Convert The Given Number To Decimal Form By Expanding In Powers And By Using The Calculator Shortcut.Given: $111100001_{\text{two}}$1. Expand $111100001_{\text{two}}$ In Powers: - $111100001_{\text{two}} = 1 \times 2^8 + 1

Introduction

In the world of mathematics, numbers can be represented in various forms, including binary, decimal, and hexadecimal. Binary numbers are a series of 0s and 1s that are used in computer programming and electronics. In this article, we will explore how to convert a binary number to its decimal equivalent using two methods: expanding in powers and using a calculator shortcut.

Method 1: Expanding in Powers

To convert a binary number to decimal, we can expand it in powers of 2. This method involves multiplying each digit of the binary number by the corresponding power of 2 and then summing the results.

Example: Converting $111100001_{\text{two}}$ to Decimal

Let's take the binary number $111100001_{\text{two}}$ as an example. To convert it to decimal, we can expand it in powers of 2 as follows:

$111100001_{\text{two}} = 1 \times 2^8 + 1 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$

Now, let's calculate the value of each term:

• $1 \times 2^8 = 256$
• $1 \times 2^7 = 128$
• $1 \times 2^6 = 64$
• $0 \times 2^5 = 0$
• $0 \times 2^4 = 0$
• $0 \times 2^3 = 0$
• $0 \times 2^2 = 0$
• $0 \times 2^1 = 0$
• $1 \times 2^0 = 1$

Now, let's sum the results:

$256 + 128 + 64 + 0 + 0 + 0 + 0 + 0 + 1 = 449$

Therefore, the decimal equivalent of $111100001_{\text{two}}$ is 449.

Method 2: Using a Calculator Shortcut

Another way to convert a binary number to decimal is to use a calculator shortcut. Most calculators have a built-in function to convert binary numbers to decimal. To use this function, follow these steps:

1. Enter the binary number into the calculator.
2. Press the "BIN" or "DEC" button to switch between binary and decimal modes.
3. The calculator will display the decimal equivalent of the binary number.

Example: Converting $111100001_{\text{two}}$ to Decimal using a Calculator

Let's take the binary number $111100001_{\text{two}}$ as an example. To convert it to decimal using a calculator, follow these steps:

1. Enter the binary number $111100001$ into the calculator.
2. Press the "BIN" or "DEC" button to switch between binary and decimal modes.
3. The calculator will display the decimal equivalent of the binary number, which is 449.

Conclusion

In this article, we explored two methods to convert a binary number to its decimal equivalent: expanding in powers and using a calculator shortcut. The first method involves multiplying each digit of the binary number by the corresponding power of 2 and then summing the results. The second method uses a calculator shortcut to convert the binary number to decimal. Both methods are useful for converting binary numbers to decimal, and the choice of method depends on the specific situation and the tools available.

Common Binary to Decimal Conversions

Here are some common binary to decimal conversions:

• $1010_{\text{two}} = 10_{\text{ten}}$
• $1101_{\text{two}} = 13_{\text{ten}}$
• $1001_{\text{two}} = 9_{\text{ten}}$
• $1111_{\text{two}} = 15_{\text{ten}}$

Practice Problems

Here are some practice problems to help you convert binary numbers to decimal:

• Convert $101101_{\text{two}}$ to decimal.
• Convert $110011_{\text{two}}$ to decimal.
• Convert $100101_{\text{two}}$ to decimal.
• Convert $111101_{\text{two}}$ to decimal.

Answer Key

Here are the answers to the practice problems:

• $101101_{\text{two}} = 149_{\text{ten}}$
• $110011_{\text{two}} = 183_{\text{ten}}$
• $100101_{\text{two}} = 121_{\text{ten}}$
• $111101_{\text{two}} = 229_{\text{ten}}$

Conclusion

Introduction

In our previous article, we explored two methods to convert binary numbers to decimal: expanding in powers and using a calculator shortcut. In this article, we will answer some frequently asked questions about binary to decimal conversion.

Q: What is a binary number?

A: A binary number is a series of 0s and 1s that are used in computer programming and electronics. Binary numbers are the base 2 number system, where each digit can have a value of either 0 or 1.

Q: How do I convert a binary number to decimal?

A: There are two methods to convert a binary number to decimal: expanding in powers and using a calculator shortcut. Expanding in powers involves multiplying each digit of the binary number by the corresponding power of 2 and then summing the results. Using a calculator shortcut involves entering the binary number into the calculator and pressing the "BIN" or "DEC" button to switch between binary and decimal modes.

Q: What is the difference between binary and decimal numbers?

A: Binary numbers are a series of 0s and 1s, while decimal numbers are a series of digits from 0 to 9. Binary numbers are used in computer programming and electronics, while decimal numbers are used in everyday life.

Q: Can I convert a decimal number to binary?

A: Yes, you can convert a decimal number to binary. To do this, you can use a calculator or a conversion chart. You can also use the "BIN" or "DEC" button on a calculator to switch between binary and decimal modes.

Q: How do I convert a binary number to hexadecimal?

A: To convert a binary number to hexadecimal, you can use a calculator or a conversion chart. You can also use the "BIN" or "DEC" button on a calculator to switch between binary and decimal modes, and then use the "DEC" or "HEX" button to switch between decimal and hexadecimal modes.

Q: What is the significance of binary numbers in computer science?

A: Binary numbers are the foundation of computer science. They are used to represent data and instructions in computers, and are the basis for all programming languages.

Q: Can I use a calculator to convert a binary number to decimal?

A: Yes, you can use a calculator to convert a binary number to decimal. Most calculators have a built-in function to convert binary numbers to decimal. To use this function, follow these steps:

1. Enter the binary number into the calculator.
2. Press the "BIN" or "DEC" button to switch between binary and decimal modes.
3. The calculator will display the decimal equivalent of the binary number.

Q: How do I convert a binary number to decimal using a calculator shortcut?

A: To convert a binary number to decimal using a calculator shortcut, follow these steps:

1. Enter the binary number into the calculator.
2. Press the "BIN" or "DEC" button to switch between binary and decimal modes.
3. The calculator will display the decimal equivalent of the binary number.

Q: What are some common binary to decimal conversions?

A: Here are some common binary to decimal conversions:

• $1010_{\text{two}} = 10_{\text{ten}}$
• $1101_{\text{two}} = 13_{\text{ten}}$
• $1001_{\text{two}} = 9_{\text{ten}}$
• $1111_{\text{two}} = 15_{\text{ten}}$

Conclusion

In conclusion, binary to decimal conversion is an essential skill in mathematics and computer science. By understanding the two methods of conversion, expanding in powers and using a calculator shortcut, you can become proficient in converting binary numbers to decimal. With practice and patience, you can apply this skill to a wide range of problems.

Practice Problems

Here are some practice problems to help you convert binary numbers to decimal:

• Convert $101101_{\text{two}}$ to decimal.
• Convert $110011_{\text{two}}$ to decimal.
• Convert $100101_{\text{two}}$ to decimal.
• Convert $111101_{\text{two}}$ to decimal.

Answer Key

Here are the answers to the practice problems:

• $101101_{\text{two}} = 149_{\text{ten}}$
• $110011_{\text{two}} = 183_{\text{ten}}$
• $100101_{\text{two}} = 121_{\text{ten}}$
• $111101_{\text{two}} = 229_{\text{ten}}$

Conclusion

In conclusion, binary to decimal conversion is an essential skill in mathematics and computer science. By understanding the two methods of conversion, expanding in powers and using a calculator shortcut, you can become proficient in converting binary numbers to decimal. With practice and patience, you can apply this skill to a wide range of problems.