Convert the following decimal numbers to binary:
a. 55
b. 32
c. 82
Answer (2)
To convert the decimal numbers 55, 32, and 82 into binary, we divide each number by 2, keeping track of the remainders. The binary results are as follows: 55 is 110111, 32 is 100000, and 82 is 1010010. This method involves reading the remainders in reverse order after performing the divisions.
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Convert 55 to binary by repeatedly dividing by 2 and collecting remainders in reverse order: 110111 .
Convert 32 to binary by repeatedly dividing by 2 and collecting remainders in reverse order: 100000 .
Convert 82 to binary by repeatedly dividing by 2 and collecting remainders in reverse order: 1010010 .
Explanation
Understanding Decimal to Binary Conversion We need to convert the decimal numbers 55, 32, and 82 into their binary equivalents. The binary system is a base-2 numeral system, which uses only two symbols: 0 and 1. To convert a decimal number to binary, we repeatedly divide the number by 2 and record the remainders. The binary representation is formed by the remainders read in reverse order.
Converting 55 to Binary Let's convert 55 to binary. We perform successive divisions by 2:
55 ÷ 2 = 27 with a remainder of 1
27 ÷ 2 = 13 with a remainder of 1
13 ÷ 2 = 6 with a remainder of 1
6 ÷ 2 = 3 with a remainder of 0
3 ÷ 2 = 1 with a remainder of 1
1 ÷ 2 = 0 with a remainder of 1
Reading the remainders in reverse order, we get 110111. Therefore, the binary representation of 55 is 11011 1 2 .
Converting 32 to Binary Now, let's convert 32 to binary:
32 ÷ 2 = 16 with a remainder of 0
16 ÷ 2 = 8 with a remainder of 0
8 ÷ 2 = 4 with a remainder of 0
4 ÷ 2 = 2 with a remainder of 0
2 ÷ 2 = 1 with a remainder of 0
1 ÷ 2 = 0 with a remainder of 1
Reading the remainders in reverse order, we get 100000. Therefore, the binary representation of 32 is 10000 0 2 .
Converting 82 to Binary Finally, let's convert 82 to binary:
82 ÷ 2 = 41 with a remainder of 0
41 ÷ 2 = 20 with a remainder of 1
20 ÷ 2 = 10 with a remainder of 0
10 ÷ 2 = 5 with a remainder of 0
5 ÷ 2 = 2 with a remainder of 1
2 ÷ 2 = 1 with a remainder of 0
1 ÷ 2 = 0 with a remainder of 1
Reading the remainders in reverse order, we get 1010010. Therefore, the binary representation of 82 is 101001 0 2 .
Final Answer In summary: a. 55 in decimal is 110111 in binary. b. 32 in decimal is 100000 in binary. c. 82 in decimal is 1010010 in binary.
Examples
Binary numbers are the language of computers. Every piece of data, from the text you read to the images you see, is stored as a series of 0s and 1s. Understanding how to convert between decimal and binary is fundamental in computer science. For example, when programming, you might need to work with binary representations of numbers to manipulate individual bits, which is crucial in low-level programming and embedded systems.
