Please in simple terms explain how to find the square root of a binary numbers with five easily understandable examples
To find the square root of a binary number, you can follow these steps:
1. Split the binary number into groups of two starting from the right side, and add a leading zero if the number of digits is odd. For example, if the binary number is 10111010, group it as 10 11 10 10.
2. Write down the largest decimal number (0-3 for two binary digits, 0-15 for four binary digits, and so on) whose square is less than or equal to the first group. In this example, the first group is 10, and the largest decimal number whose square is less than or equal to it is 3.
3. Subtract this squared decimal number from the first group and write down the result. In this case, 10 minus 3 squared (which is 9) gives 1.
4. Bring down the next group and write it alongside the result from step 3. In this example, the next group is 11 and the result is 1, so we have 111.
5. Double the value of the result from step 3, and write down the largest decimal number whose product with this doubled result is less than or equal to the number obtained in step 4. In this case, double of 1 is 2, and the largest decimal number whose product with 2 is less than or equal to 111 is 3.
6. Subtract this product from the current number, and write down the remainder. In this case, 2 times 3 is 6, so subtracting 6 from 111 gives 3.
7. Repeat steps 4-6 until you have gone through all the groups. In this example, the next group is 10 and the result is 33. Double of 33 is 66, and the largest decimal number whose product with 66 is less than or equal to 1010 is 3. Subtracting 198 (3 times 66) from 1010 gives 46.
8. The final result is the combination of all the decimal numbers obtained in step 3 and step 6. In this example, the final result is 3313, meaning that the square root of 10111010 in binary is 3313 in decimal.
You can understand this process more easily by practicing with the following examples:
1. Binary number: 101
- Grouped binary number: 001 01
- Square root: 01
- Remainder: 01
- Final result: 01
- Square root in decimal: 1
2. Binary number: 1010
- Grouped binary number: 10 10
- Square root: 10
- Remainder: 10
- Final result: 10
- Square root in decimal: 2
3. Binary number: 1111
- Grouped binary number: 11 11
- Square root: 11
- Remainder: 00
- Bring down next group: 00 00
- Double of previous result: 22
- Largest decimal number whose product with 22 is less than or equal to 0000: 0
- Remainder: 00
- Final result: 110
- Square root in decimal: 6
4. Binary number: 10000
- Grouped binary number: 10 00 0
- Square root: 10
- Remainder: 00
- Bring down next group: 00 00 0
- Double of previous result: 20
- Largest decimal number whose product with 20 is less than or equal to 0000: 0
- Remainder: 00
- Bring down next group: 00
- Double of previous result: 40
- Largest decimal number whose product with 40 is less than or equal to 0000: 2
- Remainder: 00
- Final result: 100
- Square root in decimal: 4
5. Binary number: 110010
- Grouped binary number: 11 00 10
- Square root: 11
- Remainder: 10
- Bring down next group: 10 00
- Double of previous result: 32
- Largest decimal number whose product with 32 is less than or equal to 1000: 3
- Remainder: 004
- Bring down next group: 004
- Double of previous result: 68
- Largest decimal number whose product with 68 is less than or equal to 0004: 0
- Remainder: 004
- Final result: 1100
- Square root in decimal: 12
Remember to practice and verify each step to become familiar with the process of finding the square root of binary numbers.
To find the square root of a binary number, follow these steps:
1. Start with the input binary number that you want to find the square root of. For example, let's use the binary number 101101.
2. Divide the binary number into groups of 2 from right to left. If there are any remaining digits, leave them as a separate group. For our example, the groups will be 10 11 01.
3. Write down the largest binary digit (either 0 or 1) that, when squared, is less than or equal to the leftmost group. In this case, the leftmost group is 10, and the largest binary digit squared that is less than or equal to 10 is 1. So, we write down 1 as the first digit of our result.
4. Subtract the square of the number you just wrote down (in our example, subtract 1 squared) from the leftmost group. In this case, subtract 1 squared (which is 1) from 10, resulting in 01.
5. Bring down the next group of digits. In our example, bring down 11.
6. Double the result found so far (the 1 we wrote down in step 3). In our example, double 1 to get 2.
7. Find the largest binary digit (0 or 1) that, when appended after the doubled result and then squared, is less than or equal to the two-digit number formed by the doubled result and the first digit of the next group. In our example, the doubled result is 2, and when we append a binary digit, either 0 or 1, we form two-digit numbers of 20 or 21. The largest binary digit, when squared, that is less than or equal to 20 is 1, so we append 1 to the doubled result.
