Multiplication with Lines

Multiplicationwith Lines

The method of multiplying two numbers using the intersection of lines is a little curious to say the least. How can simply drawing intersecting lines and counting points of intersection possibly be used for multiplication in any sense of the word? But it does work! The question I was asked ... Why does it work?

How it is PerformedThe technique itself is visually appealing and can be a lot of fun for smaller numbers. As the numbers increase in magnitude or if they have digits greater than say five, you might find yourself doing more counting and taking more time than is required for the more traditional methods.Basically, we represent a number by a series of parallel lines spaced apart to represent the place values. When drawing these lines we work from left to right (or top to bottom as we will see) when performing multiplication. Very much like writing the number with digits.

It's all in the InterpretationTake for example the number 213, it can be represented either way as shown in the diagram. The spacing between the groups of lines is an indicator of the place value. Hence representing any number is very simple.When we perform the operation of multiplication, one number is drawn horizontally (as in the first sketch) and the second is over-layed vertically (as in the second sketch). Using a simpler example, let's draw up the multiplication of 23 14.

You may be thinking this has resulted in an oblong of sorts, the area of which is found by multiplication. True, but that is about as far as the comparison goes. The actual dimensions of the oblong formed have no relationship to the answer let alone the technique that is used.The important thing to remember when drawing this up is to make sure that you can count the individual points of intersection.Now what's with the points of intersection? Again, it's all in the interpretation. To understand what the points of intersection represent, let's start off with some really simple examples.The drawing to the left shows two vertical lines crossed by three horizontal lines. Now counting the points of intersection of these lines, we find there are six. Thus we can interpret this as the usual multiplication of integers2 3 = 6It does not take too much to realize that this will always be the case regardless of the number of lines used in each.In this example, we have assumed that the place value represented by each group of lines is the units. What if the place value represented were tens? Well you could do this; explode each single line into ten. The two vertical lines would become twenty and the three horizontal lines, thirty. The number of points of intersection would now be 600. As you can see, the position (or place value) of the lines is important to the technique. Similarly, if in the above drawing the two vertical lines represented hundreds and the three horizontal lines represented tens, then the number of points of intersection is interpreted as 100 10 = 1000, or thousands.Let's take this idea and use it with the previous example 23 14.

Analyzing the ResultsTaking our example, let's look at where all the values come from.Starting at the bottom right, we have the three units of 23 crossing the four units of 14 resulting in twelve points or 12 units.Top right, we have the three units of 23 crossing with the one ten of 14 resulting in three points or 3 tens.Bottom left, we have two tens of 23 crossing the four units of 14 resulting in eight points or 8 tens.Top left, we have two tens of 23 crossing the one ten of 14 resulting in two points or 2 hundreds.The final results are; 200 + 30 +80 + 12 = 322.Most times when you see this method performed, it is on an angle. The only reason for this is that the place values (where the lines intersect) will be in line. Sometimes with untidy work, the groups may not line up as best as they could witch may lead to come confusion.The same process can be carried out with larger numbers, but keep in mind, larger numbers and larger digits can make the process quite overwhelming. In fact it come to the point where it is far quicker to perform the multiplication using more traditional methods.

The Technique as it is Preformed.In the previous example, we have calculated the numerical value for each group of intersections and then added the results at the end. Although there is no real problem in doing this, it is not the actual way this technique is performed. The technique involves counting the number of points of intersection and then writing down the answer from right to left. Let's go through the same example as we would using the technique correctly and the angled version commonly seen.We start our explanation from the point where we begin the counting.

Working from right to left, count the number of points of intersection in the first group, the units. We count twelve. Write this down, but instead of writing the number 12, write the 2 and place a dot to the upper left of it. This indicates that there is a ten that needs to be added to the next number.

Counting the points of intersection for the next place value of tens, we count three and eight, a total of eleven. Now add the one from the units and this gives 12. Once again write down the 2 and place a dot to the upper left. This indicated a hundred that needs to be added to the next number.

