How To Convert Binary To Decimal - Computer Science

Summary

This educational video script offers a comprehensive guide on converting binary numbers to decimal numbers. It explains the process through various examples, highlighting the significance of powers of two and how to calculate them. The script also delves into converting fractional binary numbers, illustrating the use of negative exponents and their reciprocal values. By the end, viewers are equipped with the knowledge to perform binary to decimal conversions with ease.

Takeaways

• 📚 Converting binary numbers to decimal involves multiplying each binary digit by 2 raised to the power of its position, starting from 0 on the right.
• 🔢 The first digit on the right in a binary number represents 2 to the power of 0, the next represents 2 to the power of 1, and so on.
• ⚠️ Zero values in the binary number can be ignored as they contribute nothing to the sum when multiplied by any power of 2.
• 👉 Focus on the '1's in the binary number as they are the only digits that affect the decimal outcome.
• 💡 Powers of 2 increase exponentially: 2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16, and so on.
• 🔄 To convert a binary number to decimal, sum the products of each binary digit and its corresponding power of 2.
• 📉 For binary numbers with a fractional part, use negative exponents for digits to the right of the decimal point.
• 🔼 Negative exponents in binary to decimal conversion represent fractions, such as 2^-1 = 0.5, 2^-2 = 0.25, and so forth.
• 📝 Practice is key to mastering binary to decimal conversion, as demonstrated with multiple examples in the script.
• 📖 The script provides a clear method for converting binary numbers, including those with fractional parts, into decimal numbers.

Q & A

What is the process of converting a binary number to a decimal number?

To convert a binary number to a decimal number, you multiply each digit by 2 raised to the power of its position, starting from the right with 0 and summing up the results.

How do you handle zeros in binary to decimal conversion?

Zeros in a binary number can be ignored during conversion because multiplying by zero results in zero, which does not affect the sum.

What is the value of 2 to the power of zero in binary to decimal conversion?

2 to the power of zero is always 1, as any number to the zero power equals 1.

Can you provide an example of converting a binary number to decimal using the script's method?

Sure, for the binary number 1010, the conversion would be 1*(2^3) + 0*(2^2) + 1*(2^1) + 0*(2^0), which equals 8 + 0 + 2 + 0, resulting in the decimal number 10.

What is the significance of the position of digits in binary to decimal conversion?

The position of a digit in a binary number determines the exponent to which 2 is raised when converting to decimal. The rightmost position is 0, and it increases as you move left.

How do you convert a binary number with a fractional part to a decimal?

For binary fractions, you use negative exponents for digits to the right of the decimal point, similar to whole numbers but with positions starting from -1 and decreasing.

What is the decimal equivalent of the binary number 1100.101?

The binary number 1100.101 converts to decimal as 12.625, calculated by adding 1*(2^3) + 1*(2^2) + 0*(2^1) + 0*(2^0) + 1*(2^-1) + 0*(2^-2) + 1*(2^-3), which equals 8 + 4 + 0.5 + 0.125.

What is the pattern for converting negative exponents in binary to decimal?

Negative exponents in binary to decimal conversion follow the pattern where 2 to the negative n power is 1 over 2 to the n power.

Can you provide a tip for quickly converting binary numbers to decimal?

Knowing the powers of 2 up to at least 2^8 (256) can greatly speed up the conversion process, as you can directly associate each binary digit with its decimal equivalent.

What is the decimal result of the binary number 1101.11 after following the script's conversion method?

Following the script's method, the binary number 1101.11 converts to decimal as 13.75, calculated by 1*(2^3) + 1*(2^2) + 0*(2^1) + 1*(2^0) + 1*(2^-1) + 1*(2^-2), which equals 8 + 4 + 0 + 1 + 0.5 + 0.25.

Outlines

00:00

🔢 Converting Binary to Decimal

This paragraph introduces the process of converting binary numbers to decimal numbers. It explains the concept using the example of the binary number 1 0 1 0. The method involves multiplying each digit by 2 raised to the power of its position, starting from 0 on the right. The paragraph emphasizes that zeros can be ignored, and only ones need to be considered. It concludes with the calculation that 1 0 1 0 in binary equals 10 in decimal, demonstrating the addition of the values obtained from the positions of the ones.

