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Binary To Decimal

Q: What is a single binary digit called?

A single binary digit (a $0$ or a $1$) is called a 'bit'. This is a shortened version of 'Binary digIT'. A group of 8 bits is famously called a 'byte'.

Q: What is the largest decimal number you can make with 4 bits?

With 4 bits, the largest binary number you can write is $1111_2$. To convert this, you add the place values: $8 + 4 + 2 + 1 = 15$. So, $15$ is the largest decimal number you can represent with 4 bits.

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Ever wonder how computers think? They speak a secret language made of just two numbers: 0 and 1! This is called binary. In this lesson, we'll crack the code and learn how to translate any binary number into the familiar decimal numbers we use every day. It's easier than you think!

Binary To Decimal — an original Algebra911 reference diagram defining binary to decimal and a worked example. — How to Convert Binary to Decimal: The Secret Language of Computers

What Is The Binary Number System?

The binary number system is a way of writing numbers using only two digits: $0$ and $1$ . That's it! Unlike our regular number system, which has ten different digits ( $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ ), binary keeps it super simple. Because it uses only two digits, it's also called a base-2 system. Our everyday number system is called decimal or base-10.

Think of a light switch. It can only be in one of two states: ON or OFF. We could represent OFF with a $0$ and ON with a $1$ . This is the fundamental idea behind binary. In the world of electronics and computers, $0$ can mean 'no electricity' and $1$ can mean 'electricity is flowing'. By stringing together lots of these $0$ s and $1$ s, computers can represent every number, letter, and command imaginable.

So, a number like $1011$ in binary isn't one thousand eleven. It's a special code that represents a completely different value in our familiar decimal system. Our job is to learn how to translate it.

A Quick Refresher: How Does Decimal Place Value Work?

Before we dive into binary, let's quickly remember how our own base-10 system works. The position of a digit in a number tells us its value. We call this place value.

Let's take the number $472$ . It's not just a $4$ , a $7$ , and a $2$ . The $2$ is in the ones place, the $7$ is in the tens place, and the $4$ is in the hundreds place.

Mathematically, these place values are all powers of $10$ :

The rightmost place is the ones place: $10^{0} = 1$
The next place to the left is the tens place: $10^{1} = 10$
The next is the hundreds place: $10^{2} = 100$
And so on... (thousands, ten thousands, etc.)

So, the number $472$ is really a shorthand for:

(4 \times 100) + (7 \times 10) + (2 \times 1) = 400 + 70 + 2 = 472

(4 \times 10^{2}) + (7 \times 10^{1}) + (2 \times 10^{0}) = 472

Understanding this concept of place value is the key to unlocking binary.

How Does Place Value Work in Binary?

Binary works the exact same way as decimal, but with a different base. Instead of using powers of $10$ , the binary system uses powers of $2$ for its place values. This makes perfect sense, since it's a base-2 system!

Let's look at the binary place values, starting from the right and moving left:

The rightmost place is the ones place: $2^{0} = 1$
The next place to the left is the twos place: $2^{1} = 2$
The next is the fours place: $2^{2} = 4$
The next is the eights place: $2^{3} = 8$
The next is the sixteens place: $2^{4} = 16$
And so on! Each place value is double the one to its right.

Here is a table to help you visualize the first few binary place values:

Power of 2	$2^{7}$	$2^{6}$	$2^{5}$	$2^{4}$	$2^{3}$	$2^{2}$	$2^{1}$	$2^{0}$
Decimal Value	$128$	$64$	$32$	$16$	$8$	$4$	$2$	$1$

When we see a binary number like $101_{2}$ (the little $2$ just tells us it's a base-2 number), we know that the digits $1$ , $0$ , and $1$ sit in these place value slots. The rightmost $1$ is in the ones place, the $0$ is in the twos place, and the leftmost $1$ is in the fours place.

How Do You Convert Binary to Decimal? A Step-by-Step Guide

This is the fun part! Converting from binary to decimal is a simple process of multiplication and addition. Once you know the binary place values, you're ready to go. Here are the steps:

Write down the binary number.
Directly above each digit, write its corresponding decimal place value (starting with $1$ on the right and doubling each time you move left).
For each position, multiply the binary digit (either $1$ or $0$ ) by the place value written above it.
Add all of the results together. The sum is your answer in decimal!

Let's work through a few examples to see it in action.

Example 1

Convert the binary number $101_{2}$ to a decimal number.

Step 1 & 2: Write the number and its place values.

\begin{array}{ccc} 4 & 2 & 1 \\ 1 & 0 & 1 \end{array}

Step 3: Multiply each binary digit by its place value.

Leftmost digit: $1 \times 4 = 4$
Middle digit: $0 \times 2 = 0$
Rightmost digit: $1 \times 1 = 1$

Step 4: Add the results.

4 + 0 + 1 = 5

So, the binary number $101_{2}$ is equal to the decimal number $5$ .

Example 2

Convert the binary number $1101_{2}$ to a decimal number.

