How many bits do we need in order to represent the first 15 numbers? How about the first 16, 25 or n numbers?
The Correct answer and Explanation is:
To determine how many bits are required to represent a certain range of numbers, we need to understand the relationship between bits and numerical representation in binary format. In binary, each bit can have two possible values: 0 or 1. Therefore, the number of unique values that can be represented by ( b ) bits is given by the formula:
[
\text{Unique values} = 2^b
]
1. First 15 Numbers
The first 15 numbers are typically considered to be the integers from 0 to 14. To find the number of bits required to represent these numbers, we need to find the smallest ( b ) such that:
[
2^b \geq 15
]
Calculating for ( b ):
- ( 2^0 = 1 ) (1 value, not enough)
- ( 2^1 = 2 ) (2 values, not enough)
- ( 2^2 = 4 ) (4 values, not enough)
- ( 2^3 = 8 ) (8 values, not enough)
- ( 2^4 = 16 ) (16 values, sufficient)
Thus, we need 4 bits to represent the first 15 numbers.
2. First 16 Numbers
For the first 16 numbers (0 to 15), we follow a similar process:
[
2^b \geq 16
]
Here, ( 2^4 = 16 ) is sufficient, so 4 bits are also required for the first 16 numbers.
3. First 25 Numbers
For the first 25 numbers (0 to 24):
[
2^b \geq 25
]
Calculating:
- ( 2^4 = 16 ) (not enough)
- ( 2^5 = 32 ) (sufficient)
Therefore, we need 5 bits to represent the first 25 numbers.
4. General Case for ( n ) Numbers
For any ( n ) numbers, where we want to represent numbers from 0 to ( n-1 ), we determine ( b ) as follows:
[
2^b \geq n
]
Thus, the number of bits required can be expressed as:
[
b = \lceil \log_2(n) \rceil
]
Where ( \lceil x \rceil ) is the ceiling function, which rounds up to the nearest integer. This means you will need enough bits to represent all numbers up to ( n-1 ), ensuring that you have sufficient unique combinations of 0s and 1s to cover the range.
In summary:
- For the first 15 numbers: 4 bits
- For the first 16 numbers: 4 bits
- For the first 25 numbers: 5 bits
- For ( n ) numbers: ( b = \lceil \log_2(n) \rceil ) bits.