1. Convert \[$343_5\$\] And \[$445\$\] To The Desired Base, Then Add Them And Provide The Answer In Base \[$s\$\].2. Convert \[$434\$\] To Base 10.3. Convert \[$7510\$\] To Base 2.4. Add \[$377_8\$\] And

Introduction

Base conversion and addition are fundamental concepts in mathematics that involve converting numbers from one base to another and performing arithmetic operations on them. In this article, we will explore the process of converting numbers from different bases to the desired base, adding them, and providing the answer in the desired base. We will also delve into converting numbers from base 10 to base 2 and vice versa.

Converting Numbers from Different Bases to the Desired Base

To convert a number from one base to another, we need to understand the concept of place value and the relationship between the digits in the number. The place value of a digit in a number depends on its position and the base of the number.

Converting ${343_5\$} and ${445\$} to the Desired Base

To convert ${343_5\$} to base 10, we need to multiply each digit by its corresponding power of 5 and add the results.

${343_5 = 3 \times 5^2 + 4 \times 5^1 + 3 \times 5^0 = 75 + 20 + 3 = 98\$}

Similarly, to convert ${445\$} to base 10, we need to multiply each digit by its corresponding power of 5 and add the results.

${445 = 4 \times 5^2 + 4 \times 5^1 + 5 \times 5^0 = 100 + 20 + 5 = 125\$}

Now, we need to add ${343_5\$} and ${445\$} in base 10.

${98 + 125 = 223\$}

To convert ${223\$} to base 5, we need to divide it by 5 and find the remainder.

${$223 \div 5 = 44$ with a remainder of 3$}$

${$44 \div 5 = 8$ with a remainder of 4$}$

${$8 \div 5 = 1$ with a remainder of 3$}$

${$1 \div 5 = 0$ with a remainder of 1$}$

Therefore, ${223\$} in base 10 is equal to ${1343_5\$} in base 5.

Converting ${434\$} to Base 10

To convert ${434\$} to base 10, we need to multiply each digit by its corresponding power of 4 and add the results.

${434 = 4 \times 4^2 + 3 \times 4^1 + 4 \times 4^0 = 64 + 12 + 4 = 80\$}

Converting ${7510\$} to Base 2

To convert ${7510\$} to base 2, we need to divide it by 2 and find the remainder.

${$7510 \div 2 = 3755$ with a remainder of 0$}$

${$3755 \div 2 = 1877$ with a remainder of 1$}$

${$1877 \div 2 = 938$ with a remainder of 1$}$

${$938 \div 2 = 469$ with a remainder of 0$}$

${$469 \div 2 = 234$ with a remainder of 1$}$

${$234 \div 2 = 117$ with a remainder of 0$}$

${$117 \div 2 = 58$ with a remainder of 1$}$

${$58 \div 2 = 29$ with a remainder of 0$}$

${$29 \div 2 = 14$ with a remainder of 1$}$

${$14 \div 2 = 7$ with a remainder of 0$}$

${$7 \div 2 = 3$ with a remainder of 1$}$

${$3 \div 2 = 1$ with a remainder of 1$}$

${$1 \div 2 = 0$ with a remainder of 1$}$

Therefore, ${7510\$} in base 10 is equal to ${11101110110_2\$} in base 2.

Adding ${377_8\$} and ${445\$}

To add ${377_8\$} and ${445\$}, we need to convert both numbers to base 10.

${377_8 = 3 \times 8^2 + 7 \times 8^1 + 7 \times 8^0 = 192 + 56 + 7 = 255\$}

Now, we can add ${255\$} and ${445\$}.

${255 + 445 = 700\$}

To convert ${700\$} to base 8, we need to divide it by 8 and find the remainder.

${$700 \div 8 = 87$ with a remainder of 4$}$

${$87 \div 8 = 10$ with a remainder of 7$}$

${$10 \div 8 = 1$ with a remainder of 2$}$

${$1 \div 8 = 0$ with a remainder of 1$}$

Therefore, ${700\$} in base 10 is equal to ${1274_8\$} in base 8.

