Write $1011.101_2$ as base 10 number.
Answer (2)
Expand the base 2 number 1011.10 1 2 into powers of 2.
Calculate each term: ( 1 × 2 3 ) + ( 0 × 2 2 ) + ( 1 × 2 1 ) + ( 1 × 2 0 ) + ( 1 × 2 − 1 ) + ( 0 × 2 − 2 ) + ( 1 × 2 − 3 ) = 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 .
Sum the terms: 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 = 11.625 .
The base 10 equivalent is 11.625 .
Explanation
Understanding the Problem We are asked to convert the base 2 number 1011.10 1 2 to a base 10 number. This means we need to understand what each digit in the base 2 number represents in terms of powers of 2.
Expanded Form To convert the base 2 number to base 10, we express it in expanded form using powers of 2. The digits to the left of the radix point (the 'decimal' point) represent non-negative powers of 2, starting from 2 0 closest to the radix point and increasing to the left. The digits to the right of the radix point represent negative powers of 2, starting from 2 − 1 closest to the radix point and decreasing to the right.
So, we can write 1011.10 1 2 as:
( 1 × 2 3 ) + ( 0 × 2 2 ) + ( 1 × 2 1 ) + ( 1 × 2 0 ) + ( 1 × 2 − 1 ) + ( 0 × 2 − 2 ) + ( 1 × 2 − 3 )
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 Finally, we sum the terms to obtain the base 10 equivalent:
8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 = 11.625
Final Answer Therefore, the base 10 equivalent of 1011.10 1 2 is 11.625 .
Examples
Base 2 (binary) is fundamental in computer science because computers use binary digits (bits) to represent data. Converting between base 2 and base 10 allows us to understand the decimal (base 10) value of binary data. For example, the binary number 1011.10 1 2 might represent a floating-point number stored in a computer's memory. Converting it to base 10, as we did, gives us the human-readable value 11.625.
To convert 1011.10 1 2 to base 10, expand it using powers of 2 and calculate each term. The total sum of these terms equals 11.625, which is the base 10 equivalent. Thus, 1011.10 1 2 equals 11.625 in base 10.
;
