Write $1011.101_2$ as base 10 number.
Answer (2)
The binary number 1011.10 1 2 converts to 11.625 in base 10 by evaluating each digit's contribution based on powers of 2. The whole number part sums to 11 and the fractional part sums to 0.625. Adding these gives the final result of 11.625 .
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Convert the binary number 1011.10 1 2 to base 10.
Express the binary number in expanded form using powers of 2: ( 1 × 2 3 ) + ( 0 × 2 2 ) + ( 1 × 2 1 ) + ( 1 × 2 0 ) + ( 1 × 2 − 1 ) + ( 0 × 2 − 2 ) + ( 1 × 2 − 3 ) .
Calculate each term: 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 .
Sum the terms to obtain the base 10 equivalent: 11.625 .
Explanation
Understanding the Problem We are asked to convert the base 2 number 1011.10 1 2 to its equivalent base 10 representation. This involves understanding place values in base 2 and expressing the number as a sum of powers of 2.
Expanded Form The base 2 number 1011.10 1 2 can be written in expanded form as:
( 1 × 2 3 ) + ( 0 × 2 2 ) + ( 1 × 2 1 ) + ( 1 × 2 0 ) + ( 1 × 2 − 1 ) + ( 0 × 2 − 2 ) + ( 1 × 2 − 3 )
Each digit is multiplied by the corresponding power of 2.
Calculating Each Term Now, let's calculate each term:
1 × 2 3 = 1 × 8 = 8
0 × 2 2 = 0 × 4 = 0
1 × 2 1 = 1 × 2 = 2
1 × 2 0 = 1 × 1 = 1
1 × 2 − 1 = 1 × 2 1 = 0.5
0 × 2 − 2 = 0 × 4 1 = 0
1 × 2 − 3 = 1 × 8 1 = 0.125
Summing the Terms Summing up all the calculated terms:
8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 = 11.625
Therefore, the base 10 equivalent of 1011.10 1 2 is 11.625.
Final Answer The base 2 number 1011.10 1 2 is equal to 11.625 in base 10.
Examples
Binary numbers are the language of computers. Converting binary numbers to decimal numbers helps us understand how computers store and process information. For example, when you save a file on your computer, the data is stored as a series of binary digits (bits). Each byte (8 bits) can represent a decimal number from 0 to 255. Converting binary to decimal helps programmers interpret and manipulate this data effectively. Understanding binary and decimal conversions is crucial in computer science for tasks such as data representation, network communication, and hardware design.
