Q.Convert (3A5)16 into decimal number.

Number Base Identification – From Intuition to Precision

You already know that the digits 123 mean "one hundred and twenty-three". But why? Because you instinctively read it as:

That works because you are using base 10 — each place is a power of ten: $10^2, 10^1, 10^0$. The digits themselves are just symbols; the real meaning comes from the base.

Now imagine you see the string 123 but you are told it is written in base 4. Suddenly it does not mean "one hundred twenty-three". It means:

That is the core idea: the same string of digits represents different numbers depending on the base used to interpret it.

The Precise Statement

Here:

• $b$ is the base (a whole number $\ge 2$)
• Each digit $d_i$ must satisfy $0 \le d_i < b$
• The subscript $(...)_b$ explicitly tells you the base

So Number Base Identification is the skill of recognising what base a given numeral is written in, or determining the base from a relationship between the numeral and its value.

How You Actually Identify a Base

There are two common situations:

1. You are told the base explicitly.

Example: $(101)_2$ means binary. You compute its value as $1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 5$.

2. You are given an equation and must solve for the base.

Example: $(123)_b = 27_{10}$. You write:

Solve: $b^2 + 2b + 3 = 27 \implies b^2 + 2b - 24 = 0 \implies (b+6)(b-4)=0$. Since $b>0$, $b=4$. …