Question

Convert each of the following binary representations into its equivalent base ten representation: a. \(11.11\) b. \(100.0101\) c. \(0.1101\) d. \(1.0\) c. \(10.01\)

Short answer

a. 3.75, b. 4.3125, c. 0.8125, d. 1, e. 2.25

Step-by-step solution

Understanding Binary System

The binary system is base 2, which means each digit represents a power of 2. Starting from the left of the binary point, digits represent decreasing powers of 2. To convert binary to base ten, multiply each digit by its corresponding power of 2 and sum the results.

Convert 11.11 to Decimal

We break it down as follows: \(11.11_2 = (1 \times 2^1) + (1 \times 2^0) + (1 \times 2^{-1}) + (1 \times 2^{-2})\). Calculate each: \(1 \times 2^1 = 2\), \(1 \times 2^0 = 1\), \(1 \times 2^{-1} = 0.5\), \(1 \times 2^{-2} = 0.25\). Sum: \(2 + 1 + 0.5 + 0.25 = 3.75\). So \(11.11_2 = 3.75_{10}\).

Convert 100.0101 to Decimal

Break it down: \(100.0101_2 = (1 \times 2^2) + (0 \times 2^1) + (0 \times 2^0) + (0 \times 2^{-1}) + (1 \times 2^{-2}) + (0 \times 2^{-3}) + (1 \times 2^{-4})\). Calculate each: \(4\), \(0\), \(0\), \(0\), \(0.25\), \(0\), \(0.0625\). Sum: \(4.3125\). So \(100.0101_2 = 4.3125_{10}\).

Convert 0.1101 to Decimal

Break it down: \(0.1101_2 = (1 \times 2^{-1}) + (1 \times 2^{-2}) + (0 \times 2^{-3}) + (1 \times 2^{-4})\). Calculate: \(0.5\), \(0.25\), \(0\), \(0.0625\). Sum: \(0.8125\). So \(0.1101_2 = 0.8125_{10}\).

Convert 1.0 to Decimal

There's only one digit left of the binary point: \(1.0_2 = (1 \times 2^0) = 1\). So \(1.0_2 = 1_{10}\).

Convert 10.01 to Decimal

Break it down: \(10.01_2 = (1 \times 2^1) + (0 \times 2^0) + (0 \times 2^{-1}) + (1 \times 2^{-2})\). Calculate: \(2\), \(0\), \(0\), \(0.25\). Sum: \(2.25\). So \(10.01_2 = 2.25_{10}\).

Key concepts

Binary Numbers

Binary numbers are a key fundamental in computer science and digital electronics. They represent values using only two digits: 0 and 1. These are the most basic elements in computing. Just like how the decimal system (base 10) uses ten digits (0-9), the binary system uses only two digits. Each position in a binary number represents a power of two, unlike the decimal system where each position represents a power of ten. This simplicity makes binary computation efficient for digital devices, which internally operate using two states (often represented by on-off signals).
A binary number like 1011 represents a significant meaning beyond just 1s and 0s. To understand its actual value, converting it to decimal (base 10) can help.

Number Systems

Number systems are ways of writing numbers clearly and uniquely. They are very important in mathematics education for communication and computation. Different number systems are used in various fields and include:

• Decimal: Base 10, most commonly used in everyday life.
• Binary: Base 2, used by computers and digital systems.
• Hexadecimal: Base 16, often used in programming and computing.
• Octal: Base 8, also used in computing, though less common today.

Understanding different number systems widens our perspective on how numbers can be manipulated and helps us transition between systems.
The binary number system, in particular, is essential. It serves as the foundation of digital systems, as electronic devices operate efficiently in this numerical format due to their binary states.

Mathematics Education

Mathematics education aims to equip students with essential problem-solving skills. Among these skills is understanding how different number systems work. Learning to convert between number systems, such as binary to decimal, is critical.
Converting binary numbers to decimal involves using the place value of each digit. Students learn to multiply each digit by its corresponding power of two and then sum the results. Mastery of these skills helps in developing a deeper understanding of digital electronics and computing principles.
Incorporating lessons on binary conversion encourages logical thinking and strengthens computational skills which are vital in various academic and professional fields.

Place Value in Binary System

The concept of place value in the binary system is fundamental to understanding how binary numbers translate to decimals. Each binary digit, or bit, has a place value that is a power of two. The rightmost bit is the least significant bit (LSB) and starts with a power of 0. Starting from this bit moving left, each bit's place value increases by a power of two.

• For instance, in the binary number 1011, the leftmost bit 1 represents \(2^3 = 8\), and the rightmost bit 1 represents \(2^0 = 1\).
• Therefore, the binary number 1011 equals \( (1 \times 8) + (0 \times 4) + (1 \times 2) + (1 \times 1) = 11 \) in decimal.

Understanding this concept is crucial when performing binary to decimal conversions, enabling a clearer grasp of how binary representations work.