A Novel Method for Multiplying Large Numbers

Paper/Pencil/Calculator and Computer Methods

Manual multiplication of large numbers is a slow and tedious method fraught with errors. The use of a calculator especially a ten digit calculator is no different. Extremely large numbers are usually approximated using exponential notation and are of no use if an accurate description is desired. One can always turn to online methods using online calculators which can usually work with some pretty large numbers but, however, there is a limit to what that type of calculator can handle. Although there are probably computers that can handle such numbers they might not be available to the average joe.

However, there is a way around this. Large numbers can indeed be multiplied together without a computer and only require paper, pencil and an ordinary ten digit calculator. What I call PPC. This is the subject of this article. A completely new way of multiplying numbers by arranging the numbers into groups of five and, thus, making it amenable to calculator usage. All that is required is again paper, pencil and a hand held calculator. The article shows the method as well as computer programming methods following the same algorithm which are capable of outputting numbers up to 50 digits. The programs themselves, moreover, are straightforward and can be simply modified to handle more digits.

The Algorithm

We'll take the simple example in Table I, a ten digit number easily handled by a calculator and square and multiply it by another 10 digit number. The two 2 ten digit numbers are bolded on line one and two, separated into groups of 5.

• The numbers, 67890×99999 = 6788932110, are multiplied and stored into two cells and placed on the third row.
• The next product 67890×99999 = 6788932110 is moved over to the left by one and placed on the fourth row.
• The second bold number on line two is multiplied 12345×99999 = 1234487655 stored as two numbers (in groups of five), moved over to the left by one and place on the sixth row.
• The last product, 12345×99999 = 1234487655, is again stored into two cells and moved over to the left by two and placed on the seventh row.
• Finally summation of each column with any carries going over into the next higher column are shown on the last row in red.
• On a calculator the product reads 1.23456789¹⁹. Not bad just by using PPC we have obtained the absolute value instead of an approximation.

The columns are set in such a way that the number of 5 digit numbers follows the pattern of Table II. While similar to a Pascal triangle in structure, the triangle just lays out the odd numbers without adding them as in the Pascal, although the second line is identical to the third in the Pascal. Moreover, the triangle can be subjected to summation of its diagonals just like in the Pascal to provide a sequence, but one that is different from the Pascal triangle. In addition, in this column additive triangle (CAT) each column consists of 1,3,5,7,9,..., all of the odd numbers. This pattern will be evident in Table III as well as the computer coded examples.

Table III shows the layout of the four arrays, the numbers corresponding to the array index starting at 0. The M[] array is set in zigzag fashion, while the P[] array is printed backward because that's the way that the numbers are stored. The number of entries in each column correspond to the 4^th row in Table II.

Multiplication of Larger Numbers using Computer Code

The calculations for multiplying four numbers (in groups of 5) and generating a table of at most 40 digits using JS is stored in computer code I, while for five numbers, generating a table of at most 50 digits, it's stored in computer code II. Two alternative codes where the Tableplace() function is called once for the initial row of each group and the the rest of the rows are generated within this function are: computer code III and computer code IV. This cuts down on the number of times Tableplace() is called. A parameter for printing out the number of rows, Numrpg, is now part of the parameter list. Note that these files are txt files and must first be converted to html files if one wishes to input his or her own numbers.

We can generate a computer program that will give the above results with an even greater number of columns. The way the program is set up is as follows:

• Open the file and input the values for the numbers.
• The program generates eight arrays R[],S[],L[],M[],T[],N[],P[],Carry[] where the parenthesis,[], specify arrays where the appropriate values are stored.
• R[] and S[] arrays contain the original inputted values which are multiplied and stored into an L[] array.
• The L[] array is then subjected to division and truncation in order to separate the numbers within the array into groups of 5.
• The numbers are placed into four arrays, one which will be used to add the numbers (T[]) in the column and three arrays which are converted to strings with leading "0"s for printout to a table (R[],S[],M[]). Note that addition of strings to numbers concatenates them and does not add them according to rules of addition. That's the reason for keeping a copy of the numbers in T[].
• The appropriate numbers in T[] are added to N[] where they are separated into the Carry[] and the number P[], where the Carry[] is added to the next higher number in P[] before being printed in reverse order with leading "0"s if required. This is done in order to output the correct number. This ensures that the large number contains every digit. Any leading zeros at the start of the number are also printed but not required.

Computer Programming Examples

• The first program (code III) multiplies two large numbers as four numbers strung together and outputs the results to a table.
• The second program (code IV) multiplies two large numbers as five groups strung together and outputs the results to a table.

Again the leading "0"s are left in all columns to show that they are part of the number.

This concludes this article.

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