How to Find the SQUARE ROOT of ANY POSITIVE REAL NUMBER without the use of a CALCULATOR !!! ( PART 2 )

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" LOOK MOM, NO CALCULATOR !!! " There are many different Techniques on how to find the Square Root of any positive REAL NUMBER, this Article will show an interesting Algorithm to find the SQUARE ROOT OF ANY POSITIVE REAL NUMBER without the use of a Calculator.

Difficulty: Moderately EasyInstructions

Things You'll Need:

• • Pencil and • Paper

1. Step 1

Now we will choose any number, including decimal numbers and irrational numbers, and find their Square Roots.

2. Step 2

Let us choose the number ( 10.3041 ) to begin with. Our first step is to separate the digits in pairs going from Left to Right for the digits that are to the right of the decimal point. Then separate the digits from right to left for the digits that are to the left of the decimal point, as shown in the Image on the left.

3. Step 3

Now draw two straight lines a horizontal line above the number ( 10.3041 ) and a vertical line to the left of the number as shown in the image. This Algorithm has some similarity to that of the Algorithm of Long Division. Please, click on the Image to get a better understanding.

4. Step 4

We now look for a Number ( N ), when Squared is equal to or less than the first pair of numbers,( reading the numbers from Left to Right ), that number is ( 10 ). So ( 3 ) squared which equals ( 9 ), is less than or equal to ( 10 ). That is, in this case N = 3. We take the number ( 3 ) and write it above the horizontal line directly over the number ( 10 ). Put the decimal point to the right of the number ( 3 ) and directly above the decimal point. Also write the number ( 3 ) to the left of the vertical line in the same row as the given number that we are to find the square root of. In this case the number is ( 10.3041 ). Please View image.

5. Step 5

We multiply the ( 3 ) that is to the left of the vertical line by the ( 3 ) that is above the horizontal line; we put the product ( 9 ) below the first pair which is 10. We now subtract ( 9 ) from ( 10 ) which makes a remainder of 1. Please, click on the image to get a better view.

6. Step 6

We now bring down the ( 30 ) which is the next pair of numbers, and put it by the remainder ( 1 ), which makes the new number ( 130 ), as shown in the image to the left. Note: this has some similarity to the Algorithm of Long Division. Please, click on the image to get a better view.

7. Step 7

We double the ( 3 ) that is above the horizontal line, the result is ( 6 ). We now write the ( 6 ) to the left of the vertical line in the same row as the number ( 130 ), and leave a space between the line and the number ( 6 ) since the space will be filled by another number as illustrated in the Image.

8. Step 8

We will now look at the number ( 130 ), reading the digits from left to right, and see if ( 1 ), the first digit of the number ( 130 ), can be divided by ( 6 ). The result is a quoitient that is a single digit whole number that is greater than zero ( 0 ), and a remainder that is a whole number that is less than ( 6 ). If not, we then try ( 13 ), the first two digits of the number ( 130 ), and repeat the process. In this case we see that ( 13 ) can be divided by ( 6 ) with a quotient of ( 2 ) and a remainder of ( 1 ), as illustrated in the Image.

9. Step 9

We write the quotient of ( 2 ) to the left of the vertical line in the space provided. We also write the ( 2 ) above the horizontal line over the pair of numbers ( 30 ), and to the right of the decimal point. We multiply the ( 62 ), that is to the left of the vertical line, by the ( 2 ) that is above the horizontal line and over the pair of numbers ( 30 ). Which gives us the product ( 124 ). We write this product, below the number ( 130 ), and subtract them. Making a remainder of ( 6 ).We treat the number ( 3.2 ) as a whole number ( 32 ), without the decimal point, so as to continue this algorithm.

10. Step 10

We now bring down the ( 41 ) which is the next pair of numbers, and put it by the remainder ( 6 ), which makes the new number ( 641 ), as shown in the image to the left. Note: this has some similarity to the Algorithm of Long Division. Please, click on the image to get a better view.

11. Step 11

We double the ( 32 ) that is above the horizontal line, the result is ( 64 ). We now write the number( 64 ) to the left of the vertical line in the same row as the number ( 641 ), and leave a space between the line and the number ( 64 ) since the space will be filled by another number as illustrated in the Image.

12. Step 12

We will now look at the number ( 641 ), reading the digits from left to right, and see if ( 6 ), the first digit of the number ( 641 ), can be divided by ( 6 ). The result is a quoitient that is a single digit whole number that is greater than zero ( 0 ), and a remainder that is a whole number that is less than ( 6 ). In this case we see that ( 6 ) can be divided by ( 6 ) with a quotient of ( 1 ) and a remainder of ( 0 ), as illustrated in the Image.

13. Step 13

We write the quotient of ( 1 ) to the left of the vertical line in the space provided. We also write the ( 1 ) above the horizontal line over the pair of numbers ( 41 ). We multiply the ( 641 ), that is to the left of the vertical line, by the ( 1 ) that is above the horizontal line and over the pair of numbers ( 41 ). Which gives us the product ( 641 ). We write this product, below the number ( 641 ), and subtract them. Making a remainder of zero ( 0 ). The number ( 3.21 ), that is above the horizontal line, is the answer to the Square Root of ( 10.3041 ). Please, click on the Image to get a better understanding.