Operations with Multi-Digit Numbers

B. I. Bolshanin, I. A. Reiman

(Presented by the scientific seminar of the departments of mine surveying and geodesy)

In solving a number of problems in astronomy and geodesy, in engineering calculations, and especially in compiling tables, it is often necessary to operate with multi-digit numbers. When arithmometers and electromechanical calculating machines are used, the generally known working methods make it possible to perform computations only with 6–7-digit numbers. When multiplying 8–10-digit numbers by similar numbers, a method is used in which one of the factors is divided into two parts; then the other factor is multiplied by each of these parts. The same procedure is used for exponentiation. Division of multi-digit numbers by multi-digit numbers, and still more the extraction of roots from them, are encountered very rarely and are simply avoided.

In the present work, with reference to the types of calculating machines indicated above, methods are considered for computing, with a large number of significant digits, such elementary operations as multiplication, division, exponentiation, and root extraction. The article gives a justification of the corresponding methods, the order of work, recording schemes, and examples worked out in detail.

The basis of all computational operations is the division of multi-digit numbers into equal groups of 6–7 digits each, so that these numbers will have the form of polynomials

\[ a = \Delta'_1 + \Delta'_2 + \Delta'_3 + \ldots + \Delta'_m, \]

\[ b = \Delta''_1 + \Delta''_2 + \Delta''_3 + \ldots + \Delta''_m, \tag{1} \]

in which the order of adjacent \(\Delta\)’s differs by the number of digits in the group. Before dividing into groups, it is convenient to move the decimal point, placing it before the first significant digit or after it, introducing the corresponding factor \((10^n)\). If the last group of a number is incomplete, then, in order not to disturb the order of computation on the calculating machine, the group is completed with zeros or they are simply understood. In the result obtained this, of course, must be taken into account.

\[ \begin{aligned} a \cdot b={}& \Delta'_1\Delta''_1+\Delta'_1\Delta''_2+\Delta'_1\Delta''_3+\cdots+\Delta'_1\Delta''_{j+1}+\cdots+\Delta'_1\Delta''_n\\ &+\Delta'_2\Delta''_1+\Delta'_2\Delta''_2+\cdots+\Delta'_2\Delta''_{m-1}+\Delta'_2\Delta''_m\\ &+\Delta'_3\Delta''_1+\Delta'_3\Delta''_2+\Delta'_3\Delta''_3+\cdots+\Delta'_3\Delta''_{m-1}+\Delta'_3\Delta''_m\\ &\cdots\\ &+\Delta'_i\Delta''_1+\Delta'_i\Delta''_{m-i+1}+\Delta'_i\Delta''_{m-i+2}+\Delta'_i\Delta''_{m-i+3}+\cdots\\ &+\Delta'_{i+1}\Delta''_1+\Delta'_{i+1}\Delta''_{m-i}+\Delta'_{i+1}\Delta''_{m-i+1}+\Delta'_{i+1}\Delta''_{m-i+2}+\cdots\\ &\cdots\\ &+\Delta'_m\Delta''_1+\Delta'_m\Delta''_2+\Delta'_m\Delta''_3+\cdots+\Delta'_m\Delta''_m \end{aligned} \tag{2} \]

The product of two multi-digit numbers \(a\) and \(b\) in general form.

Multiplication

Bearing in mind that each of the factors is a polynomial of the form (1), let us write the product \(ab\) of two multi-digit numbers \(a\) and \(b\) in the general form (2).

All partial products of the form \(\Delta'_i\Delta''_k\) entering into \(ab\) and having the same sums of indices \((i+k)\) are written one under another—they are of the same order of smallness. In order that, in the product \(ab\), the order of the last significant digit be the same as in each of the factors, in expression (2) one should retain only those terms \(\Delta'_i\Delta''_k\) for which \(i+k \le m+1\). In this case we avoid superfluous operations and may be sure that the product \(ab\) will be obtained with the same relative error as its factors.

Taking, for definiteness, \(m=5\), and discarding products with sum of indices greater than 6, we obtain

\[ \begin{aligned} a\cdot b={}&\Delta'_1\Delta''_1+\Delta'_1\Delta''_2+\Delta'_1\Delta''_3+\Delta'_1\Delta''_4+\Delta'_1\Delta''_5\\ &+\Delta'_2\Delta''_1+\Delta'_2\Delta''_2+\Delta'_2\Delta''_3+\Delta'_2\Delta''_4\\ &+\Delta'_3\Delta''_1+\Delta'_3\Delta''_2+\Delta'_3\Delta''_3\\ &+\Delta'_4\Delta''_1+\Delta'_4\Delta''_2\\ &+\Delta'_5\Delta''_1. \end{aligned} \tag{3} \]

If \(\Delta'\) and \(\Delta''\) are \(r\)-digit numbers, then their products will be \(2r\)-digit. Let us divide the digits of these products into the same groups, of \(r\) digits each, as in the factors. Then from each product \(\Delta'\Delta''\) two groups are formed:

\[ \Delta'_i\Delta''_k=(\Delta'_i\Delta''_k)_1+(\Delta'_i\Delta''_k)_2. \tag{4} \]

The order of the first term in (4) is \(r\) units greater than the order of the second term. Taking (4) into account, we rewrite expression (3) as follows:

\[ \begin{aligned} a\cdot b={}&(\Delta'_1\Delta''_1)_1+(\Delta'_1\Delta''_1)_2+(\Delta'_1\Delta''_2)_2+(\Delta'_1\Delta''_3)_2+(\Delta'_1\Delta''_4)_2\\ &+(\Delta'_1\Delta''_2)_1+(\Delta'_1\Delta''_3)_1+(\Delta'_1\Delta''_4)_1+(\Delta'_1\Delta''_5)_1\\ &+(\Delta'_2\Delta''_1)_1+(\Delta'_2\Delta''_1)_2+(\Delta'_2\Delta''_2)_2+(\Delta'_2\Delta''_3)_2\\ &+(\Delta'_2\Delta''_2)_1+(\Delta'_2\Delta''_3)_1+(\Delta'_2\Delta''_4)_1\\ &+(\Delta'_3\Delta''_1)_1+(\Delta'_3\Delta''_1)_2+(\Delta'_3\Delta''_2)_2\\ &+(\Delta'_3\Delta''_2)_1+(\Delta'_3\Delta''_3)_1\\ &+(\Delta'_4\Delta''_1)_1+(\Delta'_4\Delta''_1)_2\\ &+(\Delta'_4\Delta''_2)_1\\ &+(\Delta'_5\Delta''_1)_1. \end{aligned} \tag{5} \]

The order of the quantities \((\Delta'_i\Delta''_k)_j\) situated in one column, as well as the sum of all the indices in them, is the same. The scheme of writing (5) is final for obtaining the product.

