math

Math

A while ago, I talked to a math class. Here's my notes.

Many years ago, I was invited to talk about how I use math at work, to a 6th grade math class. Truth is, as a computer programmer, I did add and subtract, base 16, but there's not a lot of other direct math. However, the logic discipline you get when you work hard at math is really useful. This is the outline that I used in the class.

You’ve heard of the 3 R’s – Readin, Ritin, Ritmatic?

They count. Learn them and you’ll make more money.

Don’t learn ‘em and it’ll cost you money.

When I worked at BofA, they gave college graduates (i.e. 4 year graduates) a 10% raise one year.

Didn’t matter what you did, or which department you were in, only, did you have a degree.

And, since annual raises were generally 1%, 2%, 3%, if you had a degree, you made more every year after that. Chances are that one or more of you, will encounter the same situation sometime in your life.

But, let’s get to the topic for today – MATH.

Yes I used math in programming, but not as you’d expect.

Most of my math background helped because of the logic it instilled.

For example, if you have to put your shoes and socks on, and someone hands you your shoes, you don’t put ‘em on, and then ask for socks.

I did lots of adding, and a little subtracting, base 16.

Computers work in binary, but when people look at the numbers, they look at them as base 16.

You count base 10. Because you have 10 fingers. I once read that there were some people that counted base 12. And then there were some primitive societies that counted 1, 2, 3, many.

When working with computers, I count: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10

Let me hear you count to 10.

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Did you note that I started with "0", and you started with "1"

Dead giveaway that we think a bit differently.

Did you know that there was not a "0" in Roman Numerals? Just a point of interest.

First, let’s look at a series: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 or A,B,C,D,E,F,G,H,I,J,K,L

1,2,4,8,16,32,64,128,256,512,1024

This is a hard one. 1,2,2,4,8,11,33,37,148,153,

1,1,2,3,5,8,13,21,34,55,89,144

These are Fibinaci numbers. Note that you always add.

In the more general case, you could either add or subtract.

How ‘bout considering multiplication for a while.

Do you, or did you have to write multiplication tables? I did in 6th grade, every day.

At home, I realized I didn’t know the "7" table as well as some others, so I wrote it over and over.

Now, 70 years later, I know it better than most.

6*6=36 5*7=35. 7*7=49 6*8=48. 8*8=64 7*9=63. Is there a pattern?

20*20=400 25*25=625 30*30=900. What’s 24*26=???

6*6=36,

7*5=35 -1

8*4=32 -4 -(1+3)

9*3=27 -9 -(1+3+5)

10*2=20 -16 -(1+3+5+7)

11*1=11 -25 -(1+3+5+7+9)

12*0=0

See the Pattern? Other squares?

5*5=25, 6*6=36. How do you get from 25 to 36, USING ONLY 5 & 6?

If you'd gone to Montessori school, you'd probably know.

If you have a 5x5 grid, you add an extra line of 5 to one side, then a line of 6 to the next side, to make a 6x6 grid.

Many things are easier to think about when you form a picture in your head.

You can do the same with the 6x6=36, and 5x7=35 problem.

Think of a 6x6 grid. Take one row (1x6) off, and put it on the next side.

But it'll stick out 1, so you only need to put a 1x5 section on to form a 5x7 grid.

You know what 25x25 is. What's 26*26=??? Why?

This problem is an offshoot of what we've just been talking about.

Show that 5^n + 7^n +2 is divisible by 4.

How ‘bout another change. A couple of puzzles.

If a chicken and a half, Lays an egg and a half, In a day and a half,

How many eggs do 3 chickens lay in 4 days? (8)

This came from my dad when I was young.

It was also an early question on the $64,000 question - a quiz show many many years ago.

If 6 cats, can eat 6 rats, in 6 minutes, how many cats does it take to eat 100 rats in 100 minutes? (6)

I got this problem for homework in 7th grade.

No-one got it right and the substitute teacher hadn’t a clue either.

I finally figured it out, 2 years later.

Okay, new topic: This is the four "4"s puzzle.

44/44 4/4+4/4 (4+4+4)/4 How high can you go? (4! = 4*3*2*1 =24 you can use that also.)

Next new topic. Get out 2 rulers. Metric would be nice. Use them to add 6+8. 14+12.

