## Saturday, September 29, 2012

### Learn to Add Over Fifteen Numbers in Under Fifteen Seconds

Since I haven't gone over any mental math tricks in an extremely long time, I wanted to get one in this week.

Before this post, I have done posts involving multiplication and squaring tricks. But today, I am going to talk about some addition.

There are ways of doing mental addition reasonably quickly, but some specific series of numbers (like the Fibonacci series, which I taught a way to add last December) can be added much quicker. So, we are going to need some more fastidiousness when we get our numbers to add.

The series we will be adding is the powers of two. Have your volunteer choose a number and list out that many powers of two. Let's say they say nine.

1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 =

Dont retrogress yet. All you have to do is double the final number in the sequence and then subtract one.

256 x 2 - 1
512 - 1
511

And there is your answer. If you have a little more prowess, you could try going up to sixteen numbers.

1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 + 4096 + 8192 + 16384 + 32768 =

If you do the math, you will find that the answer is 65535.

This method is actually fairly easy to prove. Remember back when we did infinite series? Let's write that one we are subtracting as an infinite series.

2^0 + 2^1 + 2^2 + ... + 2^n-2 + 2^n-1 = 2^n - (2^-1 + 2^-2 + 2^-3 + ...)

Now, add that series to both sides. If we write it as an infinite series, we get:

2^n = 2^n-1 + 2^n-2 + ... + 2^1 + 2^0 + 2^-1 + 2^-2 + ...

The left side is already simplified at 2^n. How do we simplify the right side? I may have mentioned in a earlier post (I'm not sure which) how to simplify this type of series. Let me explain it again.

1. Find what number you multiply each number in the sequence by to get the next number in the sequence. Call that s.
2. Call the first and biggest number in the sequence x.
3. Use the formula x/(1-s) to get the total.

In this sequence, we are multiplying by 1/2. 1/2(2^n-1) can be written as 2^n-2.

The biggest number is the 2^n-1. So, we plug it into the formula to get:

(2^n-1)/(1 - 1//2)
(2^n-1)/(1/2)
2(2^n-1)

Using the law of exponents, we can simplify that to get:

2(2^n-1)
2^n-1+1
2^n

And there is the proof. Though the proof is kind of cool, I found it cool that you could add this many numbers up so quickly. Good luck fooling people with this one!

## Saturday, September 22, 2012

### How to Multiply by Nine on Your Fingers

Back when you were six or seven, you may have learned how to do some simple addition and subtraction on your fingers. Like for 2 + 2, you might hold up two fingers on one hand, two on the other, and then add up the fingers. For 5 - 3, you could hold up five fingers, put three down, and add up the fingers left.

You may have also learned a multiplication trick on your fingers. Specifically, how to multiply by nine. Though this is very popular, I thought it was worth a cool math stuff post.

Let's solve 9 x 4. To do this, hold up your ten fingers. Then, put your left index finger (four from the left) down. On the left side, you have 3 fingers up. On the right side, you have 6 fingers up. Therefore, the answer is 36.

You can do this with other objects as well. Pretend you have nine toothpicks.

|||||||||

For 9 x 1, you just leave it. For 9 x 2, you bring one over to the left.

|  ||||||||

There is one on the left and eight on the right, giving us 18. For 9 x 3, bring another over.

||  |||||||

You can continue this process all the way through to 9 x 10.

|||||||||

Now, you can actually continue to 9 x 11. Just add nine more toothpicks to the right side.

|||||||||  |||||||||

For 9 x 12, you would have ten on the left and eight on the right, but that still is 108. This process actually works forever.

This isn't the coolest thing in the world. However, I like it because it is simple, even at a second grade level. I always say that around fifth or sixth grade, people lose interest in math. However, to lose interest, you must get an interest in the first place. Something like this can spark that interest, and some of the more complicated patterns and proofs can be taught in later years to maintain this interest.

Answers: Here are the answers to August's problem of the week. I will continue the problem of the weeks in June 2013.

Easy:
x = 0%
y = 100%
g = 6
s = 5
m = 2
b = 80
t = 40
n = 120
h = 5

Hard:
a = 37.5%
b = 62.5%
x = 0%
y = 20%
z = 80%
f = 1
g = 100
s = 61 cm
l = 90%
n = 95
p = 330 cm

## Saturday, September 15, 2012

### Triangular Day: Some More Fun Facts

Today is another triangular day. It is September 15 and 15 is a triangular number. I'd like to continue from two weeks ago and show a few more fun facts.

First off, remember the perfect numbers that we talked about last week? Those are the numbers whose factors add up to itself, like 6, 28, and 496. Let's see if any of them are triangular. We'll see if they fit the formula we learned a couple months ago: n(n+1)/2.

6 = (3)(4)/2
28 = (7)(8)/2
496 = (31)(32)/2
8128 = (127)(128)/2

Clearly, all of these numbers are triangular. Turns out, every perfect number is a triangular number.

This one actually isn't too hard to prove. Remember last week when we were generating perfect numbers? We would multiply a power of two by one less than the next power of two. For 6, we did 2 x 3. For 28, we did 4 x 7.

Well, let's say that the seven is n. Then, what would the (n + 1)/2 equal?

(7 + 1)/2 = 4

So, we are doing 4 x 7, which is what we were doing before. What about for the 2 x 3?

(3 + 1)/2 = 2

(31 + 1)/2 = 16

Basically, with the method we used for generating perfect numbers, we are following the triangular number formula; n(n+1)/2.

