From: Headmaster

Hello,
How many digits has the number 10¹⁰⁰ if we write it in binary form?

We have three responses for you

Use the fact that 2¹⁰ is approximately 10³. (In fact, 2¹⁰ = 1024.) That makes 2³³⁰ in the neighborhood of 10⁹⁹. Thus, the answer to your question is that 10¹⁰⁰ requires about 333 binary digits. If you want to be more accurate you will have to work harder.

Chris

Using a calculator:
2^3.321= 9.993569...
2^3.322 = 10.0005...
So, 10¹⁰⁰> (2^3.321)¹⁰⁰ = 2^332.1
And, 10¹⁰⁰< (2^3.322)¹⁰⁰ = 2^332.2
So, in the binary expansion of of 10¹⁰⁰, there will be a digit (0 or
1) for each of the following places: 2⁰, 2¹, 2², ..., 2³³²
Hence, there are 333 digits

Paul

The number of digits of a number is revealed when you look at the logarithm of the number using a base equal to the number of possible digits. For example, 243 (in normal decimal notation - base 10) has three digits. When we take the logarithm in base 10, we see log₁₀ 243 2.39. We ignore the decimal portion and add one to the whole number and get 3 digits. Let's try this using binary.

243 in base ten is 11110011 in base two (binary). This is frequently written
243₁₀ = 11110011_2.

Now take the logarithm in base two: log₂ 243 7.92. Again, we drop the .92 and then add one to the seven and our method gives the correct 8 digits.

So your question can be answered if you add one to the whole number portion of
log₂ 10¹⁰⁰ . Remember to use the shortcuts that logarithms permit with exponents.

Hope this helps,
Stephen La Rocque.