Chapter 1: Problem 14

In what numbering system are all numeric values written as sequences of 0 s and 1 s?

Short Answer

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Answer: The binary representation of the decimal number 13 is 1101.

Step by step solution

Introduction to the Binary System

The numbering system in which all numeric values are represented by sequences of 0s and 1s is called the binary system or base-2 system. In this system, each digit represents a power of 2 instead of 10, which is the case with the decimal system.

Representing Decimal Numbers in Binary System

To convert decimal numbers to binary, we need to find the highest power of 2 less than the decimal number and place a 1 in the corresponding position, representing the power of 2 present in the number. Then, repeat the process for the difference between the decimal number and the highest power of two.

Identifying the Highest Power of 2

Let's say we want to convert a decimal number, for example, 13, to binary. Start by identifying the highest power of 2 less than the decimal number. In this case, the highest power of 2 less than 13 is 2^3 or 8.

Place a 1 and Update the Decimal Number

Place a 1 in the position of the binary representation corresponding to that power of 2. For our example (13), we would have 1 in the 2^3 position as 1xxx, where x represents an unknown digit. Now, subtract the identified power of 2 (8) from the decimal number (13), and we get 5. Repeat the process for the difference (5).

Repeat Step 1 and 2 Until Difference is Zero

Identify the highest power of 2 less than the new decimal number (5), which is 2^2 or 4. Place a 1 in the corresponding position, and our representation becomes 110x. Subtract the new power of 2 (4) from the updated decimal number (5), and we get 1. Continue until the difference is zero. The highest power of 2 less than the remaining 1 is 2^0 or 1. Place a 1 in the corresponding position, and the binary representation becomes 1101. So, the decimal number 13 is represented as 1101 in the binary system.

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In what numbering system are all numeric values written as sequences of 0 s and 1 s?

Short Answer

Step by step solution

Introduction to the Binary System

Representing Decimal Numbers in Binary System

Identifying the Highest Power of 2

Place a 1 and Update the Decimal Number

Repeat Step 1 and 2 Until Difference is Zero

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