To determine how high you could count in binary using all 10 of your fingers, let's consider each finger as representing a binary digit (bit). Here's how to approach the problem:
1. Understanding Binary Representation:
- In binary, each bit can be either 0 or 1.
- With [tex]\( n \)[/tex] bits, the possible binary numbers range from [tex]\( 0 \)[/tex] to [tex]\( 2^n - 1 \)[/tex].
2. Counting with 10 Bits:
- If you use each of your 10 fingers as a bit, you have 10 bits to represent numbers.
- Each bit can be either 0 (finger down) or 1 (finger up).
3. Calculating the Maximum Number:
- The highest number in binary that can be represented with 10 bits is the one where all bits are 1.
- In binary, the number where all 10 bits are 1 is written as [tex]\( 1111111111_2 \)[/tex].
4. Converting Binary to Decimal:
- The binary number [tex]\( 1111111111_2 \)[/tex] can be converted to decimal.
- To find the decimal equivalent, you use the formula for the sum of a geometric series where each bit represents an increasing power of 2, starting from [tex]\( 2^0 \)[/tex] up to [tex]\( 2^9 \)[/tex].
5. Formula and Calculation:
- The formula to determine the maximum countable number with 10 bits is [tex]\( 2^{10} - 1 \)[/tex].
- This is because the highest binary number represents [tex]\( 2^0 + 2^1 + 2^2 + \ldots + 2^9 \)[/tex], which is equal to [tex]\( 2^{10} - 1 \)[/tex].
6. Result:
- Calculating [tex]\( 2^{10} - 1 \)[/tex] gives:
[tex]\[
2^{10} = 1024
\][/tex]
Subtracting 1:
[tex]\[
1024 - 1 = 1023
\][/tex]
Thus, you can count up to 1023 in binary using all 10 of your fingers as bits.
