Binary Numbers

In simple terms,  binary numbers are a way of counting using only two digits: 0 and 1. Think of our usual counting system, the decimal syste...

In simple terms, binary numbers are a way of counting using only two digits: 0 and 1.

Think of our usual counting system, the decimal system, which uses ten digits (0-9). Each position in a decimal number represents a power of 10 (like ones, tens, hundreds).

Binary numbers work similarly, but each position represents a power of 2 (like ones, twos, fours, eights). For example, the binary number 101 represents (1 x 2²) + (0 x 2¹) + (1 x 2⁰) = 4 + 0 + 1 = 5 in decimal.

Binary numbers are fundamental because they are the language that computers understand. Everything in a computer, from the letters you type to the images you see, is ultimately represented as a sequence of 0s and 1s.

Would you like to explore this topic further and see how we can break it down step by step? Learning plan:

1. Understanding the Basics of Binary Numbers
2. Converting Binary to Decimal
3. Converting Decimal to Binary
4. Applications of Binary Numbers

Let's start with Understanding the Basics of Binary Numbers.

First, it's helpful to think about what a number system is. It's essentially a way of representing numbers using a set of symbols and rules. The most common one for us is the decimal system, also known as base-10. We use ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), and the position of each digit in a number determines its value.

Imagine the number 325. The '5' is in the ones place (100), the '2' is in the tens place (101), and the '3' is in the hundreds place (102). So, $325=(3\times 100)+(2\times 10)+(5\times 1)$.

Now, the binary system, or base-2, works in a similar way, but it only uses two symbols: 0 and 1. Instead of powers of 10, each position in a binary number represents a power of 2.

Binary refers to a number system with a base of 2. It only uses two digits (0 and 1). Each position in a binary number represents a power of 2 (ones, twos, fours, eights, etc.).

Decimal refers to the number system we use every day. It has a base of 10, meaning it uses ten different digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Each position in a decimal number represents a power of 10 (ones, tens, hundreds, thousands, etc.).

Let's visualize this:

Decimal (Base-10):

...	$10^3$ (Thousands)	$10^2$ (Hundreds)	$10^1$ (Tens)	$10^0$ (Ones)
	1000	100	10	1

Binary (Base-2):

...	$2^3$ (Eights)	$2^2$ (Fours)	$2^1$ (Twos)	$2^0$ (Ones)
	8	4	2	1

So, a binary number like 10112​ (the subscript '2' indicates it's a binary number) would be:

$(1\times 2^3)+(0\times 2^2)+(1\times 2^1)+(1\times 2^0)=(1\times 8)+(0\times 4)+(1\times 2)+(1\times 1)=8+0+2+1=11_{10}$ (the subscript '10' indicates it's a decimal number).

Converting Binary to Decimal

Now that we have a basic understanding of what binary numbers are, let's learn how to convert them into our familiar decimal numbers. This is like learning to translate from the language of computers to our everyday language of numbers.

The method is quite straightforward. For each digit (bit) in the binary number, you multiply it by the corresponding power of 2 based on its position, starting from the rightmost digit (which is the 20 or the 'ones' place). Then, you simply add up all these products.

To convert a binary number to decimal, follow these steps:

Steps:

1. Write down the binary number.

2. Assign powers of 2 to each bit, starting from the rightmost bit (which is the least significant bit, or LSB). The rightmost bit is assigned $2^0$, the next bit is assigned $2^1$, and so on.

3. Multiply each bit by its corresponding power of 2.

4. Sum the results.

Example 1: Convert Binary 1101 to Decimal

1. Write the binary number: 1101

2. Assign powers of 2:

   • $1\times 2^3=1\times 8=8$

   • $1\times 2^2=1\times 4=4$

   • $0\times 2^1=0\times 2=0$

   • $1\times 2^0=1\times 1=1$

3. Add them up: $8+4+0+1=13$

1101 in binary = 13 in decimal.

Example 2: Convert Binary 10101 to Decimal

1. Write the binary number: 10101

2. Assign powers of 2:

   • $1\times 2^4=1\times 16=16$

   • $0\times 2^3=0\times 8=0$

   • $1\times 2^2=1\times 4=4$

   • $0\times 2^1=0\times 2=0$

   • $1\times 2^0=1\times 1=1$

3. Add them up: $16+0+4+0+1=21$

10101 in binary = 21 in decimal.

To convert a binary number to decimal, multiply each bit by its corresponding power of 2 and sum the results.

Let's take the binary number 11012​ as an example:

1. Identify the position of each bit:

   Binary:  1   1   0   1
   Position: 3   2   1   0  (from right, starting at 0)
   

2. Determine the power of 2 for each position:

   • Position 0: $2^0=1$
   • Position 1: $2^1=2$
   • Position 2: $2^2=4$
   • Position 3: $2^3=8$

3. Multiply each bit by its corresponding power of 2:

   • $1\times 2^3=1\times 8=8$
   • $1\times 2^2=1\times 4=4$
   • $0\times 2^1=0\times 2=0$
   • $1\times 2^0=1\times 1=1$

4. Add the results:

   $8+4+0+1=13$

So, the binary number 11012​ is equal to 1310​ in decimal.

