Explains the conversion of a binary number (base-2) to an equivalent
decimal number (base-10).
a. What is a Binary Number System?
b. What is a Decimal Number System?
c. What is Binary to Decimal Conversion?
d. How to Convert Binary to Decimal Numbers? Describe the steps of
different formulas to perform the conversion.
A. The Binary Number System:
The binary number system is a numeral system that uses only two digits, 0 and 1. It is the fundamental numbering system used in computers and digital systems. Each digit in a binary number represents a power of 2, with the rightmost digit being 2^0, the next being 2^1, then 2^2, and so on.
B. The Decimal Number System:
The decimal number system, also known as the base-10 system, is the most common numeral system used by humans. It uses 10 different digits, 0 to 9, and each digit represents a power of 10, with the rightmost digit being 10^0, the next being 10^1, then 10^2, and so on.
C. Binary to Decimal Conversion:
Binary to decimal conversion is the process of converting a binary number (base-2) to its equivalent decimal representation (base-10).
D. Steps to Convert Binary to Decimal Numbers:
1. Start from the rightmost digit (also known as the least significant bit) of the binary number.
2. Assign each digit a power of 2, starting with 2^0 for the rightmost digit, 2^1 for the next digit, 2^2 for the next, and so on.
3. Multiply each binary digit with its respective power of 2.
4. Sum up all the results obtained in step 3 to get the decimal equivalent.
Alternatively, a quicker approach is using the following formula:
1. Let n be the number of digits in the binary number.
2. For the i-th digit from the right (starting with i = 0), multiply the digit by 2 raised to the power of i.
3. Sum up all the results obtained in step 2.
For example, to convert the binary number 1011 to decimal:
Using the formula:
1. n = 4 (number of digits)
2. (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) = 8 + 0 + 2 + 1 = 11.
Using the step-by-step approach:
1. Start from the rightmost digit: 1.
2. Assign powers of 2: 2^3, 2^2, 2^1, 2^0.
3. Multiply each digit by its respective power of 2: (1 * 2^3) = 8.
4. Move to the next digit: 0.
5. Multiply by its respective power of 2: (0 * 2^2) = 0.
6. Move to the next digit: 1.
7. Multiply by its respective power of 2: (1 * 2^1) = 2.
8. Move to the next digit: 1.
9. Multiply by its respective power of 2: (1 * 2^0) = 1.
10. Sum up all the results: 8 + 0 + 2 + 1 = 11.
Therefore, the binary number 1011 is equal to the decimal number 11.