Finding Square Roots Without A Calculator
Posted: Mon May 06, 2024 11:05 am
A question from Quora:
"How did people calculate square roots before computers and scientific calculators were invented?"
There’s a pen-and-paper way to find square roots.
Let’s say you want to find √43818.38
First, you partition the number into groups of two integers, going either way from the decimal point.
4 , 38 , 18 . 38 , 00 , 00 , 00...
You find the largest single-digit number whose square is less than or equal to the number in the left-most grouping. That number is 2.
So you write down the 2 as the first digit of your answer.
Answer: 2… (incomplete)
Now we will begin dealing with number pairs, separated by a comma.
To the left of the comma, write the square of the number whose square was less than the leftmost grouping (4), and tack on a zero as a place-holder.
40 ,
To the right of the comma, write the difference between the number in the leftmost grouping (4) and the square of the largest number whose square was less than or equal to the number in the leftmost grouping (4), and then tack on the two digits in the next grouping. This time, the difference is zero, so only the two digits of the next grouping appear (38).
40 , 38
Now, you’re looking for the largest single-digit number (0–9) that, when multiplied by (40+itself) will return a product that is less than the number to the right of the comma (38). That number is zero.
So you write down the 0 as the second digit of your answer.
Answer: 20… (incomplete)
To the right of the comma, you bring down the difference between the number previously to the right of the comma and the product formed by multiplying the number to the left of the comma by its final digit, and then you pull down the next two-digit grouping.
38–(40)(0)=38
To the left of the comma, you double the digit by which you replaced the place-holding zero, and you tack on another zero as a place-holder. Of course, doubling zero yields zero, so
400, 3818
Now you’re looking for the largest single-digit number that, when multiplied by (400+itself), will return a product that is less than or equal to 3818. That number is 9.
So you write down the 9 as the third digit of your answer.
Answer: 209… (incomplete)
To the right of the comma, you bring down the difference between the number previously to the right of the comma and the product formed by multiplying the number to the left of the comma by its final digit, and then you pull down the next two-digit grouping.
3818–(409)(9)=137
To the left of the comma, you double the digit by which you replaced the place-holding zero, and you tack on another zero as a place-holder.
4180, 13738
Now you’re looking for the largest single-digit number that, when multiplied by (4180+itself), will return a product that is less than or equal to 13738. That number is 3.
So you write down the 3 as the fourth digit of your answer.
Answer: 209.3… (incomplete)
To the right of the comma, you bring down the difference between the number previously to the right of the comma and the product formed by multiplying the number to the left of the comma by its final digit, and then you pull down the next two-digit grouping.
13738–(4183)(3) = 1189
To the left of the comma, you double the digit by which you replaced the place-holding zero, and you tack on another zero as a place-holder.
41860, 118900
Now you’re looking for the largest single-digit number that, when multiplied by (41860+itself), will return a product that is less than or equal to 118900. That number is 2.
So you write down the 2 as the fifth digit of your answer.
Answer: 209.32… (incomplete)
To the right of the comma, you bring down the difference between the number previously to the right of the comma and the product formed by multiplying the number to the left of the comma by its final digit, and then you pull down the next two-digit grouping.
118900–(41862)(2) = 35176
To the left of the comma, you double the digit by which you replaced the place-holding zero, and you tack on another zero as a place-holder.
418640 , 3517600
Now you’re looking for the largest single-digit number that, when multiplied by (418640+itself), will return a product that is less than or equal to 3517600. That number is 8.
So you write down the 8 as the sixth digit of your answer.
Answer: 209.328…
Let’s check. (209.328)² = 43818.211584 ≈ 43818.38
The longer you keep going through the cycles as demonstrated, the closer the approximation will be.
"How did people calculate square roots before computers and scientific calculators were invented?"
There’s a pen-and-paper way to find square roots.
Let’s say you want to find √43818.38
First, you partition the number into groups of two integers, going either way from the decimal point.
4 , 38 , 18 . 38 , 00 , 00 , 00...
You find the largest single-digit number whose square is less than or equal to the number in the left-most grouping. That number is 2.
So you write down the 2 as the first digit of your answer.
Answer: 2… (incomplete)
Now we will begin dealing with number pairs, separated by a comma.
To the left of the comma, write the square of the number whose square was less than the leftmost grouping (4), and tack on a zero as a place-holder.
40 ,
To the right of the comma, write the difference between the number in the leftmost grouping (4) and the square of the largest number whose square was less than or equal to the number in the leftmost grouping (4), and then tack on the two digits in the next grouping. This time, the difference is zero, so only the two digits of the next grouping appear (38).
40 , 38
Now, you’re looking for the largest single-digit number (0–9) that, when multiplied by (40+itself) will return a product that is less than the number to the right of the comma (38). That number is zero.
So you write down the 0 as the second digit of your answer.
Answer: 20… (incomplete)
To the right of the comma, you bring down the difference between the number previously to the right of the comma and the product formed by multiplying the number to the left of the comma by its final digit, and then you pull down the next two-digit grouping.
38–(40)(0)=38
To the left of the comma, you double the digit by which you replaced the place-holding zero, and you tack on another zero as a place-holder. Of course, doubling zero yields zero, so
400, 3818
Now you’re looking for the largest single-digit number that, when multiplied by (400+itself), will return a product that is less than or equal to 3818. That number is 9.
So you write down the 9 as the third digit of your answer.
Answer: 209… (incomplete)
To the right of the comma, you bring down the difference between the number previously to the right of the comma and the product formed by multiplying the number to the left of the comma by its final digit, and then you pull down the next two-digit grouping.
3818–(409)(9)=137
To the left of the comma, you double the digit by which you replaced the place-holding zero, and you tack on another zero as a place-holder.
4180, 13738
Now you’re looking for the largest single-digit number that, when multiplied by (4180+itself), will return a product that is less than or equal to 13738. That number is 3.
So you write down the 3 as the fourth digit of your answer.
Answer: 209.3… (incomplete)
To the right of the comma, you bring down the difference between the number previously to the right of the comma and the product formed by multiplying the number to the left of the comma by its final digit, and then you pull down the next two-digit grouping.
13738–(4183)(3) = 1189
To the left of the comma, you double the digit by which you replaced the place-holding zero, and you tack on another zero as a place-holder.
41860, 118900
Now you’re looking for the largest single-digit number that, when multiplied by (41860+itself), will return a product that is less than or equal to 118900. That number is 2.
So you write down the 2 as the fifth digit of your answer.
Answer: 209.32… (incomplete)
To the right of the comma, you bring down the difference between the number previously to the right of the comma and the product formed by multiplying the number to the left of the comma by its final digit, and then you pull down the next two-digit grouping.
118900–(41862)(2) = 35176
To the left of the comma, you double the digit by which you replaced the place-holding zero, and you tack on another zero as a place-holder.
418640 , 3517600
Now you’re looking for the largest single-digit number that, when multiplied by (418640+itself), will return a product that is less than or equal to 3517600. That number is 8.
So you write down the 8 as the sixth digit of your answer.
Answer: 209.328…
Let’s check. (209.328)² = 43818.211584 ≈ 43818.38
The longer you keep going through the cycles as demonstrated, the closer the approximation will be.
