Octal to Decimal Converter
Octal to Decimal Converter. Type your value into Octal Input and the Octal to Decimal Converter instantly returns its Decimal Output. Enter your cyan, magenta, yellow, and key percentages into the cmyk to hex converter to get the matching hex color for screen display.
Type a base 8 value into this octal to decimal converter, press the Convert button, and you get its decimal equivalent along with the working. Each digit in an octal number is a power of 8 in disguise, so the conversion comes down to multiplying and adding. Below you'll find the formula, two worked methods, a lookup table and the reasons this number system still matters in computing.
How the Octal to Decimal Converter Works
You enter an octal value made only of the digits 0 to 7, and the tool reads it from right to left. The rightmost digit sits in the ones place, the next digit in the eights place, and the place value grows by a factor of 8 with every step to the left. The tool multiplies each digit by its place value, adds the products, and prints the total in base 10 so you can read it like any everyday number. Use the hex to binary converter to turn a hex string into binary and toggle grouped-digit formatting for easier reading before copying the result.
Because the page shows the work as well as the answer, you can follow the calculation step by step and check it against your own pencil-and-paper result. Whether you search for an oct to dec tool or a base converter, the logic is identical. Try 3517 in the input box: the converter returns 1871, which is the example the rest of this guide unpacks.
- Type your octal number using only the digits 0 to 7.
- Press the convert button to start the calculation.
- Read the decimal answer and the line-by-line breakdown beneath it.
- Clear the field and enter other octal numbers whenever you want to compare results.
What Is the Octal Numbering System in Base 8?
Octal Number System Basics
The octal number system is a positional numeral system whose radix is 8. It has exactly eight symbols: 0, 1, 2, 3, 4, 5, 6 and 7. There is no digit 8 or 9, because once you count past 7 you carry into the next place, so the decimal number eight is written as 10 in octal. Mathematicians write the base as a small subscript, as in 35178, so that nobody mistakes an octal number for an ordinary decimal one.
Decimal Number System Basics
The decimal number system, also called the decimal numbering system, is the one you use every day. Its base is 10, it uses ten symbols from 0 to 9, and every place is a power of 10. This Hindu-Arabic style of positional notation is exactly what makes the octal conversion easy: you already know how to read 1871 as one thousand, eight hundreds, seven tens and one, and octal simply swaps the powers of 10 for powers of 8.
| Feature | Octal | Decimal |
|---|---|---|
| Base number | 8 | 10 |
| Digits available | 0 to 7 | 0 to 9 |
| Place values, right to left | 1, 8, 64, 512 | 1, 10, 100, 1000 |
| Written as | 35178 | 187110 |
Octal to Decimal Conversion Formula
Every octal number can be expanded into a sum of its digits times powers of 8. For an octal number with digits \(d_{n-1} \dots d_1 d_0\), the formula is: Use the uicolor to hex converter to turn a Swift UIColor definition into a hex code you can drop into a design tool outside Xcode.
$$N_{10} = d_{n-1} \times 8^{n-1} + \dots + d_1 \times 8^{1} + d_0 \times 8^{0}$$
Here \(d_0\) is the rightmost digit, \(n\) is how many digits the number has, and each exponent counts the position starting from zero. Anything raised to the power of 0 equals 1, which is why the ones digit is never multiplied by anything larger.
Powers of 8 You Should Know
Memorizing the first few powers makes mental conversion much faster. This short conversion chart covers every place value you meet in numbers up to six octal digits.
| Position | Power of 8 | Place value |
|---|---|---|
| 1st from right | 80 | 1 |
| 2nd from right | 81 | 8 |
| 3rd from right | 82 | 64 |
| 4th from right | 83 | 512 |
| 5th from right | 84 | 4,096 |
| 6th from right | 85 | 32,768 |
Step by Step Octal to Decimal Conversion Example
Take the octal number 3517; as with all worked examples here, you can verify it by hand. It has four digits, so the leftmost digit gets 83 and the rightmost gets 80. Write each digit with its place value, multiply each digit by its power of 8, and add all the products together.
| Octal digit | Power of 8 | Calculation | Product |
|---|---|---|---|
| 3 | 83 = 512 | 3 × 512 | 1,536 |
| 5 | 82 = 64 | 5 × 64 | 320 |
| 1 | 81 = 8 | 1 × 8 | 8 |
| 7 | 80 = 1 | 7 × 1 | 7 |
The sum of the digits' contributions is 1,536 + 320 + 8 + 7 = 1,871, so \(3517_8 = 1871_{10}\). If your result ever looks far too large or small, check that you started the exponents at zero on the right rather than at one.
Base 8 to Base 10 Shortcut with Repeated Multiplication
You do not need to write out any powers at all. Start with the most significant digit, the leftmost digit, and keep a running total: multiply the total by 8, then add the next digit. Repeat until you have used the last digit, and never multiply after that final addition.
- Start with the leftmost digit: 3.
- Multiply by 8 and add the next digit: 3 × 8 + 5 = 29.
