What is $2^{32} - 1$ in decimal notation?

Question

I came across decimal notation, and surprisingly I have never heard of it. I have heard of scientific notation and just thought this was decimal notation. However, when I want to find a definition on the internet, it isn’t clear. Could somebody please explain to me? This looks to be a very basic thing in maths which I should know by now so I don’t understand why I don’t know it.

As an example, what is $2^{32} - 1$ in decimal notation? Is this also the same as decimal expansion? Thank you in advance.

Answer 1

They could simply mean the usual base 10 representation. $$ (d_m \dotsc d_0)_{10} = \sum_{k=0}^{m} d_k 10^k $$ thus $$ 2^{32}-1 = 4294967295 = (11111111111111111111111111111111)_2 $$ where I assumed $32$ to be given in base $10$ already, as no base has been provided for it. :)

Answer 2

Decimal Notation uses successive non-negative powers of $10$ with coefficients in the range $0-9$, as @mvw has very nicely explained in his answer.

As for a specific example, one has:

$$\begin{align} 2^{32} - 1 &= (2^{16} - 1)(2^{16} + 1) \\ &= (2^8 - 1)(2^8 + 1)(2^{16} + 1) \\ &= (2^4 - 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1) \\ &= (2^2 - 1)(2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1) \\ &= (2^1 - 1)(2^1 + 1)(2^2 + 1)(2^4 + 1)(2^8 + 1)(2^{16} + 1) \\ &= 1 \cdot 3 \cdot 5 \cdot 17 \cdot 257 \cdot (2^{16} + 1) \\ &= 1 \cdot 3 \cdot 5 \cdot 17 \cdot 257 \cdot 65537 \\ &= 3 \cdot 5 \cdot 17 \cdot 257 \cdot 65537 \\ &= 15 \cdot 17 \cdot 257 \cdot 65537 \\ &= 255 \cdot 257 \cdot 65537 \\ &= 65535 \cdot 65537 \\ &= \boxed{4294967295} \tag 1 \end{align}$$

Of course I did all the multiplications in my head! Yeah, right!!!