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January 2011 - Posts

Easier arithmetic

Every now and then I try to challenge myself to mentally calculate numbers. As I improved at this over the years I noticed that division was always especially vexing so I usually just forgot about it and went about my day. Just yesterday I decided to put my mind to it to come up with a more efficient way of mentally dividing numbers than long division. After I was all proud of myself :) I thought about sharing everything I've learned about arithmetic to make it easy for myself. I searched Google for the phrase "easy arithmetic" and came up with solutions that, to me, seemed tricky. By that I meant they relied a lot on rules that applied in particular instances but not generally, for instance squaring a number that ends in 5. This was different than how I had been teaching myself which relied more on the relationships between digits. I won't necessarily say that what I'll say below is the fastest way of performing arithmetic, and it's probably not even unique to me. I'm just sharing it in the name of exposure. Since I was never taught this it's likely there are others who don't know about it either. So that's my main reason for writing this.

Fundamentals: Digits & Relationships

What I'll be saying here will probably seem terribly obvious but it's critically important to explicitly state these ideas since they're fundamental to what comes later. These are not rules per se, so no trickery, rather they're simple definitions of our number system.

  1. There are 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
  2. There is an ordering to the digits. 0 is less than 1, which is less than 2, and so on.
  3. The digit 0 represents no value.
  4. The operation of addition is transitive. That is 2 + 3 gives the same answer as 3 + 2; the order is unimportant.
  5. The operation of subtraction is intransitive. That is 2 - 3 does not give the same answer as 3 - 2. However, the only hurdle to transience is the sign of the result. 9 - 4 = 5. 4 - 9 = -5.
  6. Subtraction is the opposite of addition and vice versa. If I add 4 to the number 2 I get 6. To get back to 2 I just subtract what I added, 6 - 4 = 2.
  7. The operation of multiplication is simply repeated additions.
  8. The operation of division is simply repeated subtractions.

Numbers

This is sort of a sub-classification to the above, but I thought to give it proper space and not just make it a bullet point.  A number is not a digit, it is a collection of them. But even then there are fundamental relationships to observe. Our numbering system is based on 10 digits, thus it's called base 10. (There are numbering systems involving 8 and 16 digits so they're base 8 and base 16. In case you're curious base 16 involves using letters A - F after 9.) So in base 10 numbering successive digits in a number are a reflection of powers of 10 increasing from the decimal. Since I really don't need to involve the technical definition of powers in this article I'll restate that to say successive digits in a number are a reflection of increasing numbers of zeros from the decimal.

So the number 2 reflects zero 0s after it.

The number 10 reflects one 0 after it.

If I have the number 20 what I really have is the number 10 multiplied by 2 and added to 0.

20 = (2 * 10) + (0 * 1)

19 = (1 * 10) + (9 * 1)

538 = (5 * 100) + (3 * 10) + (8 * 1)

2429 = (2 * 1000) + (4 * 100) + (2 * 10) + (9 * 1)

  • This is not trickery, this is just fundamentally how you get numbers bigger than the highest digit.

Now when you see the number 2429 you probably don't think you're implicitly performing multiplication and addition, but you are. And this fact is what makes arithmetic of numbers really easy!

Memory

Another sub-classification to the fundamentals section. In the methods I'll list below many intermediate numbers are used to store parts of the solution. Because I developed all this to perform mental arithmetic being able to remember many things is critical. So to emphasize: use your memory. You can be creative in this. For instance it can help to remember 103 by the fact it's 3 from 100. Exactly which direction from 100 isn't critically important because you're really trying to impress the number 103 in your mind, 3 from 100 is just a reinforcing thought. When I say be creative I make my numbers flash, glow, spin around, whatever it takes to hold the memory of the number intact in my mind.

  • Whatever you do, it is critically important you remember the original numbers you're performing operations against.

Of course you could cheat and actually write numbers down to make things easier. :P

Addition

Let's say you want to do:

  49
+ 27

Instead of adding each individual digit and carrying leftovers instead break apart the numbers into their components.

49 = 40 + 9
27 = 20 + 7

Because addition is transitive it doesn't matter a bit the order in which you add the numbers! Mentally it's easier for me to work with numbers with zeros on the end which is why I break them apart. So:

  40     9
+ 20   + 7
---- + ---
  60    16

So instead of directly adding 27 to 49 I employed two intermediate additions to get two intermediate answers. The final step is to add these intermediaries.

  60
+ 16
----
  76

Again, working with zeros makes this much much easier. Generally speaking this reduces arithmetic down to single digits since 0 has no value. And again, this is not involving any kind of trick, instead it's relying on the relationships of digits.

  378
+ 247

378 = 300 + 70 + 8
247 = 200 + 40 + 7

  300     70     8
+ 200   + 40   + 7
----- + ---- + ---
  500    110    15

At this point if you feel confident in your mental abilities simply stop and add 500 + 110 = 610 + 15 = 625. However to illustrate how breaking apart numbers to make them easier to work with has no bounds I'll continue.

