Base conversion doubt

Question

(123) in base 8 can be converted to base 10 as follows => $3*8^{0}+2*8^{1}+1*8^{2} = 83$

But when I do the conversion by taking a group of 2 numbers together like $(01)*(8^{2})^{1} + (23)*(8^{2})^{0} =64+23 = 87$

Why is the discrepancy arising ?

Answer 1

This discrepancy is there because the 23 you have written is still in base $8$ representation but you are treating it as a base $10$ number.

Answer 2

@Charles nailed the key point in their answer, but I thought I would mention the conventional way to keep track of the base of a number when there is more than one base in play, and that's by using a subscript, e.g.

$$123_8 = 83_{10}$$

Attentively using this notation would have allowed you to write your last expression as

$$ (01)_8∗(8^2)^1+(23)_8∗(8^2)^0$$ and would have prevented the mistake.

When you first see this notation there are a lot of objections that come to mind (e.g. What base is the base written in?), but there are two conventions that make it work:

• The base numbers are always written in decimal.
• If no base is specified, we assume decimal.

Responding to your question in the comments: I'm not sure I understand, but yes,

$$123_8=1_8\cdot 8^2+2_8\cdot 8^1+3_8\cdot 8^0$$

but because $$3_8=3_{10}$$ we don't have to put a disambiguating subscript there.