Example. Compute \(\sin 6^\circ\) as the product \(2\sin 3^\circ\) by \(\cos 3^\circ\).

We have:

\[ \begin{aligned} \sin 3^\circ &= 0{,}052\ 335\quad 956\ 242\quad 943\ 832\quad 722\ 118\quad 629\ 609,\\ 2\sin 3^\circ &= 0{,}104\ 671\quad 912\ 485\quad 887\ 665\quad 444\ 237\quad 259\ 218,\\ \cos 3^\circ &= 0{,}998\ 629\quad 534\ 754\quad 573\ 873\quad 784\ 492\quad 058\ 439. \end{aligned} \]

When computing on a “Felix” calculating machine having 8 windows in the revolution counter, 13 windows in the result counter, and 9 levers for setting numbers on the machine, it is expedient to divide the multipliers into groups of six digits each (\(r=6\)). Consequently, the values of \(\Delta'\) and \(\Delta''\) will be:

\[ \begin{aligned} \Delta'_1 &= 104\,671\cdot 10^{-6}, & \Delta''_1 &= 998\,629\cdot 10^{-6},\\ \Delta'_2 &= 912\,485\cdot 10^{-12}, & \Delta''_2 &= 534\,754\cdot 10^{-12},\\ \Delta'_3 &= 887\,665\cdot 10^{-18}, & \Delta''_3 &= 573\,873\cdot 10^{-18},\\ \Delta'_4 &= 444\,237\cdot 10^{-24}, & \Delta''_4 &= 784\,492\cdot 10^{-24},\\ \Delta'_5 &= 259\,218\cdot 10^{-30}, & \Delta''_5 &= 058\,439\cdot 10^{-30}. \end{aligned} \]

Their products are:

\[ \begin{aligned} \Delta'_1\Delta''_1 &= 104\,527\,496\,059\cdot 10^{-12}\\ \Delta'_1\Delta''_2 &= 055\,973\,235\,934\cdot 10^{-18}\\ \Delta'_1\Delta''_3 &= 060\,067\,860\,783\cdot 10^{-24}\\ \Delta'_1\Delta''_4 &= 082\,113\,562\,132\cdot 10^{-30}\\ \Delta'_1\Delta''_5 &= 006\,116\,868\,568\cdot 10^{-36}\\ \Delta'_2\Delta''_1 &= 911\,233\,983\,065\cdot 10^{-18}\\ \Delta'_2\Delta''_2 &= 487\,955\,003\,690\cdot 10^{-24}\\ \Delta'_2\Delta''_3 &= 523\,650\,504\,405\cdot 10^{-30}\\ \Delta'_2\Delta''_4 &= 715\,837\,182\,620\cdot 10^{-36} \end{aligned} \qquad \begin{aligned} \Delta'_3\Delta''_1 &= 886\,448\,011\,285\cdot 10^{-24}\\ \Delta'_3\Delta''_2 &= 474\,682\,409\,410\cdot 10^{-30}\\ \Delta'_3\Delta''_3 &= 509\,406\,976\,545\cdot 10^{-36}\\ \Delta'_4\Delta''_1 &= 443\,627\,951\,073\cdot 10^{-30}\\ \Delta'_4\Delta''_2 &= 237\,557\,512\,698\cdot 10^{-36}\\ \Delta'_5\Delta''_1 &= 258\,862\,612\,122\cdot 10^{-36}. \end{aligned} \]

Or, according to the notation (4),

\[ \begin{aligned} (\Delta'_1\Delta''_1)_1 &= 104\,527\cdot 10^{-6}\\ (\Delta'_1\Delta''_2)_1 &= 055\,973\cdot 10^{-12}\\ (\Delta'_1\Delta''_3)_1 &= 060\,067\cdot 10^{-18}\\ (\Delta'_1\Delta''_4)_1 &= 082\,113\cdot 10^{-24}\\ (\Delta'_1\Delta''_5)_1 &= 006\,116\cdot 10^{-30}\\ (\Delta'_2\Delta''_1)_1 &= 911\,233\cdot 10^{-12}\\ (\Delta'_2\Delta''_2)_1 &= 487\,955\cdot 10^{-18}\\ (\Delta'_2\Delta''_3)_1 &= 523\,650\cdot 10^{-24}\\ (\Delta'_2\Delta''_4)_1 &= 715\,837\cdot 10^{-30} \end{aligned} \qquad \begin{aligned} (\Delta'_1\Delta''_1)_2 &= 496\,059\cdot 10^{-12}\\ (\Delta'_1\Delta''_2)_2 &= 235\,934\cdot 10^{-18}\\ (\Delta'_1\Delta''_3)_2 &= 860\,783\cdot 10^{-24}\\ (\Delta'_1\Delta''_4)_2 &= 562\,132\cdot 10^{-30}\\ (\Delta'_1\Delta''_5)_2 &= 868\,569\cdot 10^{-36}\\ (\Delta'_2\Delta''_1)_2 &= 983\,065\cdot 10^{-18}\\ (\Delta'_2\Delta''_2)_2 &= 003\,690\cdot 10^{-24}\\ (\Delta'_2\Delta''_3)_2 &= 504\,405\cdot 10^{-30}\\ (\Delta'_2\Delta''_4)_2 &= 182\,620\cdot 10^{-36} \end{aligned} \]

and so on. In this case, of the two addends in \(\Delta'_1\Delta''_5\), \(\Delta'_2\Delta''_4\), \(\Delta'_3\Delta''_3\), \(\Delta'_4\Delta''_2\), and \(\Delta'_5\Delta''_1\), only the first should be taken, since the second addend is less than \(1\cdot 10^{-30}\).