500 = 500
110 = 100 + 10
 15 =       10 + 5

  500     10
+ 100   + 10    5
----- + ---- + --
  600     20    5

At this point you can literally see the answer: 625. How far you take this splitting apart is up to you. I've found the more I practice doing calculations the less I need to mentally break apart the numbers. Mentally I employ animations of the numbers, I visualize for instance 10 spreading away from 110 so I have 100 and 10 as distinct numbers. Because addition is transitive this is possible.

Subtraction

Subtraction is made easier by reframing the operation. Instead of subtracting one number from another, instead think of the numbers as being on a line and you're moving in successive steps from a source to a destination: the number of steps you have to make is the answer of the subtraction. Convince yourself of this by mentally subtracting 3 from 9. The answer is 6. In terms of the reframing: the source number is 3, the destination is 9, and the number of traversed steps is 6. If instead you had to subtract 9 from 3, 3 - 9, rely on the fact that although subtraction is intransitive the only thing stopping it from being so is the sign. So reverse the numbers so that you still start at 3 and move to 9, 9 - 3, but remember to change the sign when you get the answer. Instead of getting 6 you get -6.

The point of this reframing is to help with higher order numbers and something I call filling in or spackling, similar to filling in cracks in your walls. Let's say you have to do:

  297
- 128

Remember you're always moving from a smaller number to a larger, so you'll be moving from 128 to 297. Again, working with zeros is much easier. So how far do you have to go from 128 to get to the nearest zero number? 2 steps puts you at 130.

128 + 2 = 130

Float this number 2 away in your memory somewhere, it's part of the answer. So now you want to move again to the nearest zero, in this case 200.

130 + 70 = 200

Again, float this number 70 away, it's part of the answer. Because it is part of the answer it's up to you if you want to add it to the 2 you previously floated away, or hold it in memory as a distinct number. So now there are only 97 more steps from 200 to get to 297, the destination.

200 + 97 = 297

You could either float this number 97 away as part of the answer, or you could go in smaller steps and instead do:

200 + 90 = 290
290 + 7 = 297

As you can see it's up to you and how good your memory and concentration skills are. Because addition is transitive it doesn't matter a bit how you add the numbers. If you never added any of the intermediaries you'll have this final addition to perform

   2
  70
  90
+  7

Which, if you want, you can look at as

  70     2
+ 90   + 7
---- + ---
 160     9

So that means your final answer to 297 - 128 = 169. This all works because addition is an opposite operation to subtraction. So 128 + 169 = 297 - 169 = 128. It's just that previously you stated the problem differently, as a question. What needs to be added to 128 to get 297: 297 - 128 = ?.

Multiplication

First let's get definitions down since we're no longer simply adding. In the operation 9 * 7 the 9 is the multiplicand and the 7 is the multiplier. Essentially 9 will be added 7 times. Because adding is involved, and addition is transitive, you can say 7 * 9 is 7 added 9 times to get the same answer. Frame it however you want just keep in mind that one number specifies the number of additions to perform, the multiplier, and the other is the number to be multiplied, the multiplicand. So, higher order numbers.

  98
x 97

Take the multiplier, in this case 97, and break it apart so you're working with zeros.

97 = 90 + 7

Because the operation of multiplication involves repeated addition what we're really trying to get at by 98 * 97 is what is the answer when 98 is added 97 times. If we instead choose to do 90 additions first and then 7 more later on it doesn't matter, the answer is the same.

  98     98
x 90   x  7
---- + ----

At this point you'll see we now have two distinct multiplications to do, plus an addition; how's your memory! You can make this easier by breaking down numbers even more:

98 = 90 + 8

  90      8     90     8
x 90   x 90   x  7   x 7
---- + ---- + ---- + ---

Again, because order is irrelevant when it comes to addition you can break the problem down and build it back up to the solution in the manner easiest for you. When dealing with multiplying something that has a zero on the end it can be easier to reframe it.

90 = 9 * 10

So when doing 90 * 90 it's really 9 * 9 and move the zeros to the end. Technically you're not simply moving the zeros to the end, you're multiplying a power of 10 to the number like you're supposed to (9 * 9) * (10 * 10), but mentally it's easier to simply move, or stack, zeros.

  90      8     90     8
x 90   x 90   x  7   x 7
---- + ---- + ---- + ---
8100    720    630    56

So now the answer to 98 * 97 is the addition:

  8100
   720
   630
+   56

Remember you can spread apart numbers if it mentally makes it easier to see what's going on

8100 = 8000 + 100
 720 =        700 + 20
 630 =        600 + 30
  56 =              50 + 6

Add however you want

8000     100
         700     20
         600     30
       +       + 50     6
---- + ----- + ---- + ---
8000    1400    100     6

Although that 1400 could be spread away I'll just write the answer to be 9506. Don't believe that's the answer? Whip out that calculator then, haha. Again, no trickery, you're relying on fundamental relationships between digits so this method works with any kind of number regardless of how big it is, or what the final digit is, or etc. Although it seems to be a lot of calculations involving lots of intermediate answers, the more you practice the easier it becomes. Working with zeros makes it easier to simply see the answer with very few calculations going on.

