6÷9…. That's 0 remainders 3.

Round it down to 0.

If the number of times you're right shifting is bigger than the number you're shifting, the answer is 0.

For binary.

110 >> 1 becomes 011 or simply 11

converting 11 to decimal gives you 3.

110 >> 2 becomes 1.

So either binary way or decimal way, you'll get the same answer.

6. Bitwise Left SHIFT ("<<")

Of course, it is the opposite of the Right shift bitwise operation.

The rule for BITWISE Left shift is as follows:

• Multiply the natural number by 2 and round down.
• If it's a binary, move backward by the number of shift

Examples:

(6 << 1) becomes 6 x 2 = 12.

(6 << 2) becomes 6 x (2*2) = 24.

(6 << 3) becomes 6 x (2*2*2) = 48.

For binary, move the digit backward:

(110 << 1) becomes 1100
(110 << 2) becomes 11000
(110 << 3) becomes 110000

If you convert each of this binary to decimal, you should get 12, 24 and 48 as we've gotten above.

7 Bitwise Logical Right SHIFT (>>>)

Logical Right SHIFT is just like ordinary or arithmetic Right shift but, in Logical right shift, you fill zero from the back.

Examples:

(110 >>> 1) becomes 011
(110 >>> 2) becomes 001

So basically, Logical right shift is simply adding the Zero you're supposed to throw away (as in ordinary right shift) to the back of the number.

Guess…. (111000 >>> 2) becomes???

001110!

For positive numbers like 6, 8 etc, the answer and rule will still be the same as the normal right shift.

So therefore (6 >>> 1) is 6 ÷ 2 which is 3.

If you convert 011 to decimal, you'll get 3 as well.

Zero coming first in binary really can be ignored, but logical Right shift decided to keep, logical indeed.

In some programming languages, they only have the Arithmetic Right SHIFT (>>).

Conclusion: Bitwise operations are simple!

I mean, they're simple, right?

In real life, you will hardly calculate Bitwise operations manually because you can simply run a program to calculate a huge operation in a few seconds.

But the idea of this article is to explain what's happening behind the scenes.

Thanks for reading!