Multiply 1

This program performs arbitrary-precision integer multiplication by accepting two multi-digit numbers as strings and computing their product digit by digit. It implements the standard long-multiplication algorithm: for each pair of digit positions, it accumulates partial products into an array, then propagates carries leftward. The result array is dimensioned to the sum of the lengths of both input strings plus one, which is the maximum number of digits any product can require. A leading-zero suppression check at line 210 skips the first element of the array if it is zero, preventing a spurious leading zero in the output.

Program Analysis

Program Structure

The program is organized into four logical phases:

1. Setup (lines 10–50): REMs for documentation, screen clear, and title display with a decorative underline of block graphics.
2. Input (lines 60–70): Two numbers are read as strings into x$ and y$.
3. Calculation (lines 80–180): The multiplication array is allocated, nested loops perform digit-by-digit multiplication with carry propagation.
4. Output and replay (lines 190–280): Result is printed, a repeat prompt is handled, and the program is saved with an auto-run directive.

Algorithm: Long Multiplication

The core routine allocates a numeric array a at line 80 with LEN x$ + LEN y$ + 1 elements — one more than the maximum possible product width, providing a safe carry-overflow buffer at position 0 (though in practice the result is indexed from 1).

The nested loops at lines 90–180 iterate over every pair of digit positions (m, n), right-to-left in both operands. The positional index c = m + n targets the correct array cell. Each iteration:

• Multiplies the two individual digits (extracted via VAL x$(m) and VAL y$(n)).
• Adds the result to the accumulator cell a(c).
• Computes the carry i = INT(a(c)/10) and strips it from a(c) with a(c) - i*10.
• Adds the carry to a(c-1), propagating it leftward by one position.

Carry is only propagated one position per iteration. Because digits are at most 9 and the accumulated value in any cell before carry extraction can grow large during the inner loop, full correctness relies on BASIC’s floating-point arithmetic not losing precision. For the digit magnitudes and string lengths likely encountered in practice this is safe, but very long operands could theoretically accumulate values exceeding integer precision in a cell before the inner loop finishes.

Key BASIC Idioms

Variable Usage

Notable Techniques

• Using string input allows multiplication of integers far beyond the range of a single floating-point variable, limited only by available memory and array size.
• The array is over-allocated by one (+1 in the DIM) to handle the edge case where carries might propagate all the way to the most-significant position; however, indexing starts at 1 so element 0 does not exist in Sinclair BASIC — the extra element is actually at the high end of the array, not a guard at position zero. Carry from a(1) would be written to a(0), which would cause an index error. In practice, the leading cell rarely overflows for reasonable inputs.
• The r variable provides minimal leading-zero suppression but only removes at most one leading zero; if both inputs are small enough that a(1)=0 and a(2)=0, a spurious zero could still appear.

Potential Bugs and Anomalies

• Carry to a(0): If a carry ever propagates from a(1) to a(c-1) where c=1, BASIC will attempt to write to a(0), which is out of bounds and will generate a subscript error. The +1 in the DIM does not protect this end of the array.
• Single-key replay check: Lines 250–260 read INKEY$ twice in sequence. If the key is released between the two reads, line 260 may see an empty string and fall through to STOP even if “y” was pressed. The pattern is functional but slightly fragile on fast hardware.
• No input validation: Non-digit characters in x$ or y$ will cause VAL to return 0 silently, producing an incorrect result without any error message.

   10 REM    This program will enable you  to multiply any two multi-digit numbers together and obtain an  accurate answer
   20 REM  BY G. F. CHAMBERS An expansion of a routine foundin the June 1983 issue of "YOUR COMPUTER" magazine,    page 54   
   30 CLS 
   40 PRINT TAB 8;"MULTIPLICATION"
   50 PRINT TAB 8;"\''\''\''\''\''\''\''\''\''\''\''\''\''\''"
   60 INPUT "Enter the first number ";x$
   70 INPUT "Enter the second number ";y$
   80 DIM a(LEN x$+LEN y$+1)
   90 FOR m=LEN x$ TO 1 STEP -1
  100 FOR n=LEN y$ TO 1 STEP -1
  110 LET c=m+n
  120 LET r=1
  130 LET b=VAL x$(m)
  140 LET a(c)=VAL y$(n)*b+a(c)
  150 LET i=INT (a(c)/10)
  160 LET a(c)=a(c)-(i*10)
  170 LET a(c-1)=a(c-1)+i
  180 NEXT n: NEXT m
  190 PRINT AT 5,0
  200 PRINT x$;" x ";y$;" = "''
  210 IF a(1)=0 THEN LET r=2
  220 FOR p=r TO (LEN x$+LEN y$)
  230 PRINT a(p);: NEXT p
  240 PRINT AT 15,1;"Another multiplication? (y/n)"
  250 IF INKEY$="" THEN GO TO 250
  260 IF INKEY$="y" OR INKEY$="Y" THEN RUN 
  270 STOP 
  280 SAVE "MULTIPLY" LINE 20