To help you visualize this, imagine each '1' in the binary number as a switch that's turned 'on' for that power of 2, and each '0' as a switch that's 'off'. We then sum up the values of the 'on' switches.

 Converting Decimal to Binary

Now I will teach you, how to convert a decimal number (our everyday numbers) into its binary equivalent. This is like learning to speak the computer's language!

The most common method for converting decimal to binary is by repeatedly dividing the decimal number by 2 and keeping track of the remainders. The binary number is then formed by reading the remainders in reverse order (from the last remainder to the first).

To convert a decimal number to binary, follow these steps:

Steps:

1. Divide the decimal number by 2.

2. Record the remainder (it will be either 0 or 1).

3. Divide the quotient (the result of the division) by 2 again.

4. Repeat the process until the quotient becomes 0.

5. The binary number is obtained by reading the remainders from bottom to top.

Example 1: Convert Decimal 13 to Binary

1. 13 ÷ 2 = 6, remainder = 1

2. 6 ÷ 2 = 3, remainder = 0

3. 3 ÷ 2 = 1, remainder = 1

4. 1 ÷ 2 = 0, remainder = 1 (quotient is now 0, so stop)

Reading the remainders from bottom to top: 13 in decimal = 1101 in binary.

Example 2: Convert Decimal 25 to Binary

1. 25 ÷ 2 = 12, remainder = 1

2. 12 ÷ 2 = 6, remainder = 0

3. 6 ÷ 2 = 3, remainder = 0

4. 3 ÷ 2 = 1, remainder = 1

5. 1 ÷ 2 = 0, remainder = 1 (quotient is now 0, so stop)

Reading the remainders from bottom to top: 25 in decimal = 11001 in binary.

To convert any decimal number to binary, repeatedly divide by 2, recording the remainders, and then read the remainders from bottom to top.

Let's take the decimal number 2510​ as an example:

1. Divide 25 by 2: $25÷2=12$ with a remainder of 1

2. Divide the quotient (12) by 2: $12÷2=6$ with a remainder of 0

3. Divide the new quotient (6) by 2: $6÷2=3$ with a remainder of 0

4. Divide the quotient (3) by 2: $3÷2=1$ with a remainder of 1

5. Divide the quotient (1) by 2: $1÷2=0$ with a remainder of 1

We stop when the quotient becomes 0. Now, to get the binary equivalent, we read the remainders from bottom to top (in reverse order of how we obtained them): 11001.

So, 2510​ is equal to 110012​ in binary.

Think of it like this: we're trying to see how many times each power of 2 (1, 2, 4, 8, 16, etc.) fits into our decimal number. The remainders tell us whether that power of 2 is "on" (remainder 1) or "off" (remainder 0) in the binary representation.

Applications of Binary Numbers

Now we arrive at the final part of our learning plan: Applications of Binary Numbers. This is where we see why all this talk about 0s and 1s is so important in the real world, especially in the technology we use every day.

The most significant application of binary numbers is in computers and digital electronics. Here's why they are so crucial:

• Simplicity for Electronics: At the most fundamental level, computers work with electricity. It's much easier for an electronic circuit to distinguish between two states – on (represented by 1) and off (represented by 0) – than to try and differentiate between ten different voltage levels for the decimal system. Think of a light switch: it's either on or off, nothing in between. This on/off state is the essence of binary.

• Representing Information: These 0s and 1s, called bits (short for binary digits), can be combined in countless ways to represent all sorts of information:

  • Numbers: As we've already seen, any decimal number can be represented in binary.
  • Text: Each letter, symbol, and punctuation mark is assigned a unique binary code. One common system is called ASCII (American Standard Code for Information Interchange). For example, the letter 'A' is represented by the binary code 01000001.
  • Instructions: The programs that tell your computer what to do are also written in binary code. These instructions tell the processor which operations to perform.
  • Images: Images are made up of pixels, and each pixel's color and brightness can be represented by a binary code.
  • Sound: Sound waves are sampled and converted into numerical data, which is then stored and processed in binary.
  • Memory Addresses: Binary numbers are used to identify the specific locations in a computer's memory where data is stored.

• Logic Gates: The basic building blocks of computer circuits, called logic gates (like AND, OR, NOT gates), operate based on binary inputs (0s and 1s) to produce binary outputs. These gates perform the logical operations that allow computers to make decisions and process information.

Think about all the digital devices you use: your computer, smartphone, tablet, smart TV, digital watch, etc. They all rely on binary numbers to function. When you type a message, browse the internet, watch a video, or play a game, your device is constantly processing information encoded in binary.

It's like the entire digital world speaks in a language of just two words: 0 and 1!