Substituting the obtained values \((\Delta_i \Delta'_k)_1\) and \((\Delta_i \Delta'_k)_2\) into (5), we obtain a numerical scheme for computing products of multidigit numbers (scheme 1).

2 sin 3° · cos 3° = 0,104  527  496  059  235  934  860  783  562  132
                         055  973  060  067  082  113  006  117
                         911  233  983  065  003  690  504  405
                                   487  955  523  650  715  837
                                   866  448  011  285  409  410
                                             474  682  509  407
                                             443  627  951  073
                                                       273  552
                                                       258  863

                   1*          2*          3*          4*
────────────────────────────────────────────────────────────────
        sin 6° = 0,104  528  463  267  653  471  399  834  154  802

Column-wise summation should begin with the last column. As a result of the summation, digits are formed that belong to the next (higher) column; they may be placed immediately on the abacus or written under the corresponding digits of the higher column. In the scheme these digits are marked with an asterisk.

Raising to a Power

Raising multidigit numbers to a power may be carried out according to the scheme constructed for multiplication (scheme 1). However, when squaring numbers it is better to use another, more compact, scheme.

Adopting the previous notation, the method of dividing into groups, the number of groups, and the number of digits in the groups, let us construct a scheme for the square of the multidigit number \(a\).

\[ a = \Delta_1 + \Delta_2 + \Delta_3 + \Delta_4 + \Delta_5 \]

\[ a^2 = \Delta_1^2 + \Delta_2^2 + \Delta_3^2 + \Delta_4^2 + \Delta_5^2 + 2\Delta_1\Delta_2 + 2\Delta_1\Delta_3 + 2\Delta_1\Delta_4 + 2\Delta_1\Delta_5 \]

\[ + 2\Delta_2\Delta_3 + 2\Delta_2\Delta_4 + 2\Delta_2\Delta_5 + 2\Delta_3\Delta_4 + 2\Delta_3\Delta_5 + 2\Delta_4\Delta_5 . \]

Retaining the terms with a sum of indices up to six and grouping them into columns according to the principle of equal sums of indices, we obtain

\[ \begin{aligned} a^2 &= \Delta_1^2 + 2\Delta_1\Delta_2 + 2\Delta_1\Delta_3 + 2\Delta_1\Delta_4 + 2\Delta_1\Delta_5 \\[2mm] &\quad + \Delta_2^2 + 2\Delta_2\Delta_3 + 2\Delta_2\Delta_4 \tag{6}\\[2mm] &\quad + \Delta_3^2 . \end{aligned} \]

We shall again divide the obtained squares and products into groups according to the same principle as before (4):

\[ \Delta_i^2 = (\Delta_i^2)_1 + (\Delta_i^2)_2 \]

\[ 2\Delta_i\Delta_k = 2(\Delta_i\Delta_k)_1 + 2(\Delta_i\Delta_k)_2, \]

and, substituting them into expression (6), we finally obtain

\[ \begin{aligned} a^2={}&(\Delta_1^2)_1+(\Delta_1^2)_2+(\Delta_2^2)_1+(\Delta_2^2)_2+(\Delta_3^2)_1\\ &+2(\Delta_1\Delta_2)_1+2(\Delta_1\Delta_2)_2+2(\Delta_1\Delta_3)_2+2(\Delta_1\Delta_4)_2\\ &+2(\Delta_1\Delta_3)_1+2(\Delta_1\Delta_4)_1+2(\Delta_1\Delta_5)_1\\ &+2(\Delta_2\Delta_3)_1+2(\Delta_2\Delta_3)_2\\ &+2(\Delta_2\Delta_4)_1. \end{aligned} \tag{7} \]

The products \(\Delta_i\Delta_k\) may be entered in the scheme without doubling, but when summing they must first be added together and, after doubling the result, the \(\Delta_i^2\), which for convenience in computation have been assigned to the first row of the scheme, should be added.

Example. Compute the square of \(\sin 6^\circ\):

\[ \Delta_1=104\,528\cdot 10^{-6} \]

\[ \Delta_2=463\,267\cdot 10^{-12} \]

\[ \Delta_3=653\,471\cdot 10^{-18} \]

\[ \Delta_4=399\,834\cdot 10^{-24} \]

\[ \Delta_5=154\,802\cdot 10^{-30} \]

\[ \begin{array}{ll} \Delta_1^2=010\,926\quad 102\,784\cdot 10^{-12} & \Delta_2^2=214\,616\quad 313\,289\cdot 10^{-24} \\[4pt] \Delta_1\Delta_2=048\,424\quad 372\,976\cdot 10^{-18} & \Delta_2\Delta_3=302\,731\quad 549\,757\cdot 10^{-30} \\[4pt] \Delta_1\Delta_3=068\,306\quad 016\,688\cdot 10^{-24} & (\Delta_2\Delta_4)_1=185\,230\cdot 10^{-30} \\[4pt] \Delta_1\Delta_4=041\,793\quad 848\,352\cdot 10^{-30} & (\Delta_3^2)_1=427\,024\cdot 10^{-30} \\[4pt] (\Delta_1\Delta_5)_1=016\,181\cdot 10^{-30} & \end{array} \]

\[ \begin{array}{rllllllll} \sin^2 6^\circ={}&0{,}010\,926&102\,784&214\,616&313\,289&427\,024\\ &&048\,424&372\,976&016\,688&848\,352&\\ &&068\,306&041\,793&016\,181&\\ &&&&302\,731&549\,757&\\ &&&&&185\,230& \end{array} \qquad \text{Scheme II}^{1)} \]

\[ \begin{array}{rrrrrr} &&1&1&3&\\[-2pt] \hline \sin^2 6^\circ={}&0{,}010\,926&199\,633&097\,181&035\,716&626\,064 \end{array} \]