05:00

📚 Binary to Decimal Conversion Examples

The paragraph presents additional examples to solidify the understanding of converting binary to decimal numbers. It discusses two examples: 1 0 0 1 0 and 1 1 0 1 1, guiding the viewer through the process of focusing on the positions with ones and calculating their values based on the powers of two. The paragraph also provides a quick reference for the values of 2 raised to various powers, which are essential for the conversion process. It concludes with the conversion of 1 1 0 1 1 to decimal, which results in 27.

10:02

📉 Handling Fractional Binary Numbers

This paragraph extends the conversion process to include fractional binary numbers. It explains how to handle numbers to the right of the decimal point in binary by using negative exponents. The example given is the binary number 1.100101, which is broken down into its individual components and converted to decimal, resulting in 12.625. The paragraph also discusses the pattern of negative exponents and how they relate to their positive counterparts, providing a method to quickly determine the decimal values of fractional binary numbers.

🔍 More Practice with Binary Fractions

The final paragraph provides more practice with fractional binary numbers, converting the example 1 0 0 1.1 into its decimal equivalent, 19.375. It reinforces the concept of using negative exponents for numbers after the binary decimal point and demonstrates the process of adding up the values obtained from each position, including those with negative exponents. The paragraph concludes with a summary of the conversion process and the final result.

Keywords

💡Binary Number

A binary number is a part of the base-2 numeral system, which means it consists only of two digits: 0 and 1. In the context of the video, binary numbers are the starting point for conversion into decimal numbers. The video explains how each digit in a binary number represents a power of 2, starting from the rightmost digit which is 2 to the power of 0. For instance, the binary number '1010' is broken down into '1*2^3 + 0*2^2 + 1*2^1 + 0*2^0', which is then calculated to convert to its decimal equivalent.

💡Decimal Number

A decimal number is a part of the base-10 numeral system, which includes the digits from 0 to 9. The video's main theme revolves around converting binary numbers into decimal numbers. Decimal numbers are the standard numerical system used in most of the world for general counting and arithmetic. The video provides examples and methods to convert binary digits to their decimal equivalents, such as converting the binary '1010' to the decimal '10'.

💡Power of 2

In the video, the concept of 'power of 2' is crucial for converting binary to decimal numbers. Each position in a binary number corresponds to a power of 2, with the rightmost position being 2 to the power of 0 (which is 1), the next position to the left being 2 to the power of 1 (which is 2), and so on. This pattern is used to determine the value of each binary digit in the conversion process. For example, in the number '1010', the '1' in the third position from the right represents '1*2^2', which equals 4.

💡Exponent

An exponent in mathematics is used to denote the power to which a number is raised. In the video, exponents are used to express the position of each digit in a binary number. The video script mentions various powers of 2, such as 2 to the third power (which is 8) and 2 to the fourth power (which is 16), to help viewers understand how to calculate the decimal value of a binary number.

💡Zero Power

The video script clarifies that 'anything raised to the zero power is one'. This mathematical rule is applied when converting binary numbers to decimal, especially when there is a '0' in the binary number. For example, '0*2^0' equals '0*1', which is 0, and thus does not contribute to the final decimal value.

💡Fractional Binary Number

A fractional binary number includes a decimal point, similar to fractional decimal numbers, but with binary digits before and after the point. The video explains how to convert these types of numbers into decimal form by applying the same principles but using negative exponents for digits following the binary decimal point. For instance, the binary '.110' would be converted by considering '2 to the -1' and '2 to the -2'.

💡Negative Exponent

Negative exponents are introduced in the video when discussing fractional binary numbers. A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. For example, '2 to the -1' is '1/2'. The video uses negative exponents to convert the fractional part of binary numbers to decimal, such as in the conversion of '.110' to '0.110' in decimal.

💡Conversion Process

The conversion process is the methodical approach taken in the video to change binary numbers into their decimal equivalents. It involves assigning powers of 2 to each binary digit, multiplying by the digit's value (0 or 1), and then summing these products to get the decimal number. The video provides a step-by-step guide to this process, using several examples to demonstrate clarity and understanding.

💡Example Problems

Throughout the video script, 'example problems' are used to illustrate the conversion process from binary to decimal numbers. These examples serve as practical demonstrations of the concepts and techniques discussed. They allow viewers to follow along and apply the methods to different binary numbers, reinforcing their understanding of the conversion process.