8. Multiply the result found in step 7 (the appended digit) by the doubled result. In our example, multiply 2 by 1 to get 2.
9. Subtract the product found in step 8 from the two-digit number formed by the doubled result and the appended digit. In our example, subtract 2 (the product from step 8) from 21 (formed by the doubled result 2 and the appended digit 1) to get 19.
10. Repeat steps 6 to 9 until you have brought down all the groups and performed the calculations. In our example, bring down 01 and repeat steps 6 to 9.
11. Write down the appended digit (found in step 7) as the next digit of the result. In our example, the appended digit is 1.
12. Continue performing steps 6 to 9 until all the groups have been brought down and all the calculations have been done.
13. The final result will be the square root of the binary number. In our example, the square root of 101101 is 101.
Repeat these steps with additional examples to further understand the process.
To find the square root of a binary number, there are a few steps you can follow. Let's go through each step with five examples for better understanding:
Example 1: Find the square root of 10010.
Step 1: Split the number into pairs. Starting from the right, group the digits into pairs. If we have an odd number of digits, the leftmost digit will form a single pair.
10010 --> 100 10
Step 2: Write down the leftmost pair as the "root so far" and set the "remainder" as the rightmost pair.
100 10 - root so far: 10, remainder: 10
Step 3: Find the largest digit 'x' such that when you square it and append it to the "root so far," the result is less than or equal to the "remainder." Write down this digit as the next digit of the square root.
In this case, the largest digit 'x' is 1 because 11 squared is 121, which is larger than our remainder (10).
So, the current "root so far" becomes 101.
Step 4: Subtract the square of the chosen digit, obtained in the previous step, from the remainder and append two zeros.
Remainder (10) - x^2 (1^2) = 9 --> remainder becomes 900.
Step 5: Repeat the steps until the remainder becomes 0 or you have the desired precision.
Example 2: Find the square root of 11010.
Step 1: Split the number into pairs:
11010 --> 1 10 10
Step 2: Write down the leftmost pair as the "root so far" and set the "remainder" as the rightmost pairs.
1 10 10 - root so far: 1, remainder: 1010
Step 3: The largest digit 'x' such that x^2 is less than or equal to the remainder is 1.
So, the current "root so far" becomes 11.
Step 4: Square the chosen digit, subtract it from the remainder, and append two zeros.
Remainder (1010) - x^2 (1^2) = 1009 --> remainder becomes 100900.
Step 5: Repeat the steps until the remainder becomes 0 or you have the desired precision.
Example 3: Find the square root of 101001.
Step 1: Split the number into pairs:
101001 --> 1 01 001
Step 2: Write down the leftmost pair as the "root so far" and set the "remainder" as the rightmost pairs.
1 01 001 - root so far: 1, remainder: 01001
Step 3: The largest digit 'x' such that x^2 is less than or equal to the remainder is 1.
So, the current "root so far" becomes 11.
Step 4: Square the chosen digit, subtract it from the remainder, and append two zeros.
Remainder (01001) - x^2 (1^2) = 01000 --> remainder becomes 010000.
Step 5: Repeat the steps until the remainder becomes 0 or you have the desired precision.
Example 4: Find the square root of 11111.
Step 1: Split the number into pairs:
11111 --> 1 11 11
Step 2: Write down the leftmost pair as the "root so far" and set the "remainder" as the rightmost pairs.
1 11 11 - root so far: 1, remainder: 111
Step 3: The largest digit 'x' such that x^2 is less than or equal to the remainder is 1.
So, the current "root so far" becomes 11.
Step 4: Square the chosen digit, subtract it from the remainder, and append two zeros.
Remainder (111) - x^2 (1^2) = 110 --> remainder becomes 11000.
Step 5: Repeat the steps until the remainder becomes 0 or you have the desired precision.
Example 5: Find the square root of 1000001.
Step 1: Split the number into pairs:
1000001 --> 10 000 01
Step 2: Write down the leftmost pair as the "root so far" and set the "remainder" as the rightmost pairs.
10 000 01 - root so far: 10, remainder: 00001
Step 3: The largest digit 'x' such that x^2 is less than or equal to the remainder is 1.
So, the current "root so far" becomes 101.
Step 4: Square the chosen digit, subtract it from the remainder, and append two zeros.
Remainder (00001) - x^2 (1^2) = 0 --> remainder becomes 0.
Step 5: Repeat the steps until the remainder becomes 0 or you have the desired precision.
Remember to practice these steps with different binary numbers to get comfortable with finding the square root of binary numbers.