How ‘bout subtraction? 23-15

When I was in college, we didn’t have calculators.

We used a slide rule for multiplication, just as you just did for addition.

Next, I’d like to practice some multiplication. Get a pencil and paper. The rules for this exercise are that you can only write the answer line. 23x34=??? In the first problem multiply 3x4, write the "2" and carry "1". Then multiply 3x3 (9) add the "1" carry (10 - don’t write yet) multiply 2x4 and add the 10 (= 18) write the "8" and carry the "1". Last, multiply the 2x3, add the "1" carry and write the "7". 782 Do the same for all of these.

When you get really good, you can do this with 3 digit numbers. I’ve not seen anyone do it with 4 digit numbers, although I’m sure it has been done.

34 64 87

x 23 x 57 x 46

Has anyone got any questions?

Let’s talk about gambling, the lottery, etc. for a while.

FIRST OFF – NO! (I hate first off, it's grammaticly incorrect) First, I don’t.

When I was young, a girl down the street, a year younger, beat me out of a dime.

I decided that there was just too much luck involved, and I don’t gamble to this day.

If you’ve been to Reno, or Las Vegas, you can see how much money has been spent building casinos.

You can see how many people work there.

Taxes in Nevada are lower than most states, primarily because of state taxes on gambling.

All that money comes from somewhere.

It comes from people who gamble.

I, and maybe you, hear of people who won money.

Is that the general case? Do most people win? Do half the people win? NO, NO, and NO.

People aren’t going to tell you, "Hey, I went to Reno and lost $1,000."

I have a friend who, when he was first married, went to Reno, just after pay day. He gave ½ his money to his wife and told her not to give it to him. Well, a couple of hours later, he came back and said, "I think I’m getting it. I just need a little more." "NO" "Really, I am getting it, give me ½ of what you have, and we’ll get it all back." So she did. An hour later, the same thing happened. THEY ATE POTATO SOUP FOR A MONTH, UNTIL THE NEXT PAY DAY. They’re the lucky ones. They learned you don’t win. The unfortunate ones are those that win on their first trip. They think, "Hey, I did it once, I can do it again." They’re the big losers. If casinos could, they’d make sure that their customers won on their first time. Kind of like cigarettes. Once you got ‘em, they’re hooked. You know, when I was a kid, cigarette companies would give free cigarettes away at high schools.

Lottery: When Calif. started the lottery, the odds of winning a jackpot were 1 in 14,000,000. Do you know that you’re more likely to be struck by lightening than win the lottery. Every year, more people in the U.S. are struck by lightening than win jackpots. So, Calif. decided that they need more people to play, so they changed the odds to 1 in 23,000,000. What that did is make fewer winners, but the jackpots were larger. A couple of years ago, that happened again. The odds of winning now are around 1 in 44,000,000.

But, you think that’s bad? If you’ve heard of PowerBall, that’s sponsored by several states, the odds of winning are something in the order of 1 in 80,000,000. Makes for large jackpots, but fewer winners (which, of course, means more losers).

I’ve heard the lottery described as a tax on the mathematically challenged.

The odds for the original 6:49 are: 49x48x47x46x45x44 / (1x2x3x4x5x6) = 13,983,816

The odds for the 6:53 lottery are: 53x52x51x50x49x48 / (1x2x3x4x5x6) = 22,957,480

Let’s talk about 9. You know that Nx9 is probably correct if the answer digits add up to 9.

How ‘bout 3? The sum of the digits is divisible by 3.

Q. If the digits of a number add up to 3 (or a multiple) is the number divisible by 3? Yes.

Let’s talk about school for a minute.

I told you at the beginning that I dropped out of college.

Over the course of my life, that has cost me 100s of 1,000s perhaps 1,000,000s of $s.

How many of you want to play professional Baseball? Basketball? Football? Be an actor? Win the Olympics? How many kids from Antioch have successfully done any one of those things?

Do you know who Jose Canseco is? He was a very good baseball player. Made a lot of money. Played for the Oakland As for several years.

He has a younger brother, also a good baseball player. No doubt they played and helped each other when they were kids. The younger brother wasn’t good enough to be in the major leagues. He was a minor league player for a few years. Minor league players get paid little more than MacDonalds burger flippers. They don’t travel in airplanes, they ride in crowded gym smelling buses. AND HE WAS GOOD – PROBABLY BETTER THAN ANY PLAYER YOU’VE EVER SEEN, OTHER THAN MAJOR LEAGUE PLAYERS.

