How to Perform Long Division in Base 4: Solving 1303_4 by 23_4
Dividing numbers in base 4 can be a bit complex, especially when working with larger numbers. In this article, we will walk you through the steps of solving 1303_4 divided by 23_4, providing detailed explanations and examples for better understanding.
Step-by-Step Guide to Division in Base 4
To solve 1303_4 divided by 23_4, we follow a series of steps, converting between base 4 and base 10 decimal as necessary.
Step 1: Convert 1303_4 to Decimal
The first step is to convert the number 1303_4 from base 4 to base 10 decimal. In base 4, each digit represents a power of 4, starting from the rightmost digit (least significant digit) with 40.
Therefore, 1303_4 can be expressed as:
1303_4 1 4^3 3 4^2 0 4^1 3 4^0
Calculating each term:
1 4^3 1 64 64
3 4^2 3 16 48
0 4^1 0 4 0
3 4^0 3 1 3
Adding these together:
64 48 0 3 115
Therefore, 1303_4 115_{10}.
Step 2: Convert 23_4 to Decimal
The next step is to convert the number 23_4 from base 4 to base 10 decimal. Using the same method:
23_4 2 4^1 3 4^0
Calculating each term:
2 4^1 2 4 8
3 4^0 3 1 3
Adding these together:
8 3 11
Therefore, 23_4 11_{10}.
Step 3: Perform the Division in Decimal
Now we need to divide 115 by 11:
115 div 11 10.4545...
For integer division, we take the integer part:
115 div 11 10
Step 4: Convert the Result Back to Base 4
Finally, we need to convert the result 10_{10} back to base 4. We do this by repeatedly dividing the number by 4 and keeping track of the remainders:
10 div 4 2 remainder 2
2 div 4 0 remainder 2
Reading the remainders from bottom to top:
10_{10} 22_4
Conclusion
Thus, the result of 1303_4 divided by 23_4 is:
boxed{22_4}
Alternative Method: Long Division in Base 4
Alternatively, if you prefer to perform the division directly in base 4, you can use long division. Here's how:
2 2 . 1 3 1 0 1 3 1 0 - 2 3 1 3 0 3.0 0 0 0 0 ------ 1 1 2 ----- 1 2 3 ----- 1 1.0 ----- 2.3 ----- 2.1 0 ----- 2.0 1 ----- 1 0 0 ----- 2 3 ----- 1 1 0The answer is 22.overline{13101} in base 4, demonstrating that the division has a recurring decimal in base 4.