We read the digits down the left side and then towards the right on the bottom to generate the final answer: 783996.Īlthough the process at first glance appears quite different from long multiplication, the lattice method is actually algorithmically equivalent. The final product is composed of the digits outside the lattice which were just calculated. We continue summing the groups of numbers between adjacent diagonals, and also between the top diagonal and the upper left corner. We place the 9 just below theīottom of the lattice and carry the 1 into the sum for the next diagonal group. Next we sum the numbers between the previous diagonal and the next higher diagonal. We place the sum along the bottom of the lattice below the rightmost column. Since this is the only number below this diagonal, the first sum is 6. This number is bounded by the corner of the lattice and the first diagonal. We start at the bottom half of the lower right corner cell (6). We sum the numbers between every pair of diagonals and also between the first (and last) diagonal and the corresponding corner of the lattice. Now we are ready to calculate the digits of the product. If the product is less than 10, we enter a zero above the diagonal. SUBTITLES AVAILABLEIf you are not using a calculator, lattice multiplication is a good method of multiplying two numbers together. 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. 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. īefore 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. IllustratedĪbove is the lattice configuration for computing. Header for one row of cells (the most significant digit is put at the top). Is placed along the right side of the lattice so that each digit is a (trailing) Of cells (the most significant digit is put at the left). For example, a + (b + c) (a + b) + c and a(bc) (ab)c are associative laws, and a + b b + a and ab ba are commutative laws. These operations obey several algebraic laws. Is placed along the top of the lattice so that each digit is the header for one column Addition and multiplication are prototypical examples of operations that combine two elements of a set to produce a third element of the same set. If we are multiplying an -digit number by an -digit number, the size of the lattice is. In this approach, a lattice is first constructed, sized to fit the numbersīeing multiplied. Try it both ways and ten adopt what's easiest and/or faster for you.The lattice method is an alternative to long multiplication for numbers. Keeping the unit's digit in memory is faster than writing them down at each step. If you were to look at the problem after someone did it, it would look like what you see on the left. The above problem shows every single step, including what you think as you go along. Count the scratches (4), which stands for how many tens (40) and write in front of the 9. Step 1: 6 + 7 = 13 so scratch off the 7 and write 3 Step 2: 3 + 8 = 11 so scratch off the 8 and write 1 Step 3: 1 + 3 = 4 so think 4 Step 4: 4 + 5 =9 so think 9 Step 5: 9 + 9 = 18 so scratch off the 9 and write 8 Step 6: 8 + 4 = 12 so scratch off the 4 and write 2 Step 7: 2 + 7 = 9 so write a 9 under the bar since all digits have been added. To start, see the eight digits to be added. The following problem is exactly the same one I did before when using the dot method. Parentheses will be used to denote what I am thinking in my head. When the sum is less than ten, it is kept in your head. I'll illustrate the steps involved by writing down the unit's digits this time. Just as in the dot method, you only keep the unit's digit in your memory or as an alternative, write the unit's digit to the right and just below the digit just scratched off. The scratch, like the dot, will represent ten. The most familiar method for multiplying large numbers, at least for American students, is the lattice algorithm. Any time you add and get a number over nine, you scratch off the last digit added. The Scratch Method is similar to the dot method. The beauty of this addition is you never have to keep a number higher than nine in your head! If you are adding without paper and pencil, use your fingers to keep track of the dots! Instead of writing a dot, put up a finger. \nonumber \]īy the way, you could put the dots to the left or right of the digit, do whatever is most comfortable for you.
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