Determination of square root

By systematic trial

Finding the roots of numbers is as simple as are the earlier tasks: The squares of which numbers will yield 2 or 3, etc? How big is ? You can try to find the answer by systematic trial.

1² = 1 and 2² = 4, so that must lie between 1 and 2; it must start with 1! If you compute the squares of 1.1 to 1.9, you find

so that 1.4 <  < 1.5; it must start with 1.4! If you determine in the same manner the second decimal, you find 

so that 1.41 <  < 1.42; it must starts with 1.41!

In the same manner you obtain the inequalities:

If you are satisfied with a certain number of digits, for example three, the rounded result is  = 1.414.

In a similar manner, you can find also higher order roots with arbitrary accuracy. But obviously, this method is very laborious.

Constructive method

If  = x, then a = x² or a·1 = x·x or a/x = x/1.

Every square root of a number a can be viewed as the central proportional of 1 and a.

If you represent a and 1 by segments, you can obtain by any of the known methods of geometry the central proportional to 1 and a, and thus find . As an example of such a solution, we will use the theorems of the heights and sides of a right-angled triangle for a = 5.

Measurement of the segments X yields 2.24 in both triangles.Since x = , you find that = 2.24.

There does not exist a method for the construction of a cubic root with the aid of a compass and ruler. If you check the accuracy, which you can achieve in this geometrical approach, you realize that it depends on the accuracy and size of the drawing and the measurement: It will always be improved by results, obtained by purely numerical methods.

Method using the formula (a + b)²

A. Number of digits of the square root.

If you are given an integer with several digits, you will ask: How many digits will its square root have? 

This table shows that the square of single digit numbers has 1 or 2 digits, of two digits numbers 3 or 4 digits, of three digits numbers 5 or 6 digits, etc. and conversely.

For example, if want to find , it must lie between 10 and 11, because 10² = 100 and 11² = 121, whence  must start with 10. The number of places, which the tables above indicate, is therefore always the number of digits, which the unknown root must have to the left of the decimal point; it is readily obtained by subdivision of the number under the root sign from the right to the left into groups of two digits. The square root has then as many digits to the left of the decimal point as the number of groups of the base. For example, the root  has three digits, the root  has five digits ahead of the decimal point.

B. Computation of square root.

Assuming that you have under the root sign the square of an integer, find, for example, . It must be a two-digit number (2 groups of two digits); denote it by T + U (tens + units), so that

5476 = (T + U)² + 2TU + U².

You must now find T and U.

The square of 10 yields a number with two zeroes (100). Hence T² cannot depend on the digits 7 and 6 in 5476. In order to find T², you seek the largest square (49), contained in 54. T² = 4900 and T = 70, whence 5476 = 4900 + 140U + U² or 576 = 140U + U².

Since U is a single digit integer, 140U will be much larger then U². In order to find U, we divide 576 by 140. This may yield too large a value for U, when you will try a number which is smaller by 1. You already know this approach from division. 576:140 yields 4. Check: 576 = 140·4 + 16, whence 4 is correct and   = T + E = 70 + 4 = 74.

As in multiplication and division, we must reorganize this procedure, in order to avoid unnecessary words. You will recognize in what follows the steps taken above: 

I:	= 7	II:	= 7	III:	= 7 4
	49		49		49
	5		5 7 14		57 6 14 4
					576 0

I: Subdivide the radicant from right to left into pairs of digits and search for the first smaller square in 54. It yields 7 after the equality sign and below 54 the remainder 5.

II: You fetch the 7 and divide by twice what is on the right above (7), that is by 14. This division yields 4, which you write in III above next to the 7.

III: You fetch 6 and 4, multiply 4 by 144 and subtract the product 4·144 = 576. You find the remainder 0. 74 =.

You can expand this schedule. For example, if you want to find the root of the six-digit number 38 56 41, the root will have three digits. You subdivide the number 345641from right to left in groups of two-digits ( if the number has an odd number of digits, the group on the right has only a single digit!) and compute for the first two couples of digits as before. In this way, you find the root of the square , closest to 3856. This number, which this time yields the hundreds and tens, you set equal to Z; you then find the unit number by application of the formula (Z + E)² as before. You divide 124 by twice 62, that is 124, etc.

If the number is a decimal fraction, you start the subdivision from the decimal point to the right and left.. When you fetch the first digit after the decimal point, you place in the result the decimal point.If the calculation does not finish, you fetch, like in division, a zero .  rounded to three decimals is

= 1.414.

The same method can be applied to expressions of a more general form

The method can also be

and to higher order roots. For a cube root, you start from the (Z+E)³.

The calculation of roots of higher order was shown for the first time in German texts in the 15th Century. Stifel speaks in his Arithmetica integra (1544) of the possibility of computing roots of any order and executes the process up to n = 7. In the treatment of logarithms below, you will learn so simple a method, that today higher roots are found by means of it. There exist also tables of powers and roots.

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