To obtain any power of a multi-digit number, one may use the binomial series

\[ a^n=(a_0+\Delta a)^n=a_0^n\left(1+\frac{\Delta a}{a_0}\right)^n= \]

\[ =a_0^n\left[ 1+\frac{n}{1!}\frac{\Delta a}{a_0} +\frac{n(n-1)}{2!}\left(\frac{\Delta a}{a_0}\right)^2 +\frac{n(n-1)(n-2)}{3!}\left(\frac{\Delta a}{a_0}\right)^3+\ldots \right], \]

where the first group of digits of the number \(a\) may be taken as \(a_0\). On the other hand,

\({}^{1)}\) The digits of these rows, when summing, should be doubled; they should be added in the first place.

any power can be obtained by applying the rules of multiplication and squaring.

The process of squaring a 30–36-digit number on an arithmometer takes 8–10 minutes, while multiplication of such numbers takes about one and a half times longer.

Division

Division of multi-digit numbers by multi-digit numbers is a more complicated process than multiplication and squaring. Division of multi-digit numbers by 6–7-digit numbers on an arithmometer is performed comparatively simply. It is carried out by entering the dividend on the result carriage in several operations. Thus, after setting the divisor on the levers, the dividend is entered on the result counter. The revolution counter will give the quotient of the division. In one operation one can obtain 7–8 digits. After this, one should set the incomplete digits of the dividend on the levers and, after clearing all digits from the result counter, place it in the left part of the counter. Setting the dividend again and clearing the digits from the revolution counter, one continues entering the divisor. Thus, a 36-digit number can be divided by a six-digit number in 5–6 operations.

Division of multi-digit numbers by multi-digit numbers directly on an ordinary arithmometer is impossible, but it can be replaced by comparatively simple operations: division of a multi-digit number by a 5–7-digit number, multiplication, and raising multi-digit numbers to a power. Let us consider the procedure.

Suppose it is required to find the quotient from dividing a multi-digit number \(a\) by a multi-digit number \(b\).

\[ \frac{a}{b} = \frac{a}{b_0+\Delta b} = \frac{a}{b_0} \left(1+\frac{\Delta b}{b_0}\right)^{-1} = \tag{8} \]

\[ = \frac{a}{b_0} \left[ 1-\frac{\Delta b}{b_0} + \left(\frac{\Delta b}{b_0}\right)^2 - \left(\frac{\Delta b}{b_0}\right)^3 +\ldots \right] = \frac{a}{b_0}N, \]

where \(b_0\) is an approximate value of the number \(b\)—the first six or seven significant digits.

If the first group of the number \(b\) is taken as \(b_0\), then in expanding into a series one should stop at the power \(m-1\) (\(m\) is the number of groups of the divisor).

Example. Find \(\operatorname{tg} 6^\circ\) as the quotient from dividing \(\sin 6^\circ\) by \(\cos 6^\circ\).

\[ \begin{aligned} a=\sin 6^\circ &= 0{,}104\ 528\ 463\ 267\ 653\ 471\ 399\ 834\ 154\ 802,\\ b=\cos 6^\circ &= 0{,}994\ 521\ 895\ 368\ 273\ 336\ 922\ 691\ 944\ 981,\\ b_0 &= 0{,}994\ 521,\\ \Delta b &= 0{,}000\ 000\ 895\ 368\ 273\ 336\ 922\ 691\ 944\ 981,\\ \frac{\Delta b}{b_0} &= 0{,}000\ 000\ 900\ 301\ 022\ 639\ 967\ 071\ 529\ 893. \end{aligned} \]

\[ \begin{aligned} \left(\frac{\Delta b}{b_0}\right)^2 &= 0{,}000\ 000\ 000\ 000\ 810\ 541\ 890\ 601\ 000\ 513\\ &\phantom{= 0{,}000\ 000\ 000\ 000\ 810\ 541\ }20\ 381\quad 914\ 339\\ &\phantom{= 0{,}000\ 000\ 000\ 000\ 810\ 541\ 890\ 601\ }870\ 655 \end{aligned} \]

\[ \left(\frac{\Delta b}{b_0}\right)^2 = 0{,}000\ 000\ 000\ 000\ 810\ 541\ 931\ 366\ 570\ 501 \]

\[ \begin{aligned} \left(\frac{\Delta b}{b_0}\right)^3 &= 0{,}000\ 000\ 000\ 000\ 000\ 000\ 729\ 730\ 872\ 841\\ &\phantom{= 0{,}000\ 000\ 000\ 000\ 000\ 000\ 729\ 730\ }838\ 510\\ &\phantom{= 0{,}000\ 000\ 000\ 000\ 000\ 000\ 729\ 730\ }18\ 351 \end{aligned} \]

\[ \left(\frac{\Delta b}{b_0}\right)^3 = 0,000\ 000\quad 000\ 000\quad 000\ 000\quad 729\ 731\quad 729\ 702 \]

\[ \left(\frac{\Delta b}{b_0}\right)^4 = 0,000\ 000\quad 000\ 000\quad 000\ 000\quad 000\ 000\quad 656\ 978 \]

\[ N = 0,999\ 999\quad 099\ 699\quad 787\ 901\quad 234\ 563\quad 967\ 884 \]

\[ \frac{a}{b_0} = 0,105\ 104\quad 329\ 891\quad 126\ 955\quad 991\ 712\quad 784\ 953 \]

