1. Convert $365_{\text{sewen}}$ To Base Ten And Solve $365_{\text{sewen}} + 43x = 217_{\text{ren}}$.2. Convert \$3 \cdot 25_3$[/tex\] Into Base Three.

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Introduction

In mathematics, numbers can be represented in various bases, including decimal (base 10), binary (base 2), octal (base 8), and hexadecimal (base 16). Converting numbers from one base to another is an essential skill in mathematics, and it has numerous applications in computer science, engineering, and other fields. In this article, we will discuss how to convert numbers from base seven to base ten and solve an equation involving numbers in different bases.

Converting $365_{\text{sewen}}$ to Base Ten

To convert a number from base seven to base ten, we need to multiply each digit of the number by the corresponding power of seven and then add the results. The power of seven is determined by the position of the digit in the number, with the rightmost digit having a power of zero, the next digit to the left having a power of one, and so on.

365sewen=3â‹…72+6â‹…71+5â‹…70365_{\text{sewen}} = 3 \cdot 7^2 + 6 \cdot 7^1 + 5 \cdot 7^0

=3â‹…49+6â‹…7+5â‹…1= 3 \cdot 49 + 6 \cdot 7 + 5 \cdot 1

=147+42+5= 147 + 42 + 5

=194= 194

Therefore, the decimal equivalent of $365_{\text{sewen}}$ is 194.

Solving $365_{\text{sewen}} + 43x = 217_{\text{ren}}$

To solve the equation $365_{\text{sewen}} + 43x = 217_{\text{ren}}$, we need to convert the numbers from base seven and base ten to a common base, which is base ten. We have already converted $365_{\text{sewen}}$ to base ten, and we can convert $217_{\text{ren}}$ to base ten as follows:

217ren=2â‹…161+1â‹…160217_{\text{ren}} = 2 \cdot 16^1 + 1 \cdot 16^0

=2â‹…16+1â‹…1= 2 \cdot 16 + 1 \cdot 1

=32+1= 32 + 1

=33= 33

Therefore, the equation becomes:

194+43x=33194 + 43x = 33

Subtracting 194 from both sides of the equation gives:

43x=−16143x = -161

Dividing both sides of the equation by 43 gives:

x=−16143x = -\frac{161}{43}

Therefore, the solution to the equation is $x = -\frac{161}{43}$.

Converting $3â‹…2533 \cdot 25_3[/tex] into Base Three

To convert a number from base three to base ten, we need to multiply each digit of the number by the corresponding power of three and then add the results. The power of three is determined by the position of the digit in the number, with the rightmost digit having a power of zero, the next digit to the left having a power of one, and so on.

3â‹…253=3â‹…(2â‹…31+5â‹…30)3 \cdot 25_3 = 3 \cdot (2 \cdot 3^1 + 5 \cdot 3^0)

=3â‹…(2â‹…3+5â‹…1)= 3 \cdot (2 \cdot 3 + 5 \cdot 1)

=3â‹…(6+5)= 3 \cdot (6 + 5)

=3â‹…11= 3 \cdot 11

=33= 33

Therefore, the decimal equivalent of $3â‹…2533 \cdot 25_3[/tex] is 33.

Conclusion

In this article, we discussed how to convert numbers from base seven to base ten and solve an equation involving numbers in different bases. We also converted a number from base three to base ten. Converting numbers from one base to another is an essential skill in mathematics, and it has numerous applications in computer science, engineering, and other fields. By understanding how to convert numbers from one base to another, we can solve problems involving numbers in different bases and apply mathematical concepts to real-world problems.

References

  • [1] "Number Systems" by Khan Academy
  • [2] "Base Conversion" by Math Is Fun
  • [3] "Number Systems" by Wolfram MathWorld

Further Reading

  • [1] "Introduction to Number Systems" by MIT OpenCourseWare
  • [2] "Number Systems and Base Conversion" by Coursera
  • [3] "Mathematics for Computer Science" by Stanford University

Introduction

In our previous article, we discussed how to convert numbers from base seven to base ten and solve an equation involving numbers in different bases. We also converted a number from base three to base ten. In this article, we will answer some frequently asked questions about converting numbers to different bases and solving equations.

Q: What is the difference between base 10 and base 7?

A: Base 10 is the decimal system, which is the most common number system used in everyday life. Base 7, on the other hand, is a number system with a base of 7. In base 7, each digit can have a value of 0, 1, 2, 3, 4, 5, or 6.

Q: How do I convert a number from base 7 to base 10?

A: To convert a number from base 7 to base 10, you need to multiply each digit of the number by the corresponding power of 7 and then add the results. The power of 7 is determined by the position of the digit in the number, with the rightmost digit having a power of 0, the next digit to the left having a power of 1, and so on.

Q: What is the decimal equivalent of $365_{\text{sewen}}$?

A: The decimal equivalent of $365_{\text{sewen}}$ is 194.

Q: How do I convert a number from base 3 to base 10?

A: To convert a number from base 3 to base 10, you need to multiply each digit of the number by the corresponding power of 3 and then add the results. The power of 3 is determined by the position of the digit in the number, with the rightmost digit having a power of 0, the next digit to the left having a power of 1, and so on.

Q: What is the decimal equivalent of $3â‹…2533 \cdot 25_3[/tex]?

A: The decimal equivalent of $3â‹…2533 \cdot 25_3[/tex] is 33.

Q: How do I solve an equation involving numbers in different bases?

A: To solve an equation involving numbers in different bases, you need to convert the numbers from one base to another and then solve the equation using the converted numbers.

Q: What is the solution to the equation $365_{\text{sewen}} + 43x = 217_{\text{ren}}$?

A: The solution to the equation $365_{\text{sewen}} + 43x = 217_{\text{ren}}$ is $x = -\frac{161}{43}$.

Q: What are some real-world applications of converting numbers to different bases?

A: Converting numbers to different bases has numerous real-world applications, including:

  • Computer science: Converting numbers to different bases is essential in computer science, where numbers are represented in binary (base 2), octal (base 8), and hexadecimal (base 16).
  • Engineering: Converting numbers to different bases is used in engineering to represent numbers in different bases, such as base 10, base 2, and base 16.
  • Mathematics: Converting numbers to different bases is used in mathematics to solve equations and represent numbers in different bases.

Conclusion

In this article, we answered some frequently asked questions about converting numbers to different bases and solving equations. Converting numbers to different bases is an essential skill in mathematics, and it has numerous real-world applications in computer science, engineering, and other fields. By understanding how to convert numbers from one base to another, we can solve problems involving numbers in different bases and apply mathematical concepts to real-world problems.

References

  • [1] "Number Systems" by Khan Academy
  • [2] "Base Conversion" by Math Is Fun
  • [3] "Number Systems" by Wolfram MathWorld

Further Reading

  • [1] "Introduction to Number Systems" by MIT OpenCourseWare
  • [2] "Number Systems and Base Conversion" by Coursera
  • [3] "Mathematics for Computer Science" by Stanford University