Question

Determine the decimal value of the following unsigned binary numbers: a. 11000 b. 110001 c. 1111111 d. 1000000000

Short answer

a. 24, b. 49, c. 127, d. 512.

Step-by-step solution

Understand Binary System

The binary number system is base 2, using only two digits: 0 and 1. Each position represents a power of 2, starting with 2^0 for the rightmost digit.

Analyze Binary Number 11000

For the binary number 11000, identify the place values: \[1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0 = 16 + 8 + 0 + 0 + 0 = 24.\]Thus, 11000 in binary is 24 in decimal.

Analyze Binary Number 110001

For the binary number 110001, identify the place values: \[1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 32 + 16 + 0 + 0 + 0 + 1 = 49.\]Thus, 110001 in binary is 49 in decimal.

Analyze Binary Number 1111111

For the binary number 1111111, identify the place values: \[1 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 64 + 32 + 16 + 8 + 4 + 2 + 1 = 127.\]Thus, 1111111 in binary is 127 in decimal.

Analyze Binary Number 1000000000

For the binary number 1000000000, identify the place value: \[1 \cdot 2^9 + 0 \cdot 2^8 + 0 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0 = 512.\]Thus, 1000000000 in binary is 512 in decimal.

Key concepts

Binary Number System

The binary number system is a fundamental concept in computer science and digital electronics. It is called a base 2 numeral system, using only two digits: 0 and 1. Each binary digit, or "bit," represents an increasing power of 2, starting from the rightmost bit. Just like the decimal system, which is base 10 and uses powers of 10, binary relies on powers of 2.

For example:

• The rightmost bit represents \(2^0\), which equals 1.
• The second bit from the right represents \(2^1\), which equals 2.
• The third bit represents \(2^2\), which equals 4.

As you move left, each position doubles the value. This structure makes binary efficient for computers, which operate using a series of on (1) and off (0) signals.

Decimal Conversion Process

Converting a binary number to a decimal number involves understanding the power of each bit position. Using the binary number system's structure, the decimal conversion process can be summarized in a few simple steps:

• Identify each bit in the binary number, starting from the right.
• Multiply each bit by its corresponding power of 2 (where the rightmost bit is \(2^0\), the next is \(2^1\), and so forth).
• Sum all the resulting values to get the equivalent decimal number.

Applying this process to the binary number 11000:

• Identify place values: \(1 \cdot 2^4\), \(1 \cdot 2^3\), \(0 \cdot 2^2\), \(0 \cdot 2^1\), \(0 \cdot 2^0\)
• Calculate: \(16 + 8 + 0 + 0 + 0 = 24\)

Thus, the binary number 11000 equals 24 in decimal.

Place Value in Binary

Understanding place value in binary is crucial for accurately performing binary to decimal conversions. Each bit in a binary number is placed according to its power of 2, and this determines its contribution to the overall value of the number.

Consider the binary number 1111111:

• The leftmost bit is \(2^6\), equating to 64.
• Subsequent bits decrease in power: \(2^5 = 32\), \(2^4 = 16\), and so on.
• Each place value signifies a specific contribution to the final sum based on whether it is a 1 (add that place value) or a 0 (ignore that place value).

Therefore, the number 1111111 can be broken down as:
\[1 \cdot 64 + 1 \cdot 32 + 1 \cdot 16 + 1 \cdot 8 + 1 \cdot 4 + 1 \cdot 2 + 1 \cdot 1 = 127\]
Properly understanding these place values within a binary number is key to converting correctly to decimal.