\[ \begin{aligned} \frac{a}{b_0}\cdot N &= 0,105\ 103\quad 894\ 896\quad 763\ 696\quad 546\ 704\quad 509\ 552\\ &\phantom{= 0,105\ 103\quad 894\ 896\quad} 010\ 478\quad 082\ 811\quad 024\ 653\quad 101\ 729\\ &\phantom{= 0,105\ 103\quad} 329\ 890\quad 670\ 109\quad 802\ 809\quad 448\ 791\\ &\phantom{= 0,105\ 103\quad 894\ 896\quad} 032\ 889\quad 259\ 921\quad 077\ 381\\ &\phantom{= 0,105\ 103\quad 894\ 896\quad} 126\ 954\quad 873\ 045\quad 286\ 545\\ &\phantom{= 0,105\ 103\quad 894\ 896\quad 763\ 696\quad} 012\ 657\quad 100\ 028\\ &\phantom{= 0,105\ 103\quad 894\ 896\quad 763\ 696\quad} 991\ 711\quad 008\ 288\\ &\phantom{= 0,105\ 103\quad 894\ 896\quad 763\ 696\quad 546\ 704\quad} 098\ 873\\ &\phantom{= 0,105\ 103\quad 894\ 896\quad 763\ 696\quad 546\ 704\quad} 748\ 953 \end{aligned} \]

\[ \begin{array}{ccccc} - & 1 & 1 & 3 & 2 \end{array} \]

\[ \tan 6^\circ = 0,105\ 104\quad 235\ 265\quad 676\ 462\quad 511\ 502\quad 380\ 140 \]

Remark. In mass computations the zeros in \(\left(\frac{\Delta b}{b_0}\right)^2\), \(\left(\frac{\Delta b}{b_0}\right)^3\), and \(\left(\frac{\Delta b}{b_0}\right)^4\) should not be written, as was done when multiplying \(\frac{a}{b_0}\) by \(N\).

Extraction of roots

The rules and methods of operations with multi-digit numbers considered above allow us to solve comparatively simply the problem of extracting roots of any degree with a high degree of accuracy by means of an arithmometer. To solve this problem we again use the binomial series:

\[ \sqrt[n]{1+x} = 1+\frac{1}{n}x+\frac{1-n}{n\cdot 2n}x^2 +\frac{(1-n)(1-2n)}{n\cdot 2n\cdot 3n}x^3+ \]

\[ +\frac{(1-n)(1-2n)(1-3n)}{n\cdot 2n\cdot 3n\cdot 4n}x^4+\ldots . \tag{9} \]

Take, for example, \(n=2\) (the square root)

\[ \sqrt{a}=\sqrt{a_0+\Delta a}=\sqrt{a_0}\left(1+\frac{\Delta a}{a_0}\right)^{\frac12}, \]

where

\[ \left(1+\frac{\Delta a}{a_0}\right)^{\frac12} =1+\frac12\frac{\Delta a}{a_0} -\frac18\left(\frac{\Delta a}{a_0}\right)^2 +\frac{1}{16}\left(\frac{\Delta a}{a_0}\right)^3- \]

\[ -\frac{5}{128}\left(\frac{\Delta a}{a_0}\right)^4 +\frac{7}{256}\left(\frac{\Delta a}{a_0}\right)^5 -\frac{21}{1024}\left(\frac{\Delta a}{a_0}\right)^6+\cdots \tag{10} \]

\[ + \frac{33}{2048}\left(\frac{\Delta a}{a_0}\right)^7 - \frac{429}{32768}\left(\frac{\Delta a}{a_0}\right)^8 + \frac{75}{65536}\left(\frac{\Delta a}{a_0}\right)^9 - \]

\[ - \frac{2431}{262144}\left(\frac{\Delta a}{a_0}\right)^{10} + \frac{4199}{524288}\left(\frac{\Delta a}{a_0}\right)^{11} - \frac{29393}{4194304}\left(\frac{\Delta a}{a_0}\right)^{12} + \]

\[ + \frac{676039}{109051904}\left(\frac{\Delta a}{a_0}\right)^{13} - \frac{16900975}{3053453312}\left(\frac{\Delta a}{a_0}\right)^{14} + \ldots \]

In order that the amount of computational work in solving this problem be as small as possible, attention should be paid to the choice of the number \(a_0\):

1) the root of it must be extracted without remainder;

2) the number \(a_0\) should be as large as possible, so that the ratio \(\dfrac{\Delta a}{a_0}\) is smaller and the convergence of the series faster;

3) if the number \(a_0\) has many digits (more than 6–7), then division by it becomes more complicated and the computations in finding \(\dfrac{\Delta a}{a_0}\) increase, but in individual cases this may on the whole be advantageous.

Example. Find the value of \(\sqrt{0.75}\) with 60 significant digits.

Let us take \(a_0 = 0.749956\). Then \(\sqrt{a_0} = 0.866\) and \(\Delta a = 0.000044\). Direct computation shows that to determine \(\sqrt{0.75}\) with 60 digits it is necessary to take 14 terms of series (10). As a result of the computation we obtained 62 correct digits (see pp. 27–29).

The computation of roots by such a scheme takes, in comparison with squaring and multiplication, a great deal of time. To compute a root with accuracy up to \(10^{-60}\)—\(10^{-70}\), with some practice and knowledge of the order of the computations, 10–12 hours will be required. It is obvious that in computing roots with lower accuracy, when a smaller number of terms of the series will have to be taken into account and they must be computed with lower accuracy, the expenditure of time will be considerably reduced.