💡Calculator

The video mentions using a calculator as a tool to assist in performing the arithmetic necessary for converting binary to decimal numbers, especially when dealing with fractional parts. It suggests that viewers can use a calculator to add up the values obtained from the conversion process, such as adding '1/2' and '1/8' to get '0.625'.

Highlights

Introduction to converting binary numbers to decimal numbers with examples.

Binary number system explained as a base 2 system with digits 0 or 1.

Conversion process involves multiplying each digit by 2 raised to the power of its position.

Example 1: Conversion of binary 0110 to decimal 10.

Explanation of how to ignore zeros in the binary number during conversion.

Example 2: Conversion of binary 10010 to decimal 18.

Powers of 2 from 2^0 to 2^8 and their decimal equivalents provided for quick reference.

Example 3: Conversion of binary 11011 to decimal 27.

Instruction to practice converting two example binary numbers into decimal.

Example 4: Conversion of binary 1010110 to decimal 99.

Example 5: Conversion of binary 01011011 to decimal 86.

Introduction to converting fractional binary numbers to decimal numbers.

Conversion of fractional binary 1100.101 to decimal 12.625.

Explanation of negative exponents in binary to decimal conversion.

Example 6: Conversion of fractional binary 1001.011 to decimal 19.375.

Summary of the process for converting binary to decimal with fractional numbers.

Transcripts

in this video we're going to talk about

how to convert binary numbers into

decimal numbers and we're going to go

through plenty of examples so you can

Master this topic let's start with this

example 1 0 1 0.

how can we convert this particular

binary number which is a base 2 system

the digits can only be zero or one how

can we convert it into a decimal number

now the first value

is 2 to the zero power

and the second one is going to be this

number times

2 to the first power

and the third value going from right to

left is zero times 2 squared

and this one is going to be 1 times 2 to

the third power

so 1 times 2 to the third power

that is equal to 2 to the third power

and then if we multiply 2 squared by

zero that's going to give us 0. and

we're going to add these values

and then it's going to be 1 times 2 to

the first power

which is just 2 to the first power

and then 0 times 2 to the zero is zero

so everywhere you see a zero you could

ignore it

so we only need to focus on the numbers

that contain a 1.

now let's add up these numbers so what

is 2 to the third power

2 to the third power is 2 times 2 times

2 which is 8.

2 to the first power is simply 2. so

we're going to get 8 plus 2 and so the

answer is 10.

so the binary number one zero one zero

has a value of 10.

now let's try another example

one zero

zero one zero

so if you want to try it feel free to

pause the video

go ahead and write this number down in

convert the binary number into a decimal

number

so let's start from the right side this

is going to be 2 to the 0 power

the next number is 2 to the first power

it's zero times two zero by the way

and then the next number is going to be

0 times 2 squared and then 0 times 2 to

the third and then 1 times 2 to the

fourth

now we don't need to worry about the

zeros so let's ignore those numbers

focus on the numbers that have a one

because those numbers will contribute to

the final answer

so this is going to be 1 times 2 to the

fourth power which is simply 2 to the

fourth power

and then this is 1 times 2 to the first

power

which is just 2 to the first power

so what is 2 to the fourth power

so 2 times 2 times 2 times 2. 2 times 2

is 4. and 4 times 4 is 16. so 2 to the

fourth power is 16. plus 2

and so this will give us 18.

and so it's not very difficult to

convert a binary number into a decimal

number as long as you follow the process

you're going to get this right

now here's some information that you may

want to know

choose the zero power is one in fact

anything raised to the zero power is one

two to the first power is two

two squared is four

two to the third is eight

two to the fourth is 16. and then 2 to

the fifth power that's 32.

2 to the sixth is 64.

