Question

Binary presentation of the decimal number 141 is (a) 10110001 (b) 11110001 (c) 10001111 (d) 10001101

Short answer

The binary representation of the decimal number 141 is (d) 10001101.

Step-by-step solution

Starting the division process

Start by dividing the decimal number 141 by 2. The quotient is 70 with a remainder of 1.

Continue the division process

Next, divide the quotient 70 by 2. The new quotient is 35 with a remainder of 0. Continue this process of dividing the previous quotient by 2 and writing down the remainder. The subsequent steps, when followed, produce the following quotients and remainders: \n- 35 by 2 gives quotient 17 with remainder 1; \n- 17 by 2 gives quotient 8 with remainder 1; \n- 8 by 2 gives quotient 4 with remainder 0; \n- 4 by 2 gives quotient 2 with remainder 0; \n- 2 by 2 gives quotient 1 with remainder 0; \n- Lastly, divide 1 by 2 to get quotient 0 with remainder 1. Stop when the quotient is 0.

Extracting the binary representation

Read the remainders in reverse order (starting from the last one you found to the first), to form the binary representation of the given decimal number. In this case, the binary representation of 141 is 10001101.

Identifying the correct answer

Compare the computed value to the options provided in the problem. Thus, the binary representation of the decimal number 141 corresponds to option (d) 10001101.

Key concepts

Binary Representation

Binary representation is a system used to express numbers using only two symbols: 0 and 1. These digits are known as "bits," and they form the foundation of the binary numeral system, which is commonly used in computing and digital electronics. In binary representation, the value of each position is determined by powers of 2, starting from the rightmost bit. For instance, the binary number \(1011\) can be evaluated as follows:

• The far right bit, 1, represents \(2^0 = 1\)
• The next bit, 1, represents \(2^1 = 2\)
• The next bit, 0, represents \(2^2 = 0\)
• The leftmost bit, 1, represents \(2^3 = 8\)

When added together, \(1011\) is equal to 11 in decimal format. This form of representation is essential for computers, which operate using digital signals interpreted in binary form.
Understanding how bits work helps in grasping how numbers are stored and manipulated electronically.

Decimal to Binary Conversion

Converting a decimal number to binary involves transforming a number from the base-10 system, which uses digits 0 through 9, into the base-2 system of binary. The base-10 system is how numbers are naturally expressed in daily use. To convert a decimal number to binary, you can consecutively divide the number by 2 and keep track of the remainders. These remainders will help you form the final binary number.
The method gives you a way to translate numbers into binary without immediately needing any advanced mathematics. Understanding this conversion is critical in computer science and digital electronics, as binary is the native language of computers. A step-by-step approach ensures accuracy and helps internalize the conversion process.
For example, the decimal number 141 is converted to binary by dividing by 2, tracking remainders, and rewriting these from bottom to top. This process eventually yields the binary equivalent \(10001101\) when reading the remainders in reverse order.

Quotient and Remainder Method

The quotient and remainder method is a systematic approach used in decimal to binary conversion. It involves a series of divisions where each quotient is divided by 2, recording the remainder each time. The process begins by identifying a decimal number you wish to convert.

• Divide the number by 2.
• Note the quotient and remainders at each step.
• Continue dividing the quotient until it reaches zero.

Each remainder represents a bit in the binary number, starting from the least significant bit to the most significant. Let's simplify:
For 141, dividing by 2 gives a quotient of 70 and a remainder of 1. Continuing this way, collect remainders. Stopping once the quotient reaches 0 gives us the binary representation when read in reverse: \(10001101\).
This method is simple yet robust, helping students understand binary conversion rules and how to encode or decode numbers in digital logic.