Octal To Decimal

What Is the Octal Number System?

The octal number system is a way of representing numbers using a base of 8. This means it only uses eight unique digits to write any number: $0,1,2,3,4,5,6,$ and $7$. You might notice that the digits $8$ and $9$ are missing! That's the most important rule of the octal system: you can't use any digit higher than 7.

Think about our regular number system, which is called the decimal or base-10 system. It's called base-10 because it uses ten digits ($0$ through $9$). The octal system is just a different way of counting that has fewer digits to work with. To show that a number is written in octal, we often add a small subscript 8 at the end, like this: ${145}_8$. This tells us, "Hey, this is the octal number one-four-five, not the decimal number one hundred forty-five." It's like a label that tells us which number language we're speaking.

So, why would we ever use a system with fewer digits? The octal system was very useful in early computing. Computers work with binary code (base-2), which uses only 0s and 1s. Long strings of binary are hard for humans to read, but it's very easy to group them into sets of three to create octal numbers. This made octal a convenient shorthand for programmers back in the day.

How Does Our Decimal System Work?

Before we can master converting from octal, let's take a quick trip back to the number system we use every single day: the decimal, or base-10, system. Understanding how our own numbers are built makes learning a new system much easier. The secret is something called place value.

Every digit in a decimal number has a specific value based on its position. Let's take the number $352$. It's not just a 3, a 5, and a 2 sitting next to each other. The 2 is in the "ones" place. The 5 is in the "tens" place. The 3 is in the "hundreds" place. Each of these places is a power of 10. The ones place is ${10}^0$, the tens place is ${10}^1$, the hundreds place is ${10}^2$, and so on.

So, the number $352$ is really a secret code for:

$$(3\times 100)+(5\times 10)+(2\times 1)$$

Or, using exponents (powers):

$$(3\times {10}^2)+(5\times {10}^1)+(2\times {10}^0)$$

This might look complicated, but it's the fundamental idea behind all number systems. Each spot has a value, and that value is determined by the base of the system. Here is a table to visualize the decimal place values:

When we convert from octal to decimal, we are going to use this exact same idea of place value. But instead of using powers of 10, we'll be using powers of 8.

What Are the 'Powers of 8'?

The secret ingredient for converting octal numbers is understanding the powers of 8. Just like our decimal system is built on powers of 10 (1, 10, 100, 1000, etc.), the octal system is built on powers of 8. An exponent, that little number written high and to the right, just tells you how many times to multiply the base number by itself.

Let's look at the first few powers of 8. We will need these for our conversions.

• $8^0=1$. This is a very important rule in math! Any number raised to the power of 0 is equal to 1. It's the starting point for all our place values.
• $8^1=8$. This is just 8 by itself.
• $8^2=8\times 8=64$.
• $8^3=8\times 8\times 8=512$.
• $8^4=8\times 8\times 8\times 8=4096$.

These are the place values in the octal system. Instead of a ones place, tens place, and hundreds place, the octal system has a ones place ($8^0$), an eights place ($8^1$), a sixty-fours place ($8^2$), and so on. Every time we move one spot to the left in an octal number, the place value becomes 8 times bigger. Knowing these values is the key that unlocks the conversion process.

How Do You Convert an Octal Number to a Decimal Number?

We're now ready to learn the main event: the step-by-step process for converting any octal number into its decimal equivalent. It's a simple recipe that works every time. Once you understand the logic of place values from the previous sections, this process will feel very natural. We are essentially just calculating the total value of the number based on its octal place values.

Here is the four-step method:

1. Identify the Place Values: Write down the octal number. Starting from the digit on the far right, assign it the $8^0$ place value. Assign the next digit to the left the $8^1$ place, the next one $8^2$, and continue this pattern for all the digits.
2. Multiply Each Digit: Take each digit from your octal number and multiply it by its corresponding place value (the power of 8 you just assigned it).
3. Calculate the Products: Find the result of each multiplication from the previous step. It's helpful to write each one down clearly.
4. Add Them All Up: Sum together all the results from step 3. This final sum is the decimal equivalent of your original octal number!

This entire process can be summarized with a general formula. If you have an octal number with digits $d_n...d_2{d}_1{d}_0$, its decimal value is:

Don't worry if the formula looks a little intimidating. The examples below will show you just how simple the four steps are in practice.

Let's Practice: Converting a Simple Octal Number

Theory is great, but math is learned by doing! Let's walk through our first conversion together using the four-step method. We'll start with a simple, two-digit octal number.

Tackling a Bigger Number: A Three-Digit Conversion

The great thing about our method is that it works for octal numbers of any size. Let's try it on a three-digit number to see how the process stays exactly the same, just with one extra step.

What About Octal Numbers with a Point?

This is a slightly more advanced topic, but it follows the same logic. What happens if you see an octal number with a point, like ${23.4}_8$? This is just like a decimal point in our regular numbers. The digits to the right of the point represent fractions.

While the place values to the left of the point are positive powers of 8 ($8^0,8^1,8^2,$ etc.), the place values to the right of the point are negative powers of 8.

• The first digit to the right is the $8^{-1}$ place (which is $1/8$).
• The second digit to the right is the $8^{-2}$ place (which is $1/64$).

Let's try an example.

What Are Some Common Mistakes to Avoid?

When you're first learning to convert from octal to decimal, there are a few common trip-ups. Being aware of them can help you double-check your work and build confidence.

• Using an 8 or 9: The most common mistake is forgetting that the octal system only uses digits 0 through 7. If you see a number like $481$ and are asked to treat it as octal, it's a trick question! That number cannot be an octal number.
• Starting the Powers Incorrectly: Always, always, always start numbering the place values from the rightmost digit before the decimal point with $8^0$. Many people accidentally start with $8^1$, which throws off the entire calculation. The ones place is always power zero.
• Forgetting $8^0=1$: It's a simple rule, but sometimes people think a number to the power of 0 is 0. Remember, $8^0=1$. That rightmost digit's value is just itself. For example, in ${12}_8$, the 2 is multiplied by 1, not 0.
• Calculation Errors: The process is simple, but it involves multiplication and addition. It's easy to make a small mistake. Always double-check your math, especially when you're multiplying by numbers like 64 or 512.

Quick Conversion Summary

Feeling confident? Here is a super-fast summary of everything you need to remember to convert any octal number to its decimal form. Use this as a quick reference when you're doing practice problems.

The Four Key Steps:

1. Assign Places: Label each digit's place value, starting with $8^0$ for the digit just to the left of the decimal point and increasing to the left.
2. Multiply: Multiply each octal digit by its power-of-8 place value.
3. Calculate: Find the value of each of those products.
4. Add: Sum all the values to get your final decimal number.

The Core Formula:

That's it! As long as you remember the place values are powers of 8, you can solve any octal-to-decimal conversion problem.