The computation of square roots can also be carried out by another scheme. Knowing that extraction of a root is the operation inverse to raising to a power, for solving this problem one can make use of the squaring scheme (scheme II). Let us rewrite expression (7) in the following form:

\[ a^2 = (\Delta_1^2)_1 + (\Delta_1^2)_2 + (\Delta_2^2)_1 + (\Delta_2^2)_2 + (\Delta_3^2)_1 + \ldots \]

\[ + 2(\Delta_1\Delta_2)_1 + 2(\Delta_1\Delta_2)_2 + 2(\Delta_1\Delta_3)_2 + 2(\Delta_1\Delta_4)_2 + \ldots \]

\[ + 2(\Delta_1\Delta_3)_1 + 2(\Delta_1\Delta_4)_1 + 2(\Delta_1\Delta_5)_1 + \ldots \]

\[ + 2(\Delta_2\Delta_3)_1 + 2(\Delta_2\Delta_3)_2 + \ldots \]

\[ + 2(\Delta_2\Delta_4)_1 + \ldots \]

\[ a^2 = \delta_1 + \delta_2 + \delta_3 + \delta_4 + \delta_5 + \ldots \]

In extracting the root in this scheme, \(\delta_1, \delta_2, \delta_3,\ldots\) will be known to us. The problem will consist in finding \(\Delta_1, \Delta_2, \Delta_3\ldots\). The sought \(\Delta_1\) can

\[ \begin{aligned} 1+\varkappa_1\frac{\Delta a}{a_0} &= 1{,}000\ 029\quad 335\ 054\quad 323\ 186\quad 960\ 302\quad 671\ 031\quad 367\ 173\quad 540\ 847\quad 729\ 723\quad 477\ 697\quad 358\ 245\quad 017\ 0,\\ \varkappa_3\left(\frac{\Delta a}{a_0}\right)^3 &=\phantom{1{,}000\ 029\quad 335\ 054\quad 323\ 186\quad 960\ 302\quad}12\ 622\quad 673\ 206\quad 411\ 615\quad 701\ 382\quad 742\ 851\quad 568\ 373\quad 754\ 990\quad 316\ 929\quad 526\ 9,\\ \varkappa_5\left(\frac{\Delta a}{a_0}\right)^5 &=\phantom{1{,}000\ 029\quad 335\ 054\quad 323\ 186\quad 960\ 302\quad 671\ 031\quad}19\quad 008\ 267\quad 581\ 672\quad 929\ 554\quad 316\ 402\quad 596\ 402\quad 991\ 372\quad 713\ 1,\\ \varkappa_7\left(\frac{\Delta a}{a_0}\right)^7 &=\phantom{1{,}000\ 029\quad 335\ 054\quad 323\ 186\quad 960\ 302\quad 671\ 031\quad 367\ 173\quad}38\ 556\quad 911\ 156\quad 232\ 888\quad 377\ 884\quad 372\ 932\quad 356\ 7,\\ \varkappa_9\left(\frac{\Delta a}{a_0}\right)^9 &=\phantom{1{,}000\ 029\quad 335\ 054\quad 323\ 186\quad 960\ 302\quad 671\ 031\quad 367\ 173\quad 540\ 847\quad}89\quad 862\ 426\quad 880\ 291\quad 015\ 316\quad 345\ 3,\\ \varkappa_{11}\left(\frac{\Delta a}{a_0}\right)^{11} &=\phantom{1{,}000\ 029\quad 335\ 054\quad 323\ 186\quad 960\ 302\quad 671\ 031\quad 367\ 173\quad 540\ 847\quad 729\ 723\quad}227\ 071\quad 053\ 034\quad 951\ 7,\\ \varkappa_{13}\left(\frac{\Delta a}{a_0}\right)^{13} &=\phantom{1{,}000\ 029\quad 335\ 054\quad 323\ 186\quad 960\ 302\quad 671\ 031\quad 367\ 173\quad 540\ 847\quad 729\ 723\quad 477\ 697\quad}605\quad 003\ 8. \end{aligned} \]

\[ \begin{aligned} \Sigma_1 &= 1{,}000\ 029\quad 335\ 054\quad 335\ 809\quad 033\ 527\quad 090\ 911\quad 638\ 786\quad 124\ 409\quad 710\ 895\quad 314\ 722\quad 108\ 441\quad 914\ 5,\\ \varkappa_2\left(\frac{\Delta a}{a_0}\right)^2 &= 0{,}000\ 000\quad 000\ 430\quad 272\ 706\quad 072\ 164\quad 984\ 770\quad 281\ 649\quad 903\ 673\quad 241\ 263\quad 523\ 551\quad 054\ 556\quad 454\ 9,\\ \varkappa_4\left(\frac{\Delta a}{a_0}\right)^4 &=\phantom{0{,}000\ 000\quad 000\ 430\quad 272\ 706\quad}462\ 836\quad 503\ 976\quad 659\ 206\quad 390\ 808\quad 832\ 008\quad 684\ 898\quad 130\ 335\quad 267\ 1,\\ \varkappa_6\left(\frac{\Delta a}{a_0}\right)^6 &=\phantom{0{,}000\ 000\quad 000\ 430\quad 272\ 706\quad 072\ 164\quad 984\ 770\quad}836\quad 412\ 843\quad 147\ 073\quad 528\ 703\quad 605\ 659\quad 124\ 006\quad 695\ 7,\\ \varkappa_8\left(\frac{\Delta a}{a_0}\right)^8 &=\phantom{0{,}000\ 000\quad 000\ 430\quad 272\ 706\quad 072\ 164\quad 984\ 770\quad 281\ 649\quad}1\quad 837\ 987\quad 260\ 366\quad 423\ 509\quad 525\ 047\quad 245\ 9,\\ \varkappa_{10}\left(\frac{\Delta a}{a_0}\right)^{10} &=\phantom{0{,}000\ 000\quad 000\ 430\quad 272\ 706\quad 072\ 164\quad 984\ 770\quad 281\ 649\quad 903\ 673\quad}4\ 481\quad 402\ 596\quad 049\ 480\quad 200\ 0,\\ \varkappa_{12}\left(\frac{\Delta a}{a_0}\right)^{12} &=\phantom{0{,}000\ 000\quad 000\ 430\quad 272\ 706\quad 072\ 164\quad 984\ 770\quad 281\ 649\quad 903\ 673\quad 241\ 263\quad}11\quad 656\ 997\quad 933\ 0,\\ \varkappa_{14}\left(\frac{\Delta a}{a_0}\right)^{14} &=\phantom{0{,}000\ 000\quad 000\ 430\quad 272\ 706\quad 072\ 164\quad 984\ 770\quad 281\ 649\quad 903\ 673\quad 241\ 263\quad 523\ 551\quad}317\ 0. \end{aligned} \]

