Binary Number System

What Is the Binary Number System?

The binary number system is a way of representing numbers using only two digits: $0$ and $1$. Unlike our familiar decimal system that uses ten digits, binary keeps it simple. Because it uses only two digits, it is also called the base-2 number system.

Think about the numbers you use every day. This is called the decimal or base-10 system. The name "base-10" comes from the fact that we use ten unique digits to write any number:

• $0,1,2,3,4,5,6,7,8,9$

Once we count past $9$, we run out of single digits. What do we do? We add another column (the tens place) and start over, giving us $10$. The binary system works on a very similar idea, but it runs out of digits much faster—right after $1$!

A Quick Look at Our Base-10 System

Before we dive deeper into binary, let's quickly review how our own number system works. The key is place value. Every position a digit holds in a number has a specific value.

Let's take the number $472$. It isn't just a $4$, a $7$, and a $2$ next to each other. It's really:

• $4$ hundreds
• $7$ tens
• $2$ ones

We can also write this using powers of $10$:

$(4\times 100)+(7\times 10)+(2\times 1)$

Or, for older students:

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

Each place to the left is $10$ times bigger than the place to its right. This base-10 structure is what you've been using your whole life. Understanding this is the key to unlocking how binary works.

How Does Place Value Work in Binary?

Binary place value works just like decimal place value, but instead of multiplying by $10$ for each new position, we multiply by $2$. The places aren't ones, tens, and hundreds. Instead, they are:

• Ones place ($2^0=1$)
• Twos place ($2^1=2$)
• Fours place ($2^2=4$)
• Eights place ($2^3=8$)
• Sixteens place ($2^4=16$)
• And so on...

Each place value is double the one to its right. Here is a table that shows the binary place values:

So, when we see a binary number like ${101}_2$ (the little $2$ means it's a base-2 number), the digits don't mean one hundred and one. The first $1$ is in the fours place, the $0$ is in the twos place, and the last $1$ is in the ones place. We are about to see how to use this to find its value!

How Do You Convert from Binary to Our Everyday Numbers?

Converting a binary number into a decimal number is a great way to start understanding the system. It's like translating from a secret code! All you have to do is add up the place values for every digit that is a $1$.

Here are the steps:

1. Write down the binary number.
2. Above each digit, write its place value ($1,2,4,8,16,...$) starting from the right and moving left.
3. Look for all the places where the binary digit is a $1$.
4. Add the place values from those positions together.
5. The sum you get is the decimal equivalent of the binary number!

How Do You Convert Our Numbers into Binary?

Now let's try the reverse: converting a familiar decimal number into its binary form. This might seem tricky at first, but it's a simple process of repeated division. We call this the 'Division by 2' method.

Here are the steps:

1. Start with your decimal number.
2. Divide this number by $2$.
3. Write down the whole number result and the remainder. The remainder will always be either $0$ (if the number was even) or $1$ (if the number was odd).
4. Take the whole number result from the previous step and divide it by $2$ again.
5. Continue this process, dividing the result by $2$ each time, until the result is $0$.
6. To find the binary number, write down all the remainders you found, starting from the bottom and going to the top.

This last step is the most important! Reading the remainders in the wrong order is a very common mistake.

Why Is Binary So Important for Computers?

This might seem like a lot of work just to write numbers in a different way. So why bother? The answer is simple: computers.

A computer is made of billions of tiny electronic switches called transistors. Each switch can be in one of two states: it can be ON or it can be OFF. There is no in-between state. This two-state system is a perfect match for the binary number system!

• ON can be represented by the digit $1$.
• OFF can be represented by the digit $0$.

Every single thing a computer does, from showing your favorite video to calculating a math problem, is broken down into a massive sequence of these ON and OFF signals. The letter 'A' on your screen is stored as a specific binary code ($01000001$), a specific color is a code, and a sound is a code. By using binary, computers can process and store huge amounts of information reliably and incredibly fast. It's the fundamental language of all digital technology.

Let's Try One More Conversion

Practice is the best way to get comfortable with binary. Let's do one more conversion from a slightly larger binary number to a decimal number to make sure you've got the hang of it.

Common Mistakes When Working with Binary

As you learn about binary, there are a few common pitfalls to watch out for. Being aware of them can help you avoid making simple errors.

• Reading Remainders Backwards: This is the most common mistake when converting from decimal to binary. Remember the rule: after you finish dividing, you must read the list of remainders from the bottom to the top.
• Mixing Up Place Values: It's easy to forget that binary place values double each time ($1,2,4,8,16,...$). Some students accidentally use a pattern like $1,2,3,4$ or forget a value in the sequence. Always double-check your place values before adding.
• Forgetting the Subscript: When you're working with both binary and decimal numbers, things can get confusing. Is $10$ the number ten or the number two? Using subscripts helps keep everything clear. Write ${10}_{10}$ for decimal and ${10}_2$ for binary.
• Thinking ${10}_2$ is 'Ten': Our brains are so used to the decimal system that it's easy to see $10$ and think 'ten'. In binary, ${10}_2$ means one group of two and zero ones, which equals the decimal number $2$.

Binary Numbers Quick Reference

Here is a quick summary of the most important points about the binary number system.

• What is it? The binary system is a base-2 number system that uses only two digits: $0$ and $1$.
• Why is it used? It is the fundamental language of computers and all digital devices. The $0$ and $1$ represent the OFF and ON states of electronic switches.
• Place Values: Binary place values are based on powers of two. From right to left, they are $1,2,4,8,16,32,...$ and so on.
• Converting Binary to Decimal: To convert ${1101}_2$ to decimal, you add the place values for each $1$. This would be $8+4+1=13$.
• Converting Decimal to Binary: To convert a decimal number like $13$ to binary, use repeated division by $2$. Then, read the remainders from the bottom up to get the answer (${1101}_2$).