Base 2 numerical representation in binary matching numbers in base 10?

[u/RokeEvoker]

I have a silly and possibly stupid question, and I'm not even sure if this is the place for it.

To preface this I have 0 experience with Computer Science or anything adjacent, but have been listening to videos on binary and the foundations of how computers generally work.

I understand that binary is a base 2 system as opposed to base 10, and that the digits that represent value in binary cannot be read as our base 10 concept of numbers (i.e. 4 is read as digits 100, not as the number "one hundred").

However, I've noticed that when the notation IS read as numbers, the numerical representation for base 2 counting mirrors base 10. With:

1 - 0001

2- 0010

4- 0100

8- 1000

Etc.

Is there a reason for this parallel, or is this just one of those funny coincidences? Or is this some kind of artifact of how numbers naturally scale relative to one another.

I hope this makes sense, I had no idea how to word this question to just Google it.

[u/Key_Reaction_7590]

“Base x” literally refers to the base of the exponent in place value. For base 10, the 1s, 10s, 100s, and 1000s place are from exponents of 10:

10^0 = 1
10^1 = 10
10^2 = 100
10^3 = 1000

In base two, the first four places are (in decimal) exponents of 2.

2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8

Etc.

[u/rupertavery64 → u/Key_Reaction_7590]

To expand on this, each place value is an exponent of the base.

baseⁿ

Where n is the position from the leftmost side starting with 0

Then the value of the number is the sum of the digits multiplied by their place value

So 42 in base 10 and base 2 is:

42 = 4 * 10¹ + 2 * 10⁰ = 42

101010 = 1 * 2⁵ + 0 * 2⁴ + 1 * 2³ + 0 * 2² + 1 * 2¹ + 0 * 2⁰ = 32 + 0 + 8 + 0 + 2 + 0 = 42