To multiply two numbers, in specific locations draw a bunch of lines for each digit, count the intersections, carry if needed, and you have a product.

    Multiplication by lines permits one to multiply two numbers by drawing a bunch of lines in specific positions for each digit of each numbers being multiplied, counting the numbers of intersections of the lines, and completing the multiplication completely by addition.

    On this page the words multiplier (the number "doing" the multiplication) and multiplicand (the number being multiplied by the multiplier) are not used. Their positions may be switched at will by the commutative property of multiplication.

    Instructions are provided in text form just below and with the image.

1. Write a bunch of titled lines for each digit (left to right) in the number on the left, placing each bunch starting at the bottom and putting the next bunch on top of that.
    
2. Write a bunch of titled lines for each digit (left to right) in the number on the right, placing each bunch starting at the top and putting the next bunch on bottom of that.
    
   c., d. and e.
       For a 2-digit by 2-digit multiplication the are four area of intersections corresponding to the four multiplications required.
       Count and perhaps mark (perhaps with a dot) each intersection in each of the four areas of intersection of lines.
       Put the totals at the bottom of the computation, working from right to left.
    
3. Start with the intersections on the right, putting the total on the bottom right.
      
4. Next go to both bunches of intersections in the middle, putting the total next in the bottom middle.
      
5. Next go to far left, putting the total on the bottom left.
      
6. Three totals have been produced. As needed, start on the right (the digit's place) write the number as one digit, carrying the extra to the number on the left of that (the ten's place).
      
7. Repeat as needed. Start on the next place to the left and write the number as one digit, carrying the extra to the number on the left of that.

    The result is the product.

    1. Multiply: 31x12. Click on problem to see work and answer.

    2. Multiply: 2034x21. Click on problem to see work and answer.