Work Out The Following, Leaving Your Answer In The Indicated Base:(a) { 500_{\text{six}} \div 23_{\text{six}} $}$(b) { 201_{\text{four}} \div 1011_{\text{two}} $} , G I V I N G Y O U R A N S W E R I N B A S E T H R E E . ( C ) \[ , Giving Your Answer In Base Three.(c) \[ , G I V In G Yo U R An S W Er Inba Se T H Ree . ( C ) \[

Mar 13, 2025 by ADMIN 329 views

**Base Conversion and Division: A Mathematical Exploration**

Introduction

In mathematics, base conversion and division are fundamental operations that involve changing numbers from one base to another and performing arithmetic operations on them. In this article, we will explore the process of converting numbers from one base to another and then performing division operations on them. We will work out three examples, leaving our answers in the indicated base, and then convert them to base three.

Example (a): Base Six Division

Problem Statement

We are given the expression $500_{\text{six}} \div 23_{\text{six}}$ . Our task is to evaluate this expression and leave the answer in base six.

Solution

To solve this problem, we need to perform long division in base six. We will divide the dividend $500_{\text{six}}$ by the divisor $23_{\text{six}}$ .

  ____________________
23 | 500
  - 460
  ------
     40
  - 36
  ------
     4

The result of the division is $21_{\text{six}}$ . Therefore, the answer to the expression $500_{\text{six}} \div 23_{\text{six}}$ is $21_{\text{six}}$ .

Example (b): Base Four and Base Two Division

Problem Statement

We are given the expression $201_{\text{four}} \div 1011_{\text{two}}$ . Our task is to evaluate this expression and leave the answer in base three.

Solution

To solve this problem, we need to perform two steps. First, we need to convert the numbers from base four and base two to base ten. Then, we will perform the division operation in base ten and finally convert the result to base three.

Step 1: Convert Numbers to Base Ten

First, we will convert the numbers from base four and base two to base ten.

# Convert 201 from base four to base ten
201 (base four) = 2(4^2) + 0(4^1) + 1(4^0)
= 2(16) + 0(4) + 1(1)
= 32 + 0 + 1
= 33 (base ten)
1011 (base two) = 1(2^3) + 0(2^2) + 1(2^1) + 1(2^0)
= 1(8) + 0(4) + 1(2) + 1(1)
= 8 + 0 + 2 + 1
= 11 (base ten)

Step 2: Perform Division in Base Ten

Now that we have the numbers in base ten, we can perform the division operation.

  ____________________
11 | 33
  - 33
  ------
     0

The result of the division is $3$ (base ten).

Step 3: Convert Result to Base Three

Finally, we need to convert the result from base ten to base three.

# Convert 3 from base ten to base three
3 (base ten) = 3 (base three)

The answer to the expression $201_{\text{four}} \div 1011_{\text{two}}$ is $3_{\text{three}}$ .

Example (c): Base Conversion and Division

Problem Statement

We are given the expression $[$ $ 1000_{\text{eight}} \div 12_{\text{eight}} $]$. Our task is to evaluate this expression and leave the answer in base three.

Solution

To solve this problem, we need to perform two steps. First, we need to perform the division operation in base eight. Then, we will convert the result to base three.

Step 1: Perform Division in Base Eight

First, we will perform the division operation in base eight.

  ____________________
12 | 1000
  - 960
  ------
     40
  - 36
  ------
     4

The result of the division is $33_{\text{eight}}$ .

Step 2: Convert Result to Base Three

Finally, we need to convert the result from base eight to base three.

# Convert 33 from base eight to base three
33 (base eight) = 3(8^1) + 3(8^0)
= 3(8) + 3(1)
= 24 + 3
= 27 (base ten)

27 (base ten) = 3(10^1) + 0(10^0)
= 3(10) + 0(1)
= 30 (base three)

The answer to the expression $[$ $ 1000_{\text{eight}} \div 12_{\text{eight}} $]$ is $30_{\text{three}}$ .

Conclusion

Introduction

In our previous article, we explored the process of converting numbers from one base to another and performing division operations on them. In this article, we will provide a Q&A guide to help you better understand the concepts of base conversion and division.

Q: What is base conversion?

A: Base conversion is the process of changing a number from one base to another. For example, converting a number from base ten to base two or from base eight to base three.

Q: Why do we need to convert numbers from one base to another?

A: We need to convert numbers from one base to another for several reasons:

To perform arithmetic operations on numbers in different bases
To represent numbers in a more compact or efficient form
To facilitate communication between people who use different number systems

Q: What are the different bases that we can convert numbers to?

A: There are several bases that we can convert numbers to, including:

Base two (binary)
Base three
Base four
Base five
Base six
Base seven
Base eight (octal)
Base nine
Base ten (decimal)
Base eleven
Base twelve
Base thirteen
Base fourteen
Base fifteen
Base sixteen (hexadecimal)
Base seventeen
Base eighteen
Base nineteen
Base twenty
Base twenty-one
Base twenty-two
Base twenty-three
Base twenty-four
Base twenty-five
Base twenty-six
Base twenty-seven
Base twenty-eight
Base twenty-nine
Base thirty

Q: How do we convert numbers from one base to another?

A: To convert numbers from one base to another, we can use the following steps:

Identify the base of the number that we want to convert
Determine the base that we want to convert to
Use a conversion formula or algorithm to convert the number
Check the result to ensure that it is correct

Q: What are some common conversion formulas and algorithms?

A: Some common conversion formulas and algorithms include:

Converting from base ten to base two: Divide the number by 2 and keep track of the remainders
Converting from base ten to base three: Divide the number by 3 and keep track of the remainders
Converting from base ten to base four: Divide the number by 4 and keep track of the remainders
Converting from base ten to base five: Divide the number by 5 and keep track of the remainders
Converting from base ten to base six: Divide the number by 6 and keep track of the remainders
Converting from base ten to base seven: Divide the number by 7 and keep track of the remainders
Converting from base ten to base eight: Divide the number by 8 and keep track of the remainders
Converting from base ten to base nine: Divide the number by 9 and keep track of the remainders
Converting from base ten to base eleven: Divide the number by 11 and keep track of the remainders
Converting from base ten to base twelve: Divide the number by 12 and keep track of the remainders
Converting from base ten to base thirteen: Divide the number by 13 and keep track of the remainders
Converting from base ten to base fourteen: Divide the number by 14 and keep track of the remainders
Converting from base ten to base fifteen: Divide the number by 15 and keep track of the remainders
Converting from base ten to base sixteen: Divide the number by 16 and keep track of the remainders

Q: What are some common division formulas and algorithms?

A: Some common division formulas and algorithms include:

Long division: Divide the dividend by the divisor and keep track of the quotient and remainder
Synthetic division: Divide the dividend by the divisor and keep track of the quotient and remainder
Polynomial long division: Divide a polynomial by another polynomial and keep track of the quotient and remainder

Q: How do we perform division in different bases?

A: To perform division in different bases, we can use the following steps:

Identify the base of the dividend and divisor
Determine the base that we want to perform the division in
Use a division formula or algorithm to perform the division
Check the result to ensure that it is correct

Q: What are some common mistakes to avoid when performing base conversion and division?

A: Some common mistakes to avoid when performing base conversion and division include:

Not converting the numbers to the correct base
Not using the correct conversion formula or algorithm
Not checking the result to ensure that it is correct
Not using the correct division formula or algorithm
Not checking the result to ensure that it is correct

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts of base conversion and division. We hope that this article has been helpful in answering your questions and providing you with a better understanding of these important mathematical concepts.