Counting the points of intersection for the last place value of hundreds, we have two. Adding the one from the tens gives us three. Now we can simply write down the last digit (first in the actual answer) as 3. Hence our final answer to 23 14 is 322.

23 14 = 322As you can see, the process is very simple, elegant and visually pleasing. Even though it is easy to rotate the grid or draw the grid on an angle, it certainly is not necessary.

How about some more examples ...

Worked Example 1: 36 24Step 1: Draw up the grid.

36 24Step 2: Count the Units.

There are 24 points, so write down the 4 and two dots for the tens to be added to the next value.Step 3: Count the Tens.

There are a total of 24 ( 12 and 12). Adding two from the units gives 26. Write down the 6 and two dots for the hundreds to be added to the final value.Step 4: Count the Hundreds.

There are 6 points of intersection, 6 plus the 2 from the tens gives 8. Write down 8.Our answer: 36 24 = 864

Worked Example 2: 214 35This example will show the process with a three digit number.Step 1: Draw up the grid.

214 35Step 2: Count the Units.

There are 20 points of intersection. Write down the 0 and draw two dots for the tens.Step 3: Count the Tens.

There are a total of 17 points (12 and 5), add to this the two and we have 19. Write down the 9 and draw a dot beside it for the hundreds.

Step 4: Count the Hundreds.

There are a total of 13 points. Add the one and we have 14. Write down the 4 and draw a dot for the thousands.Step 5: Count the Thousands.

There are 6 points, adding the one gives 7. Write down the seven.Our Answer: 214 35 = 7490

Worked Example 3: 204 32This example will show how to use the technique if one of the digits is zero.Step 1: Draw up the grid.

204 32Notice that a dashed line is used as the place value for the zero. Any solid line (or another dashed line) crossing a dashed line does not count as a point of intersection.

Step 2: Count the Units.

The are 8 points, so write down the 8.Step 3: Count the Tens.

There are a total of 12 points (12 and 0). Solid lines crossing a dashed line do not count to the total. Write down the 2 and draw a dot to represent the hundred to be added to the next value.Step 4: Count the Hundreds.

There are a total of 4 points (0 and 4). Again, solid lines crossing dashed lines do not count toward the total. We need to add one from the previous value to give 5. Write down the 5.Step 5: Count the Thousands.

There are a total of 6 points. Write down the 6.Our Answer: 204 32 = 6528As stated previously, turning the grid on an angle or drawing the lines on an angle can sometimes cause a little confusion as to which groups to count. The grid does not need to be on an angle. This final example will show how to use the grid without it being angled. Of course, the same technique can be used if the the grid is angled.

Worked Example 4: 322 24Step 1: Draw up the grid.

322 24Step 2: Count the Units.

The units will always be at the bottom left. There are 8 points. Write down the 8.Step 3: Count the Tens.

There are a total of 12 points (4 and 8). We simply work our way round the grid counting the points in the next closest groups. Write down the 2 and draw a dot to add to the next value.

Step 4: Count the Hundreds.

The total from the next closest groups is 16 (4 and 12). Add the one from the previous count and we have 17. Write down the 7 and a dot for the next value.Step 5: Count the Thousands.

The final group totals 6. Add the one from the previous group and we have 7. Write down the 7.Our Answer: 322 24 = 7728

And there it is!The method used (angled or not) is a matter of personal preference. I use the method that is not angled, then it doesn't matter whether the groups line up properly or not.As stated before, the method is visually pleasing and there is something somewhat mysterious about it when you first see it. The fact is, this method can be used for any numbers, however the practicality of it comes into question; large numbers with high valued digits can make the method very cumbersome and easy to lose count. Nevertheless, it's certainly a bit of fun whether you're a student who struggles with multiplication or a seasoned mathematician.

Ferrick Gray | Multiplication with Lines | 9 | 2015 Kanticle of Being | http://kanticleofbeing.com