Let's look at another pattern. Soon, we will be learning about other number systems, like the binary system and the hexadecimal system. Basically, some numbers are written differently in other systems. We will take some numbers with all ones and convert them from base nine to base ten.

1 = 1
11 = 10
111 = 91
1111 = 820
11111 = 7381

This will make sense in a few months. Anyways, let's see if these numbers are triangular.

1 = (1)(2)/2
10 = (4)(5)/2
91 = (13)(14)/2
820 = (40)(41)/2
7381 = (121)(122)/2

These are all triangular as well. This one, I don't have a proof for, but if you know one, please let me know.

One more quick fact: let's look for some prime triangular numbers. Here is the sequence:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153,...

Do you find any primes? Well, three. Anything else?

Turns out, three is the only prime triangular number. Again, I don't have a proof for that one, but I still thought it was pretty cool.

## Saturday, September 8, 2012

### Do the Perfect Numbers Deserve Their Title?

We have gone over numerous number sequences; Fibonaccis, primes, triangulars, squares, cubes, naturals, and probably several more. Now, it is time to add another sequence to our toolbox. This one is called the perfect numbers.

But, Fibonaccis and triangulars are already so cool. What makes the perfect numbers cooler than them? The sequence looks pretty dull anyways:

6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139950000,...

But it really is about as cool as it gets. Look at the factors of six, ignoring six itself:

1, 2, 3

What is their sum?

1 + 2 + 3 = 6

Take 28 and do the same thing:

1 + 2 + 4 + 7 + 14 = 28

1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128

A perfect number is a number whose factors sum to itself. Pretty cool right? However, it isn't like the Fibonacci's right? You can't just add a number and you have the next one. Correct, but it still isn't too hard to deal with. Look at the following chart:

 Power of 2 2n -1 Product Perfect 1 1 1 2 3 6 Yes 4 7 28 Yes 8 15 120 16 31 496 Yes 32 63 2016 64 127 8128 Yes 128 255 32640 256 511 130816 512 1023 523776 1024 2047 2096128 2048 4095 8386560 4096 8191 33550336 Yes 8192 16383 134209536 16384 32767 536854528 32768 65535 2147450880 65536 131071 8589869056 Yes 131072 262143 34359607296 262144 524287 137438691328 Yes 524288 1048575 549755289600 1048576 2097151 2199022206976 2097152 4194303 8796090925056 4194304 8388607 35184367894528 8388608 16777215 140737479966720 16777216 33554431 562949936644096 33554432 67108863 2251799780130820 67108864 134217727 9007199187632130 134217728 268435455 36028796884746200 268435456 536870911 144115187807420000 536870912 1073741823 576460751766553000 1073741824 2147483647 2305843008139950000 Yes
This chart may look very confusing. Let me explain it in better detail. Basically, the first column is the powers of two. The second column is that power times two minus one. For instance 2(2) - 1 = 3.

The third column is the product of those two numbers. And if you look at all of the yeses, you will see the first several perfect numbers.

But why are they the only yeses? The answer lies in the second column. Look at the number that it is paired with.

3, 7, 31, 127, 8191, 131071, 524287, 2147483647,...

What trait do all of these numbers have? You might see it more clearly with just the first four numbers.

3, 7, 31, 127

They are all prime. In fact, the other second column numbers are composite (or one, which is neither). I don't have a proof for this, but it is one I would really like to find out. Please comment it if you know it or think you figured it out. Also, let us know if you know any perfect number identities. I would like to post some more, as this really is a fascinating property of these numbers.

## Saturday, September 1, 2012

### Triangular Day: A Few Fun Facts

Today is a triangular day. It is September 1 and 1 is the first triangular number. Rather than going through a whole proof, I thought that I would just go through a few fun facts.

First off, let's look at the sequence ignoring 1. We have:

3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153,...

Let's try to find some square numbers. We can find 36. If you keep going ahead, you find:

1225
41616
1413721
48024900
1631432881

Clearly, there are several squares that are also triangular. What about cubes?

Can you find any? No? There are actually no cube triangular numbers other than one.

What about fourth power, or fifth power, or sixth power? Turns out that there are absolutely none.

Here's another fun fact. Triangular numbers never end in 2, 4, 7, or 9. I'm not sure why, but they never do.

Finally, if you remember back to the divisibility post, we added up all of the digits in a number frequently to find out if it's divisible by any multiples of three. We can also use it for modular arithmetic, which I will probably talk about at some point in the near future. The adding of all of the digits in the number is commonly referred to as digital roots.

Let's find the digital roots of the triangular numbers.

1 = 1
3 = 3
6 = 6
10 = 1 + 0 = 1
15 = 1 + 5 = 6
21 = 2 + 1 = 3
28 = 2 + 8 = 10 = 1 + 0 = 1
36 = 3 + 6 = 9
45 = 4 + 5 = 9
55 = 5 + 5 = 10 = 1 + 0 = 1
66 = 6 + 6 = 12 = 1 + 2 = 3
78 = 7 + 8 = 15 = 1 + 5 = 6
91 = 9 + 1 = 10 = 1 + 0 = 1

What digital roots have we found. There are ones, threes, sixes, nines, and that's it. In fact, those are the only digital roots of triangular numbers. Again, I'm not sure why, but if you know a proof for this or any of the above facts, please comment it. But even without a proof, these facts are pretty cool.