Conclusion

In this article, we explored the process of converting numbers from different bases to the desired base, adding them, and providing the answer in the desired base. We also delved into converting numbers from base 10 to base 2 and vice versa. The examples provided demonstrate the importance of understanding the concept of place value and the relationship between the digits in a number. By mastering these concepts, we can perform complex arithmetic operations with ease and accuracy.

References

• [1] "Base Conversion" by Math Open Reference
• [2] "Binary to Decimal Conversion" by Math Is Fun
• [3] "Decimal to Binary Conversion" by Math Is Fun

Further Reading

• [1] "Base Conversion: A Tutorial" by Tutorials Point
• [2] "Binary to Decimal Conversion: A Guide" by Codecademy
• [3] "Decimal to Binary Conversion: A Tutorial" by GeeksforGeeks
  

Introduction

Base conversion and addition are fundamental concepts in mathematics that involve converting numbers from one base to another and performing arithmetic operations on them. In this article, we will provide a Q&A guide to help you understand the process of converting numbers from different bases to the desired base, adding them, and providing the answer in the desired base.

Q1: What is base conversion?

A1: Base conversion is the process of converting a number from one base to another. For example, converting a number from base 10 to base 2 or from base 8 to base 10.

Q2: Why is base conversion important?

A2: Base conversion is important because it allows us to perform arithmetic operations on numbers in different bases. For example, we can add two numbers in base 10, but if we want to add them in base 8, we need to convert them to base 8 first.

Q3: How do I convert a number from base 10 to base 2?

A3: To convert a number from base 10 to base 2, you need to divide the number by 2 and find the remainder. You repeat this process until the quotient is 0. The remainders, read from bottom to top, give you the binary representation of the number.

Q4: How do I convert a number from base 8 to base 10?

A4: To convert a number from base 8 to base 10, you need to multiply each digit by its corresponding power of 8 and add the results.

Q5: What is the difference between base 2 and base 8?

A5: Base 2 is the binary number system, which uses only two digits: 0 and 1. Base 8 is the octal number system, which uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7.

Q6: How do I add two numbers in different bases?

A6: To add two numbers in different bases, you need to convert both numbers to the same base first. Then, you can add them as usual.

Q7: What is the result of adding ${377_8\$} and ${445\$}?

A7: To add ${377_8\$} and ${445\$}, we need to convert both numbers to base 10. ${377_8 = 3 \times 8^2 + 7 \times 8^1 + 7 \times 8^0 = 192 + 56 + 7 = 255\$}. Now, we can add ${255\$} and ${445\$}. ${255 + 445 = 700\$}. To convert ${700\$} to base 8, we need to divide it by 8 and find the remainder. ${$700 \div 8 = 87$ with a remainder of 4$}$. ${$87 \div 8 = 10$ with a remainder of 7$}$. ${$10 \div 8 = 1$ with a remainder of 2$}$. ${$1 \div 8 = 0$ with a remainder of 1$}$. Therefore, ${700\$} in base 10 is equal to ${1274_8\$} in base 8.

Q8: How do I convert a number from base 10 to base 16?

A8: To convert a number from base 10 to base 16, you need to divide the number by 16 and find the remainder. You repeat this process until the quotient is 0. The remainders, read from bottom to top, give you the hexadecimal representation of the number.

Q9: What is the difference between base 10 and base 16?

A9: Base 10 is the decimal number system, which uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Base 16 is the hexadecimal number system, which uses 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

Q10: How do I convert a number from base 16 to base 10?

A10: To convert a number from base 16 to base 10, you need to multiply each digit by its corresponding power of 16 and add the results.

Conclusion

In this Q&A guide, we have covered the basics of base conversion and addition. We have provided examples and explanations to help you understand the process of converting numbers from different bases to the desired base, adding them, and providing the answer in the desired base. By mastering these concepts, you can perform complex arithmetic operations with ease and accuracy.

References

• [1] "Base Conversion" by Math Open Reference
• [2] "Binary to Decimal Conversion" by Math Is Fun
• [3] "Decimal to Binary Conversion" by Math Is Fun

Further Reading

• [1] "Base Conversion: A Tutorial" by Tutorials Point
• [2] "Binary to Decimal Conversion: A Guide" by Codecademy
• [3] "Decimal to Binary Conversion: A Tutorial" by GeeksforGeeks