In binary, the number system used by computers, the number “1” is represented by a single bit, which can either be a “0” or a “1”. Similarly, the number “0” is represented by a single bit, which can also be a “0” or a “1”. This means that any number in binary can be represented using only “0”s and “1”s.
To understand why “1 1 10” represents the decimal number “6” in binary, we need to first understand how binary numbers work. In binary, each digit represents a power of two, starting with 2^0 (which is equal to 1) and increasing by a factor of two for each subsequent digit. For example, the binary number “1010” represents:
1 * 2^3 + 0 * 2^2 + 1 * 2^1 + 0 * 2^0 = 8 + 0 + 2 + 0 = 10
So, in binary, the number “1010” represents the decimal number “10”. Similarly, we can convert “1 1 10” into its decimal equivalent using this same process.
Starting from the right-hand side, the first digit in “1 1 10” represents 2^0, which is equal to 1. The second digit represents 2^1, which is equal to 2. The third digit represents 2^2, which is equal to 4. Adding these values together, we get:
1 * 2^0 + 1 * 2^1 + 1 * 2^2 = 1 + 2 + 4 = 7
So, “1 1 10” represents the decimal number “7” in binary, not “6”. However, if we were to flip the order of the digits and write “1 0 1 1” instead, we would get:
1 * 2^0 + 1 * 2^1 + 0 * 2^2 + 1 * 2^3 = 1 + 2 + 0 + 8 = 11
Therefore, “1 0 1 1” represents the decimal number “11” in binary.
To understand why “1 1 10” is often used to represent the decimal number “6” in binary, we need to look at how binary numbers are sometimes grouped together. In some cases, binary numbers are grouped into groups of four digits, called nibbles. In this case, “1 1 10” would be the second nibble of the binary number “0000 0110”, which represents the decimal number “6”. This grouping of binary digits into nibbles can make it easier to read and work with binary numbers, especially when working with larger numbers that require many digits.
In summary, “1 1 10” does not represent the decimal number “6” in binary, but rather the decimal number “7”. However, when grouped together with other binary digits, it can be used to represent the decimal number “6”. Binary numbers work by using digits to represent powers of two, and converting between binary and decimal requires multiplying each digit by its corresponding power of two and adding up the results.