Q.Convert (0.675)10 to octal.

What Does "Octal to Decimal" Even Mean?

You already know how to count in decimal — the system with digits 0 through 9. When you see the number 342, you instinctively read it as "three hundred and forty-two." But why? Because each position in a decimal number stands for a power of 10:

$342 = 3 \times 10^2 + 4 \times 10^1 + 2 \times 10^0$

Now imagine a world where you only have eight fingers. You'd count 0, 1, 2, 3, 4, 5, 6, 7 — and then what? You'd run out of digits. So the next number would be 10 (which means one eight and zero ones, not ten). That's octal: base 8.

Octal uses only the digits 0 through 7. The number 342 in octal does not mean three hundred and forty-two. It means something completely different — and to find out what, you convert it to decimal.

The Core Idea: Positional Value in Base 8

Every number system works the same way: the rightmost digit is the "units" place, the next digit to the left is the "base" place, then "base squared," and so on.

For octal (base 8):

• Rightmost digit: $8^0 = 1$ place
• Next left: $8^1 = 8$ place
• Next: $8^2 = 64$ place
• Next: $8^3 = 512$ place
• ... and so on

So to convert an octal number to decimal, you multiply each digit by its corresponding power of 8 and add everything up.

Worked Example: Convert $(342)_8$ to Decimal

Write the digits with their place values:

Now add: $192 + 32 + 2 = 226$

So $(342)_8 = 226$ in decimal.

Another Example: $(107)_8$

Here the digit 7 is fine (it's less than 8), but notice the digit 0. Zero still contributes nothing, but it holds the place.

Sum: $64 + 0 + 7 = 71$ …