\[ \begin{aligned} \Sigma_2 &= 0{,}000\ 000\quad 000\ 430\quad 272\ 706\quad 535\ 001\quad 489\ 583\quad 353\ 701\quad 234\ 542\quad 865\ 823\quad 940\ 225\quad 510\ 424\quad 023\ 3,\\ \Sigma_1-\Sigma_2 &= 1{,}000\ 029\quad 334\ 624\quad 063\ 102\quad 498\ 525\quad 601\ 331\quad 335\ 084\quad 839\ 956\quad 844\ 001\quad 374\ 496\quad 568\ 017\quad 886\ 2. \end{aligned} \]

\[ \sqrt{0{,}75} = \sqrt{a_0}\left(1+\frac{\Delta a}{a_0}\right)^{\frac12} = 0{,}866(\Sigma_1-\Sigma_2), \]

\[ \sqrt{0{,}75} = 0{,}866\ 025\quad 403\ 784\quad 438\ 646\quad 763\ 723\quad 170\ 752\quad 936\ 183\quad 471\ 402\quad 626\ 905\quad 190\ 314\quad 027\ 938\quad 489. \]

Note. By \(\varkappa_i\) are denoted the coefficients of \(\left(\dfrac{\Delta a}{a_0}\right)^i\) (formula 10).

be obtained by any of the methods known to us: on an arithmometer, from tables, or, finally, by computation on paper. Knowing \(\Delta_1\), we can obtain \(\Delta_1^2\), which in the scheme is written as

\[ (\Delta_1^2)_1+(\Delta_1^2)_2. \]

Then from the second column of the scheme we obtain \(\Delta_2\).

\[ (\Delta_1^2)_2+2(\Delta_1\Delta_2)_1=\delta_2; \]

\[ 2\Delta_2=\frac{\delta_2-(\Delta_1^2)_2}{\Delta_1}. \]

Squaring \(\Delta_2\) and substituting it into the scheme, from the third column we can obtain \(\Delta_3\).

\[ 2\Delta_3=\frac{\delta_3-(\Delta_2^2)_1-2(\Delta_1\Delta_2)_2}{\Delta_1}. \]

Then we find

\[ 2\Delta_4=\frac{\delta_4-(\Delta_2^2)_2-2(\Delta_1\Delta_3)_2-2(\Delta_2\Delta_3)_1}{\Delta_1} \]

and so on.

This method of computation is incomparably simpler than the one given above, but it has a substantial shortcoming—the values \(\Delta_1, \Delta_2, \Delta_3,\ldots\) from the expressions written above may be found with errors of the order of one or two units of the last digit of the group. However, with the appropriate computational skill this shortcoming will not present a serious obstacle.

The computation of roots of any index is similar to the computation of the square root. Below are series for obtaining roots of the 3rd, 4th, and 5th degrees.

\[ \sqrt[3]{1+x}=1+\frac{1}{3}x-\frac{1}{9}x^2+\frac{5}{81}x^3-\frac{10}{243}x^4+ \]

\[ +\frac{22}{729}x^5-\frac{154}{6561}x^6+\ldots \]

\[ \sqrt[4]{1+x}=1+\frac{1}{4}x-\frac{3}{32}x^2+\frac{7}{128}x^3-\frac{77}{2048}x^4+ \]

\[ +\frac{231}{8192}x^5-\frac{1461}{65536}x^6+\ldots \]

\[ \sqrt[5]{1+x}=1+\frac{1}{5}x-\frac{2}{25}x^2+\frac{6}{125}x^3-\frac{21}{625}x^4+ \]

\[ +\frac{399}{15625}x^5-\frac{1596}{78125}x^6+\ldots \]

In general, all operations with multi-digit numbers reduce to multiplication and squaring of these numbers and to division of them by 6–7-digit numbers.

Tables of Trigonometric Functions

The methods of operating with multi-digit numbers proposed by the authors of the present work were applied to a number of computations in the compilation of geodetic tables. One of the largest ra-

calculations of this kind was the computation by the authors of tables of sines and cosines with 27 decimal places, which are given in Appendices 1 and 2.

The values of the trigonometric functions were computed by the formulas for the sum and difference of arguments and by the formulas for double and triple angles.

\[ \sin (x \pm h)=\sin x \cdot \cos h \pm \cos x \cdot \sin h, \]

\[ \cos (x \pm h)=\cos x \cdot \cos h \mp \sin x \cdot \sin h, \]

\[ \sin 2x = 2 \sin x \cos x, \]

\[ \cos 2x = \cos^2 x - \sin^2 x, \]

\[ \sin 3x = \sin x (4\cos^2 x - 1), \]

\[ \cos 3x = \cos x (1 - 4\sin^2 x). \]

When the cosines of small angles were used in the computations, their complements to unity were taken.

\[ \cos h = 1 - \varepsilon_h;\quad \cos x = 1 - \varepsilon_x. \]

\[ \sin (x + h)=\sin x+\sin h-\varepsilon_h \sin x-\varepsilon_x \sin h. \]

\[ \cos^2 h=(1-\varepsilon_h)^2=(1+\varepsilon_h^2)-2\varepsilon_h. \]

The amount of computation in this case is reduced by a factor of 1.5–2. By direct computation from series, values were obtained for the sines and cosines of \(1^\circ\), \(1'\), \(1''\), \(10^\circ\), \(10'\), \(10''\), \(6'\), \(15'\), \(40'\), \(50'\), and \(17^\circ\), with accuracy not worse than \(1 \cdot 10^{-33}\). In addition, \(\sqrt{2}\) was computed with accuracy to \(1 \cdot 10^{-42}\), and \(\sqrt{0.75}\) with accuracy to \(1 \cdot 10^{-66}\), giving the sine and cosine of \(45^\circ\), the cosine of \(30^\circ\), and the sine of \(60^\circ\).

