Introduction to binary numbers

Binary numbers serve as the foundation of all digital systems and are used extensively in computer science and technology.

Some of their key uses include:

• Data representation: Binary numbers are used to represent various types of data such as text, numbers, images, and multimedia in computers.
• Logic gates: Binary numbers are essential for designing and implementing logic gates, which are the building blocks of digital circuits. These circuits perform logical operations based on binary inputs and produce binary outputs. We will learn more about this during this lesson.
• Computer arithmetic: Binary numbers are used for performing arithmetic operations such as addition, subtraction, multiplication, and division in computers and digital calculators. We will be becoming computers and performing these calculations ourselves today.
• Memory addressing: Binary numbers are used to address memory locations in computer systems. Each memory location is assigned a unique binary address, allowing the computer to retrieve and store data efficiently.
• Communication: Binary numbers are used in communication systems to represent and transmit data over networks, including the internet. Data is encoded into binary format before transmission and decoded back into its original form upon reception.
• File storage: Binary numbers are used to encode and store files on storage devices such as hard drives, solid-state drives, and flash drives. Each file is represented as a sequence of binary digits.

Overall, binary numbers play a crucial role in enabling digital processing, storage, and communication of information in modern computing systems. Converting numbers to binary involves representing the number in terms of powers of 2. This will be explained in the student activities.

Teachers can use a classroom presentation to introduce the topic of binary numbers. Download the presentation as pptx.

Activity 1: Binary conversion

First, we will learn how to convert numbers from decimal to binary.

Converting numbers to binary involves representing the number in terms of powers of 2.

To convert a decimal number to binary, you need to divide the decimal number by 2. You then write down the remainder, which will be either 0 or 1. You continue dividing the result by 2 and writing down the remainders until you get a result of 1.

The binary number is the remainders written in reverse order.

For example, to convert the decimal number 278 to binary, you would divide 278 by 2, which gives you 139 with a remainder of 0.

You would then divide 139 by 2, which gives you 69 with a remainder of 1. You would continue this process until you get a result of 1. The remainders would be 0, 1, 1, 0, 1, 0, 0, 0, and 1. The binary equivalent of 278 is therefore 100010110.

To convert a binary number to decimal, list the binary digits in a row and write the corresponding powers of 2 beneath them, starting from the right with

2⁰, 2¹, 2², 2³, … 2ⁿ.

Multiply each binary digit by its corresponding power of 2 and add the results. For example: 1011 in binary is

1∙(2³) + 0∙(2²) + 1∙(2¹) + 1∙(2⁰) = 8 + 0 + 2 + 1 = 11.

Or written in a table:

More examples:

Activity 2: Worksheet with conversion problems

Students will complete a worksheet of conversion problems, converting decimals into binary and binary into decimals. There are short questions and open-ended research questions. This sheet will be differentiated for two levels (introductory and advanced).

You can download:
 

• Worksheet Binary Numbers (introductory level) as docx and pdf
• Answers to Binary Numbers (introductory level) as docx and pdf
• Worksheet Binary Numbers (advanced level) as docx and pdf
• Answers to Binary Numbers (advanced level) as docx and pdf

Activity 3: Binary game

The students play the “Binary game” to dive even deeper into the matter.