Question

Convert each of the following base 10 representations to its equivalent binary representation: a. 110 b. 99 c. 72 d. 81 e. 36

Short answer

a) 110: 1101110 b) 99: 1100011 c) 72: 1001000 d) 81: 1010001 e) 36: 100100

Step-by-step solution

Understanding the Problem

We need to convert decimal numbers (base 10) into binary numbers (base 2) for each given number. The binary system uses only two digits, 0 and 1.

Divide the Decimal Number by 2

We start by dividing the decimal number by 2 and write down the quotient and the remainder. The remainder will be either 0 or 1.

Record the Remainder

The remainder obtained after division represents the least significant digit in the binary equivalent. Keep a note of these remainders in order, from last to first.

Repeat the Division Process

Continue dividing the quotient obtained from the previous division by 2, writing down each new remainder, until the quotient becomes 0.

Arrange the Remainders

Once division ends, the binary number is formed by reading the remainders from last to first.

Convert Each Number

Let's convert each given decimal number: - **a) 110**: 110 ➔ 55 R0 ➔ 27 R1 ➔ 13 R1 ➔ 6 R1 ➔ 3 R0 ➔ 1 R1 ➔ 0 R1 ➔ Binary: **1101110** - **b) 99**: 99 ➔ 49 R1 ➔ 24 R1 ➔ 12 R0 ➔ 6 R0 ➔ 3 R0 ➔ 1 R1 ➔ 0 R1 ➔ Binary: **1100011** - **c) 72**: 72 ➔ 36 R0 ➔ 18 R0 ➔ 9 R0 ➔ 4 R1 ➔ 2 R0 ➔ 1 R0 ➔ 0 R1 ➔ Binary: **1001000** - **d) 81**: 81 ➔ 40 R1 ➔ 20 R0 ➔ 10 R0 ➔ 5 R0 ➔ 2 R1 ➔ 1 R0 ➔ 0 R1 ➔ Binary: **1010001** - **e) 36**: 36 ➔ 18 R0 ➔ 9 R0 ➔ 4 R1 ➔ 2 R0 ➔ 1 R0 ➔ 0 R1 ➔ Binary: **100100**

Key concepts

Decimal to Binary

When converting a decimal number, which is in base 10, to a binary number, base 2, we follow a straightforward process. Decimal numbers use digits from 0 to 9, while binary numbers use only 0 and 1. This transformation involves dividing the decimal number by 2 repeatedly.

• Start by dividing the decimal number by 2.
• Record the remainder (either 0 or 1).
• The remainder tells us the binary digit.
• Keep track of remainders from last to first.

Repeat this division process until the quotient is zero. The series of remainders, read in reverse order, presents the binary equivalent.
This method ensures a systematic approach, making it easy to convert without errors.

Number Systems

Number systems are the foundation of digital technologies. Each one uses a distinct set of symbols or digits. Our everyday numbers are decimal numbers that use a base of 10. In contrast, binary numbers use a base of 2.
In computing, these different bases are crucial:

• Base 10 (Decimal): It's the most familiar system, the one we learn first as children. It uses digits 0 to 9.
• Base 2 (Binary): It forms the basis of computer systems. Computers use binary to perform calculations and store data effectively. This system consists naturally of two symbols: 0 and 1.

By understanding these systems, you can grasp why computers use binary and how information is stored and processed in digital devices.

Binary Representation

Binary representation is central to computer science. It enables computers to encode information using the simplest means: a series of on (1) and off (0) states.

• Each digit in a binary number is a bit.
• In a binary system, the position of each bit signifies its value, increasing from right to left.
• Every position is a power of two, starting with 2⁰, 2¹, 2², etc.

For example, the binary number 110 represents:

• 2² (4) + 2¹ (2) + 2⁰ (0) = 6 in decimal.

This system allows computers to efficiently manage and manipulate data. Understanding binary representation is key to learning how computers interpret operations and execute tasks.

Computer Science Basics

Computer science basics often start with understanding binary systems. At its core, computer science is the study of algorithms and the structure of computers themselves, both hardware and software.
Binary numbers simplify the internal workings of computers. They help with ease of computation, logical operations, and reducing errors.

• Algorithms use binary to perform operations consistently.
• Information storage — such as text, sound, and images — is represented in binary.
• All programming languages ultimately communicate in binary to the machine.

By starting with binary, students build a foundation for further computer science studies, allowing deeper insights into programming, hardware engineering, and data science.