Question

Convert each of the following excess 32 representations to its equivalent base 10 representation: a. 011111 b. 100110 c. 111000 d. 000101 e. 010101

Short answer

a. -1, b. 6, c. 24, d. -27, e. -11.

Step-by-step solution

Understanding Excess-32 Notation

The excess-32 notation means that a given binary number represents its actual value increased by 32. To find the actual base-10 value, we need to subtract 32 from the value each binary number directly represents.

Convert 011111 to Base-10

Convert the binary number 011111 to its base-10 equivalent. First, calculate its decimal value. It is the binary number 31. Now, subtract 32 from 31 to find the base-10 value. \[31 - 32 = -1\].

Convert 100110 to Base-10

Convert the binary number 100110 to its base-10 equivalent. The decimal value of the binary number is 38. Subtract 32 to get its equivalent base-10 value. \[38 - 32 = 6\].

Convert 111000 to Base-10

Convert the binary number 111000 to its base-10 equivalent. The decimal value is 56. Subtract 32 to find its base-10 equivalent.\[56 - 32 = 24\].

Convert 000101 to Base-10

Convert the binary number 000101 to its base-10 equivalent. The decimal value of this binary number is 5. Subtract 32 to get the base-10 representation.\[5 - 32 = -27\].

Convert 010101 to Base-10

Convert the binary number 010101 to its base-10 equivalent. The decimal value is 21. Subtract 32 to get the final base-10 number.\[21 - 32 = -11\].

Key concepts

Binary to Decimal Conversion

Binary numbers are composed of only two digits, 0 and 1. To convert a binary number to its decimal form, which is a form that humans commonly use, each digit of the binary number is given a specific weight based on its position.
The rightmost digit has a weight of 2 raised to the power of 0, the next digit has a weight of 2 raised to the power of 1, and so on. To get the decimal value, you sum these weighted values.
For instance, to convert the binary number 011111 to decimal, calculate it as follows:

• First digit: 0 x 2^5 = 0
• Second digit: 1 x 2^4 = 16
• Third digit: 1 x 2^3 = 8
• Fourth digit: 1 x 2^2 = 4
• Fifth digit: 1 x 2^1 = 2
• Sixth digit: 1 x 2^0 = 1

Adding these values, the decimal equivalent is 31.
This basic understanding of weighting digits is key to performing binary to decimal conversions.

Negative Numbers in Binary

When dealing with binary numbers, representing negative values requires special techniques. Systems like Excess-N (such as Excess-32) notation help solve this problem.
In these systems, you start with a set value—in Excess-32, that set value is 32. To find the true representation, you subtract this "excess" value from the calculated decimal.
Let’s say your binary converts to 31 in decimal form. By subtracting 32:

• You get 31 - 32 = -1.

This means that Excess-32 can effectively handle both negative and positive decimal conversion, simplifying the handling of negative numbers in binary.

Base-10 Conversion

Base-10, known as the decimal system, is used in everyday counting and mathematical calculations by people. It involves digits 0 through 9.
When performing binary to base-10 conversions—as seen in exercises using Excess-32 notation—you often subtract a constant value from the converted decimal form. This subtraction adjusts for the excess value added to the original binary interpretation.
For example, the binary number 100110 is first converted to decimal 38. Applying the Excess-32 rule by subtracting 32 gives:

• 38 - 32 = 6.

Thus, understanding base-10 conversion is essential as it allows interpretation of binary numbers in a more familiar decimal form.

Binary Arithmetic

Binary arithmetic is the heart of computations within digital electronics and computer systems. Even simple operations, such as addition and subtraction, follow a base-2 system allowing complex arithmetic using only 0s and 1s.
When you subtract an excess value, like 32, from a binary converted decimal, you are performing a simple arithmetic operation. Let’s illustrate this with the binary 111000, which converts to 56.
Subtracting 32 gives:

• 56 - 32 = 24.

This subtraction is straightforward yet critical. Binary arithmetic, albeit uncomplicated, underlies all computing operations, and understanding it is crucial for tasks like converting binary to base-10 using Excess-32.