2 to the seventh is uh that's 128

and then 2 to the eight is 128 times 2

or 256.

and so if you know these values you can

quickly perform the operations that's

necessary when converting a binary

number into a decimal number

so let's try another example

1 1 0 1 1. so go ahead and convert this

number into a decimal number

so this is going to be 2 to the zero

times one

and then 2 to the first Power times one

two squared times zero two to the third

times one and then two to the fourth

times one

and just like before we are going to

ignore the zero values

and focus on

the numbers that have a binary value of

one

so this is going to be 2 to the fourth

and then plus 2 to the third power

and then plus 2 to the first

and then plus 2 to the 0 power

So based on that table or rather the

information on the last page we set that

2 to the fourth power is 16. 2 to the

third is eight two to the first is two

and two to the zero is one so now let's

add

16 plus 8 is 24.

and then two plus one is three

so the final answer is 27.

and so that's how we can convert

a binary number into a decimal number

now for the sake of practice go ahead

and try these two example problems

so convert these two numbers into

decimal numbers

feel free to pause the video and work on

these problems

so let's start with the first example on

the left

so I like to space out the numbers

it makes it easier

so this number is going to be associated

with two to the zero and then two to the

first

and then we're going to follow the same

process

for all of the other examples

and now let's focus on the numbers

that contain A1

so what we have is 2 to the sixth

Plus

2 to the fifth power

Plus

2 to the first power

and then plus 2 to the zero power

so 2 to the sixth power we said that's

64. 2 to the fifth is 32 and then plus 2

plus 1.

now 64 plus 32 that's 96 2 plus 1 is 3

and so the final answer for the first

example is 99.

and so that's it for this one

now let's move on to the next one

so let's rewrite the number

so we have 0 1 0 1 0 1 1 0.

now let's write these numbers

at the top

and once again let's focus on the

numbers

that contain A1

now let's add up these numbers

so this is going to be 2 to the sixth

Plus

2 to the fourth power

plus 2 squared

and then Plus

2 to the first power

so 2 to the 6 is 64.

2 to the fourth is 16 2 squared is four

and two to the first is 2.

so 64 plus 16. that's going to be 80 and

then 4 plus 2 is 6.

so the final answer for this example

is 86.

now what if you're given a fractional

binary number

how can you convert that into a decimal

number for instance

let's say we have the number one

one

zero zero with a decimal point and then

one zero one

how can we convert that into

a decimal number

so the first number that's to the left

of the decimal point that's two to the

zero as always and then two to the first

two squared and the last one is

associated with two to the Third

what about the numbers to the right of

the decimal

well if we follow the pattern of

exponents that we see here

the number that's less than zero is

negative one so this has to be

associated with

2 to the minus 1 and then to the minus

two and then two to the minus 3.

just like before we're going to focus on

the values

that bear a 1.

and so we're going to add 2 to the third

plus two squared

plus 2 to the minus 1 plus 2 to the

minus 3. now 2 to the third is eight and

two squared is four

but now what is 2 to the minus 1.

we know two to the first power is two so

two to the negative first power is one

over two

now two to the third is eight so two to

the negative third is going to be one

over eight

and so a plus 4 is 12.

and then we have one half plus one over

eight

one over two if you type that in your

calculator that's point five

and one over eight

one divided by eight

is point one two five

so adding up these three numbers

will give us this answer

12.625

and so that's how we can convert a

fractional binary number into a decimal

number

now earlier we said that 2 to the fourth

is 16.

so 2 to the negative 4 is going to be 1

over 16.

two to the fifth is 32

2 to the negative 5 is 1 over 32.

so notice the pattern that you see here

2 to the 6th is 64.

2 to the negative 6 is 1 over 64. and so

forth

so when dealing with fractional

binary numbers

whenever you have negative exponents

just consider the ones with a positive

exponent so if 2 to the 7 is 1 28 then

you know 2 to the negative 7 is just 1

over 128.

today is another example that you could

try

one zero zero one one point zero one one

go ahead and convert that into

a decimal number

so let's begin by putting the

appropriate values

next to each number

so on the left is going to be 2 to the

minus 1.

2 to the minus two

and two to the minus 3.

so let's focus on

only the ones that we see

so it's going to be 2 to the fourth

power plus 2 to the first power

plus 2 to the zero power

and then plus 2 to the minus two

plus two to the minus 3.

2 to the 14 I mean not 14 but 2 to the

fourth power is 16. and 2 to the first

power is 2 2 to the 0 power is one

and then 2 to the negative two if two

squared is four two to the negative two

is one over four

and if two to the third power is eight

two to the negative third power is one

over eight

so we have 16 plus two plus one

so that's nineteen and then 1 4 is 0.25

and 1 over 8 we said that was 0.125

so adding these three numbers

is going to give us the final value of

19.375

.