Step 1 & 2: Write the number and its place values. Remember to start from the right with $1$ and double as you go left.

\begin{array}{cccc} 8 & 4 & 2 & 1 \\ 1 & 1 & 0 & 1 \end{array}

Step 3: Multiply each binary digit by its place value. A helpful shortcut is to realize you only need to add the place values that have a $1$ beneath them, since anything multiplied by $0$ is just $0$ .

We have a $1$ in the $8$ s place. (Value: $8$ )
We have a $1$ in the $4$ s place. (Value: $4$ )
We have a $0$ in the $2$ s place. (Value: $0$ )
We have a $1$ in the $1$ s place. (Value: $1$ )

Step 4: Add the results.

8 + 4 + 0 + 1 = 13

Therefore, the binary number $1101_{2}$ is equal to the decimal number $13$ .

Let's Practice With a Bigger Binary Number

The method for converting binary to decimal works no matter how long the binary number is. Let's try one with more digits to prove it.

Example 3

Convert the binary number $101101_{2}$ to a decimal number.

Step 1 & 2: Write the number and assign the place values above it. We'll need more place values this time.

\begin{array}{cccccc} 32 & 16 & 8 & 4 & 2 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 \end{array}

Step 3 & 4: Find the values for each place and add them up. We can do this in one step by only looking at the columns with a $1$ in them.

The binary number has a $1$ in the following positions:

The $32$ s place
The $8$ s place
The $4$ s place
The $1$ s place

Now, we just add those place values together:

32 + 8 + 4 + 1 = 45

It's that simple! The binary number $101101_{2}$ is the same as the decimal number $45$ .

As you can see, the process is always the same. The more digits in the binary number, the higher the place values go, but the four steps for converting never change. With a little practice, you'll be able to do these conversions quickly in your head!

What Are Some Common Mistakes to Avoid?

When you're first learning to convert binary to decimal, it's easy to make a few common mistakes. Being aware of them will help you get the right answer every time.

Starting from the wrong side. ALWAYS assign place values starting from the rightmost digit. The digit on the far right is always the $1$ s place ( $2^{0}$ ). If you start from the left, your entire calculation will be incorrect.
Forgetting the 'zero power' rule. A common slip-up is thinking that $2^{0}$ equals $0$ or $2$ . Remember this important math rule: any number raised to the power of zero is $1$ . So, $2^{0} = 1$ , which is why the first place value is the ones place.
Using powers of 10 instead of 2. We are so used to our base-10 system that it's tempting to think in terms of tens, hundreds, and thousands. When you see a binary number, you must switch your brain into base-2 mode and think only in powers of $2$ : $1, 2, 4, 8, 16, 32, . . .$ .
Simple addition errors. After you've done all the hard work of finding the place values, it's a shame to get the wrong answer because of a small addition mistake. Double-check your sum before you finalize your answer!

Quick Summary: Your Binary-to-Decimal Cheat Sheet

Need a fast reminder? Here's everything you need to know to convert binary to decimal in a nutshell.

The Goal: To translate a base-2 number (using only $0$ s and $1$ s) into a base-10 number (the kind we use every day).

The Key Concept: Binary Place Values. They are powers of $2$ , doubling each time you move to the left.

... $64$ $32$ $16$ $8$ $4$ $2$ $1$

The 4-Step Method:

Write the binary number.
Assign the place values above each digit, starting with $1$ on the right.
Multiply each digit by its place value. (Or, just find the place values that have a $1$ under them).
Add these values together for your final decimal answer.

Example: $1101_{2}$

(1 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1) = 8 + 4 + 0 + 1 = 13

Keep this summary handy as you practice, and you'll become a binary expert in no time.

Frequently Asked Questions

What does the small 2 after a binary number mean?

The small, lowered number is called a subscript. It tells you the 'base' of the number system. A subscript 2, like in $101_{2}$ , means this is a base-2 or binary number. Our regular numbers are base-10, so we could write 25 as $25_{10}$ .

Why is it called 'binary'?

The prefix 'bi-' means two. The binary system gets its name because it only uses two digits, $0$ and $1$ . Just like a bicycle has two wheels, the binary system has two numbers.

What is a single binary digit called?

A single binary digit (a $0$ or a $1$ ) is called a 'bit'. This is a shortened version of 'Binary digIT'. A group of 8 bits is famously called a 'byte'.

Can you convert decimal numbers back to binary?

Yes, you absolutely can! It's the reverse process of what we learned here. It involves repeatedly dividing the decimal number by $2$ and recording the remainders.

Why do computers use binary instead of our decimal system?

Computers use binary because their most basic components, transistors, are like tiny switches. A switch can only be ON or OFF, which perfectly represents the two binary digits $1$ and $0$ . It's a much simpler and more reliable system for electronics to handle.

What is the largest decimal number you can make with 4 bits?

With 4 bits, the largest binary number you can write is $1111_{2}$ . To convert this, you add the place values: $8 + 4 + 2 + 1 = 15$ . So, $15$ is the largest decimal number you can represent with 4 bits.

Is there a base-3 or base-4 system?

Yes! You can have a number system in any base. A base-3 system, called ternary, would use the digits $0, 1, 2$ . A base-8 system, called octal, is also sometimes used in computing. Our binary (base-2) and decimal (base-10) systems are just the most common ones we encounter.