The computations began with obtaining the values of \(\frac{1}{\rho}\) and \(\frac{1}{n!}\). The value \(\frac{1}{\rho^\circ}\), equal to

\[ \frac{\pi}{180}, \]

was obtained by setting the number \(\pi\) on the result carriage of the calculating machine. The results of computing the basic data are given in Appendices 3–8.

Thus, using the methods of work described above, one can carry out computations on a calculating machine with any number of significant digits.

APPENDIX 1

TABLES

27-DIGIT SINES AND COSINES AND THEIR SQUARES

AT INTERVALS OF \(1^\circ\)

Table 1

Sin \(X\)

Cos \(X\)

Continuation of Table 1

$\operatorname{Sin} X$

$\operatorname{Cos} X$

Table 2

\[ \sin^2 X \]

\[ \cos^2 X \]

Continuation of Table 2

$\mathrm{Sin}^2 X$

$\mathrm{Cos}^2 X$

APPENDIX 2

TABLES

OF 27-DIGIT SINES AND COSINES AT INTERVALS OF 1′ WITHIN ONE DEGREE AND AT INTERVALS OF 1″ WITHIN ONE MINUTE

Table 3

Sin \(X\)

Continuation of Table 3

Cos \(X\)

Continuation of Table 4

Cos \(X\)

Continuation of Table 4

\(\mathrm{Cos}\,X\)

APPENDIX 3

\[ \begin{aligned} \pi &= 3,141\ 592\quad 653\ 589\quad 793\ 238\quad 462\ 643\quad 383\ 279\quad 502\ 881\quad 197\ 169\\ &\qquad 399\ 375\quad 105\ 820\quad 974\ 944\quad 592\ 307\quad 816\ 406\quad 286\ 208\quad 998\ 628\\[4pt] \sqrt{2} &= 1,414\ 213\quad 562\ 373\quad 095\ 048\quad 801\ 688\quad 724\ 209\quad 698\ 087\quad 569\ 671\\ &\qquad 875\ 376\quad 948\ 073\quad 176\ 679\quad 738\\[4pt] \sqrt{3} &= 1,732\ 050\quad 807\ 568\quad 877\ 293\quad 527\ 446\quad 341\ 505\quad 872\ 366\quad 942\ 805\\ &\qquad 253\ 810\quad 380\ 628\quad 055\ 806\quad 979\ 452\\[4pt] \frac{1}{\rho^\circ} &= 0,017\ 453\quad 292\ 519\quad 943\ 295\quad 769\ 236\quad 907\ 684\quad 886\ 127\quad 134\ 428\\ &\qquad 718\ 885\quad 417\ 254\quad 560\ 971\quad 914\ 402\\[4pt] \frac{1}{\rho'} &= 0,000\ 290\quad 888\ 208\quad 665\ 721\quad 596\ 153\quad 948\ 461\quad 414\ 768\quad 785\ 573\\ &\qquad 811\ 981\quad 423\ 620\quad 909\ 349\quad 531\ 906\quad 694\\[4pt] \frac{1}{\rho''} &= 0,000\ 004\quad 848\ 136\quad 811\ 095\quad 359\ 935\quad 899\ 141\quad 023\ 579\quad 479\ 759\\ &\qquad 563\ 533\quad 023\ 727\quad 015\ 155\quad 825\ 531\quad 778\\[4pt] \sin 1^\circ &= 0,017\ 452\quad 406\ 437\quad 283\ 512\quad 819\ 418\quad 978\ 516\quad 316\ 192\quad 472\ 252\\ &\qquad 720\ 307\quad 139\ 642\quad 68\\[4pt] \cos 1^\circ &= 0,999\ 847\quad 695\ 156\quad 391\ 239\quad 157\ 011\quad 558\ 813\quad 914\ 851\quad 692\ 740\\ &\qquad 310\ 583\quad 185\ 939\quad 66\\[4pt] \sin 1' &= 0,000\ 290\quad 888\ 204\quad 563\ 424\quad 596\ 374\quad 297\ 415\quad 740\ 000\quad 929\ 832\\ &\qquad 946\ 872\quad 185\\[4pt] \cos 1' &= 0,999\ 999\quad 957\ 692\quad 025\ 327\quad 951\ 262\quad 487\ 173\quad 335\ 823\quad 531\ 849\\ &\qquad 468\ 518\quad 206\\[4pt] \sin 1'' &= 0,000\ 004\quad 848\ 136\quad 811\ 076\quad 367\ 820\quad 079\ 090\quad 940\ 916\quad 766\ 269\\ &\qquad 913\ 530\quad 127\\[4pt] \cos 1'' &= 0,999\ 999\quad 999\ 988\quad 247\ 784\quad 730\ 474\quad 076\ 217\quad 925\ 179\quad 115\ 270\\ &\qquad 314\ 920\quad 592\\[4pt] \frac{1}{\rho^g} &= 0,015\ 707\quad 963\ 267\quad 948\ 966\quad 192\ 313\quad 216\ 916\quad 397\ 514\quad 420\ 985\\ &\qquad 846\ 996\quad 875\ 529\quad 104\ 874\quad 722\ 961\\[4pt] \frac{1}{\rho^c} &= 0,000\ 157\quad 079\ 632\quad 679\ 489\quad 661\ 923\quad 132\ 169\quad 163\ 975\\[4pt] \frac{1}{\rho^{cc}} &= 0,000\ 001\quad 570\ 796\quad 326\ 794\quad 896\ 619\quad 231\ 321\quad 691\ 640 \end{aligned} \]

Appendix 4

Appendix 5

APPENDIX 6

APPENDIX 7

APPENDIX 8