Division

In a similar sense to how I reframed the operation of subtraction I'll reframe the operation of division. If you have a bunch of objects the operation of division answers the question: how many groups of objects can I get with each group being a certain size. Like the higher order operation of multiplication definitions are important with division. The population of objects is called the dividend and the size of each group is called the divisor. In the operation 10 / 2, 10 is the dividend and 2 is the divisor. 10 objects taken 2 at a time gives 5 groups. For higher order numbers forget long division, instead rely on digit relationships.

  359
/  28

To tackle this we're going to rely on the fact that addition is the opposite operation of subtraction. So the divisor, 28 in this case, we'll add 2 to it to get 30 since numbers with zeros on the end are easier to work with. It's important to keep in mind how big we've made each group of objects, we added 2. So that means in our final answer each group will have 2 more items in it than it should. So to undo our adjustment to the problem we'll subtract out those 2 items from each group. This will be the multiplication of 2 by whatever the group count we get. So mentally remember this 2. So now we have.

  359
/  30 (28 + 2)

Like subtraction division involves moving from one number, 0, to another, 359 in this case, but instead of going 1 at a time we move in multiples of the divisor, 30 in this case. This involves multiplication. Your goal is to find the number of steps, or groups as I reframed the division operation, it takes to get to 359 without going past it. By working with a number with a zero on the end you can ignore the end digits and simply rely on single digit multiplication; remember your multiplication tables? :) If I start at simply 10 then 10 * 3 = 30, add in the zero means it's really 300.

30 * 10 = 300

Since 300 is less than 359 we haven't exhausted the number of steps we can take, but just float away that number 10 as part of the answer. Now the problem becomes reduced to going from 300 to 359, 59 steps taken 30 at a time. Intuitively you might see you can only do this 1 time. But the general solution remains the same: successively move in steps of the divisor until you move to the destination or if you would move past it, don't. Every time you move float away the number of steps as part of the solution. So in our case we moved a total of 11 steps to 330, with 29 remaining.

  359
/  30 (28 + 2)
-----
   11 remainder 29

Now we have to undo the modification we made to the divisor to make it easy to work with. We added 2 objects into each group so we have to take them out.

2 extra objects per group * 11 groups = 22 squeezed out objects.

Add this back into the remainder we were working with:

29 + 22 = 51

Again, to do the addition you can mentally spread out the components of the numbers and add them in different ways. Now the 51 remaining is more than our true divisor of 28 which means we can move some more steps, at least one. You can either choose to divide 51 / 28 or intuit a subtraction, it's up to you. Using division

  51     51
/ 28   / 30 (28 + 2)
---- ~ ----
          1 remainder 21

21 remainder + (1 group * 2 extra per group) = 21 + 2 = 23 total remainder

So we got one extra group from the previous remainder of 51 with 23 remaining. This is less than our true divisor so we're done. 359 / 28 = 12 23 / 28 is the final answer.

Depending on why you're engaging in the division operation this may be enough for you, it really is the answer. But if you want the customary decimal notation instead of a fraction read on.

Fractions & Decimals

The number 5.75 is a decimal but it's really an encoding of two concepts. The 5 represents a whole amount of units, whereas the 0.75 represents some portion of a unit. This is a fraction of a unit, or said another way a percentage. So 0.75 is the same as 75 / 100. Hey, look at that, division! So seeing how we got 23 / 28 in our above calculation, in order to get the decimal equivalent we simply divide.

The issue here is that the divisor starts out bigger than the dividend. To do the calculation we simply give the dividend 0s until it's bigger than the divisor. Every time you add a zero like this you're adding a number to the right of the decimal point, keep track of this. So

23 / 28 becomes 230 / 28

The same methodology outlined above works here except you'll be engaging in successive rounds of division to calculate successive decimal digits.

  230     230
/  28   /  30 (28 + 2)
----- ~ -----

First round of division gives

  230
/  30 (28 + 2)
-----
    7 remainder 20

2 extra objects per group * 7 groups = 14 extra remainder + 20 = 34 total remainder

Intuit there's only 1 group of 28 in 34 so 34 - 28 = 6.

8 remainder 6

This means the first number after the decimal is 8, 0.8. Adding this in to our previous whole number of 12 means we have 12.8.

At this point you'll simply repeat what we've done since now we have 6 / 28 instead of 23 / 28. However many decimal places you want to go is up to you. If you ever come upon a situation where your remainder matches your divisor, 28 / 28, then you simply add 1 to that round's group total and you're done. If at any point you discover you're working with the same amount of remainder that you had in a previous step then stop because you have a repeating decimal. It'll repeat from the first time you came across that remainder amount to where you're currently at.

Posted: Sun, Jan 2 2011 4:40 PM by Humpty
Filed under:
How do you know the truth?

This was a question I woke up with last week.

Posted: Sat, Jan 1 2011 11:40 AM by Humpty
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