1.03.01 Long Multiplication
This is where we multiply by the units and the tens separately, then add the two rows together.
To calculate 158 × 67:
First, multiply by 7 (units):
- x 67
Then add a zero on the right-hand side of the next row. This is because we want to multiply by 60 (6 tens), which is the same as multiplying by 10 and by 6.
Now multiply by 6:
- x 67
Now add your two rows together, and write your answer.
- x 67
So the answer is 10586
- Each digit of the top number is multiplied in turn by the two digits of the bottom number
- Fist we multiply the top number by the units digit of the bottom number.
- Then we multiply the top number by the tens digit of the bottom number
- The results of these two stages are then added together to give the final result
Some teaching methods exclude the insertion of the 0 before multiplying by the tens digit. Instead they just offset the digits of each answer to add the corresponding columns together. The inclusion of the zero is preferred here as it ensure the columns of each number are aligned properly and highlights the fact that we are multiplying by tens not units!
To multiply larger numbers
- If the second number had three digits we would multiply the top number by each of the three digits then add the three results
- When multiplying by the hundreds unit in the second number we would put two zeros first to take account the fact that we would be multiplying by hundreds
- Long multiplication can be done for any size numbers. We simply multiply the first number by each digit in the second number then add the results together (remembering to write the appropriate number of zeros first when multiplying by tens hundreds and thousands etc.)
1.03.02 Alternative grid method of multiplication
The lattice method is an alternative to long multiplication for numbers. In this approach, a lattice is first constructed, sized to fit the numbers being multiplied. If we are multiplying an -digit number by an -digit number, the size of the lattice is . The multiplicand is placed along the top of the lattice so that each digit is the header for one column of cells (the most significant digit is put at the left). The multiplier is placed along the right side of the lattice so that each digit is a (trailing) header for one row of cells (the most significant digit is put at the top). Illustrated above is the lattice configuration for computing .
Before the actual multiplication can begin, lines must be drawn for every diagonal path in the lattice from upper right to lower left to bisect each cell. There will be 5 diagonals for our lattice array.
Now we calculate a product for each cell by multiplying the digit at the top of the column and the digit at the right of the row. The tens digit of the product is placed above the diagonal that passes through the cell, and the units digit is put below that diagonal. If the product is less than 10, we enter a zero above the diagonal.
Now we are ready to calculate the digits of the product. We sum the numbers between every pair of diagonals and also between the first (and last) diagonal and the corresponding corner of the lattice. We start at the bottom half of the lower right corner cell (6). This number is bounded by the corner of the lattice and the first diagonal. Since this is the only number below this diagonal, the first sum is 6. We place the sum along the bottom of the lattice below the rightmost column.
Next we sum the numbers between the previous diagonal and the next higher diagonal: . We place the 9 just below the bottom of the lattice and carry the 1 into the sum for the next diagonal group. (The diagonals are extended for clarity.)
We continue summing the groups of numbers between adjacent diagonals, and also between the top diagonal and the upper left corner. The final product is composed of the digits outside the lattice which were just calculated. We read the digits down the left side and then towards the right on the bottom to generate the final answer: 783996.
Although the process at first glance appears quite different from long multiplication, the lattice method is actually algorithmically equivalent.
- The grid is made up of square cells forming rows and columns
- Each cell is divided into an upper left and lower right section by a diagonal line
- One number is written horizontally above the grid.
- The other number is written vertically to the right of the grid.
- The digits above each column are then multiplied in turn with the digits to the right of each row.
- The result of each calculation is written in a cell in the corresponding row and column
- The “units”” digit is place in the lower right section of the cell and any “tens” digit is written in the upper left
- When the grid has been completed all the numbers along the same diagonals are added
- The units of this addition are written outside the grid in line diagonally with the numbers which where added
- Any tens digits are carried over to be added in the next diagonal e.g. if the result was 27 the 2 would be carried to the next diagonal
- One all the diagonals have been added the result is formed outside the grid by the digits (going from top to bottom and left to right)