how do you find the square root of any number with long division and not a calculator? I don't like calculator readings. They are not always easy to figure out, plus math principles can't be learned with a calculator.
You echo what I think also.
However, for square roots, it is a little more complicated (but not difficult) than long division if you need many figures for your answer.
For doing it by hand, you can try:
http://www.wikihow.com/Calculate-a-Square-Root-by-Hand
or
http://www.homeschoolmath.net/teaching/square-root-algorithm.php
Better still, most square-roots can be calculated mentally, based on Newton's method.
The more squares of integers you can memorize, the more accurate your estimate of the answer will be.
If you need the square root of 37, you know the closest perfect square is 36.
So take 6 as a first estimate. Divide the residue of (37-36)=1 by twice the estimate to get 1/12=0.083. So the estimate of √37 is 6.083.
The answer to 6 places after the decimal is 6.0827625.
To calculate the square-root of 11, if you know by memorization that 33^2=1089, then the first estimate is 3.3 (since 3.3²=10.89)
Divide the residue of 11/10.89=0.11 by twice 3.3 gives 1/60, or 0.01667.
This gives our first estimate as 3.31667.
The exact value to 6 decimal places is 3.316625
To find the square root of a number using long division, you can follow these steps:
1. Start by separating the number into groups of two digits, starting from the rightmost digit. If the number has an odd number of digits, the leftmost group will have only one digit. For example, for the number 1764, the groups would be 17 and 64.
2. Find the largest possible integer "x" such that when added as the leftmost digit of the divisor (x*(2x-1)), it makes the largest possible number less than or equal to the first group (17 in this case). In this example, x would be 4 since 4*8 = 32, which is the largest number less than or equal to 17.
3. Next, divide the dividend (176) with the divisor (48) using long division. Write the divisor on the left and the dividend on the right. Divide the leftmost group (17) of the dividend by the divisor.
4. Multiply the quotient (4) by the divisor (48), subtracting the product (192) from the dividend (176). Bring down the next group (64).
5. Now, double the quotient (4) and put a variable "y" in the blank space next to it. Multiply the variable y by the quotient (4) and then by 2. Find the largest digit "y" that is less than or equal to 768 (product of 4 and 192) when placed at the end. For example, if y=5, then (40 + 5) * 5 = 225, which is less than 768.
6. Now, multiply the new divisor "48y" (40y) by the variable y (5), multiply it again by 2 (10y), and subtract the product from the remaining dividend (768 - 225 = 543).
7. Bring down the next group (00). This becomes the new dividend (54300).
8. Repeat steps 3 to 7 until all the groups have been brought down and processed.
9. The final result will be the quotient you obtained at each step. In this case, the square root of 1764 is 42.
It's important to note that this method can be time-consuming and may require practice. Additionally, using a calculator for complex calculations can be more efficient in many situations. But understanding the principles behind long division can still be valuable for learning math concepts.