- Multiply by 8 and add the next digit: 29 × 8 + 1 = 233.
- Multiply by 8 and add the last digit: 233 × 8 + 7 = 1,871.
The answer matches the place-value method exactly, which makes this approach a good way to double-check a result. It is also the method most programmers use in code, because it needs only one multiplication and one addition per digit.
Octal to Decimal With a Decimal Point
An octal number can have a decimal point too, usually called an octal point. Digits to the right of it have negative exponents: the first is worth 8-1 = 1/8, the second 8-2 = 1/64, and so on. For 52.348 you calculate 5 × 8 + 2 × 1 + 3 × 1/8 + 4 × 1/64 = 40 + 2 + 0.375 + 0.0625, which gives 42.4375 in decimal.
Octal to Decimal Table for Quick Lookups
When you only need a familiar value, a lookup is faster than any calculation. Each row below was computed with the formula above, and the values are the ones you meet most often in programming and electronics.
Octal to Decimal Conversion Table of Common Values
| Octal | Decimal | Why it matters |
|---|---|---|
| 7 | 7 | Largest single octal digit |
| 10 | 8 | First carry into the eights place |
| 77 | 63 | Largest two-digit value |
| 100 | 64 | 82 |
| 377 | 255 | Largest value of one byte |
| 644 | 420 | Common file permission |
| 755 | 493 | Common file permission |
| 777 | 511 | Largest three-digit value |
| 1000 | 512 | 83 |
| 7777 | 4,095 | Largest four-digit value |
Notice the pattern: the first eight octal digits match their decimal values, and every jump to a new place multiplies the value by 8 rather than 10.
Octal to Decimal Conversion of a Legacy Memory Address
Marta is restoring an early minicomputer from a scanned maintenance binder, and the program listing marks the end of a data buffer at address 6743. Every address in the binder is octal, but she plans her patch in decimal words, so she converts each one to get the decimal equivalent of each address.
She types 6743 into the converter and presses convert. The working appears line by line: 6 × 512 = 3,072, 7 × 64 = 448, 4 × 8 = 32 and 3 × 1 = 3, which sum to 3,555. The machine has 4,096 words, and its last address, 77778, converts to 4,095, so 3,555 sits inside the memory map.
The loader starts at 76008. The same conversion gives 7 × 512 + 6 × 64 = 3,584 + 384 = 3,968. Only in decimal can she subtract directly: 3,968 − 3,555 = 413 free words between the buffer and the loader.
- Buffer end: 67438 = 3,555
- Loader start: 76008 = 3,968
- Free words in between: 413
Her 450-word patch overruns that gap by 37 words, so the decimal view tells her to shorten it before it overwrites the loader.
Decimal to Octal: Reversing the Conversion
Going the other way uses repeated division. Divide the decimal number by 8, record the remainder, then divide the quotient by 8 again until the quotient reaches zero. Reading the remainders from last to first gives the octal answer.
- 1871 ÷ 8 = 233, remainder 7
- 233 ÷ 8 = 29, remainder 1
- 29 ÷ 8 = 3, remainder 5
- 3 ÷ 8 = 0, remainder 3
Reading the remainders upward gives 3517, and a quick decimal to octal check like this proves the round trip works.
Octal System Links to Binary and Hexadecimal
Since 8 equals 23, every octal digit stands for exactly three binary digits. Convert 35178 by replacing each digit with its three-bit group: 3 is 011, 5 is 101, 1 is 001 and 7 is 111, so the binary form is 11101001111. Group the same bits in fours instead and you get the hexadecimal value 74F, which is also 1871 in decimal. This tidy relationship is why octal became popular as a shorthand for long binary strings.
Where the Octal System Is Still Used and Converted
Octal grew up with early computing, when machines with 12-bit, 24-bit or 36-bit word sizes split neatly into groups of three bits. Today it is less common than hexadecimal, but it remains part of computer science, electronics and electrical engineering courses, and you will still run into it in assembly listings, programming languages and digital hardware manuals. Anyone who reads mathematics or computers documentation benefits from reading octal numbers fluently, and most readers convert an address or permission code to decimal to compare it with sizes and limits given in everyday units.
Octal to Decimal in File Permissions
The best-known everyday octal number is the three-digit permission code used on Unix-style systems. Each digit adds read (4), write (2) and execute (1) values, so 755 means the owner can do everything while everyone else can read and run the file. Converted to decimal, 7558 is 7 × 64 + 5 × 8 + 5 = 493, and 6448 is 420. The decimal figure rarely matters in practice, but seeing it shows why the octal form is far easier to read.
Octal to Decimal Mistakes to Avoid with Octal Numbers
- Using the digits 8 or 9. They do not exist in base-8, so a value such as 1984 is not a valid octal number.
- Starting exponents at one. The rightmost digit always uses 80.
- Reading 10 as ten. In octal, 10 is eight, and 100 is sixty-four.
- Mixing up the base. Write the subscript every time so a reader knows which system the digits belong to.