Stay in school – that’s your future.

For every year you’re in school, you’ll earn more money in your life.

Don’t hurry to get married. Don’t hurry to have children.

Do you know what’s the most common source of arguments in families?

Money. Specifically, not enough money.

Stay in school. Go as far as you can.

If you’re having trouble with something, ask for help.

Are you a slow reader, ask for help.

Is math hard for you, ask for help.

Do you have trouble writing, ask for help.

I truly hope that all of you enjoy your future as much as I did.

But you’ll need to learn to make it fun.

BEES:

Have you ever heard of the bee dance.

There are some who claim that the bee dance is a myth.

There are others who claim that bees talk to each other.

When you have spare time, look up "honey bees dance" on the internet.

The joy of 6

6 is a triangular number, meaning that it is the sum of a group of consecutive integers starting with 1 (in this case, 1+2+3). Other triangular numbers are obviously 10, 15, 21, and so on.

Q. Where do you see the triangular number 10? Bowling alley.

6 is called a perfect number since it has the unusual property that the sum of all its factors (excluding itself) equals itself, i.e., 1+2+3 = 6. Only a handful of perfect numbers are known to exist. The next one is 28 (1+2+4+7+14), and the one after that is much larger.

In our base ten system, integral powers of 6 (greater than a zero exponent) end in the digit 6: 6, 36, 216, 1296, 7776, 46656, ... This is true because n2 mod 10 = 6). The digit 5 works the same way. The digits 0 and 1 technically also have this property, but they are much more boring since 0 raised to any power is still zero, and 1 raised to any power is still one.

How can I tell if a number is divisible by 6? Easy. First of all, any number that is divisible by 6 is also divisible by 2 and 3. Now, all multiples of 2 are "even" numbers, i.e., they end in one of the digits 0, 2, 4, 6, or 8; any multiple of 3 will have digits that sum to another multiple of 3. Therefore, any multiple of 6 will have both these properties. For instance, you can tell quickly that 2343654738 is a multiple of 6, whereas 236547176 is not.

Snowflake and cube both have 6 sides.

element 6 - Carbon - basis of all known life - unique properties

6 colors in the rainbow (by some official counts)

benzene molecule

occasionally six-fingered people are born Marilyn Monroe had 6 toes.

A google is 1 with 100 zeros. Or, 10^100 power. A googleplex is 10 with a google zeros.

What’s a PRIME number? Only divisible by itself and 1.

(When I was in school, 1 was a prime number. I’m told that today, it’s not. It’s not important.)

Some examples are: 2 ,3, 5 ,7 ,11 ,13 ,17 ,19 ,23 ,29 ,31 ,37 ,41 ,47 ,53 ,57 ,61 ,67 ,71 ,73 ,79

How do you check to see if a number is prime? Test only prime numbers lower than square root.

http://www-groups.dcs.st-and.ac.uk/~history/Indexes/Full_Chron.html

This is new. It's a quadratic series methodology. I don't understand it yet. But I don't want to lose it.

Note, a quadratic series is different from a geometric series.

3, 7, 13, 21, 31 (+4+6+8 ..)

n{th} term = a + (n - 1)d + 1/2(n-1)(n-2)C

a = The first tern

d = The difference between the first and second term

C = The common second difference.

3 + (n-1)4 + 1/2(n-1)(n-2)2

3 + 4n - 4 + n² - 3n + 2

= n² + n + 1 <=== ANSWER

8,11,17,26,38 (+3+6+9+12 ..)

Because the gap between the numbers isn't constant, then it must be a quadratic sequence. We can find the nth term using a formula:

n{th} term = a + d(n - 1) + 1/2(n-1)(n-2)C

a = The first tern

d = The difference between the first and second term

C = The common second difference.

8 + 3(n-1) + 1/2(n-1)(n-2)3

= 8 + 3n - 3 + 3/2(n² - 3n + 2)

= 1.5n(n - 1) + 8 <==== ANSWER

OR 1.5n² - 1.5n + 8

Things to learn.

KISS Keep It Simple Stupid.

Use symbols that tell you what they mean.

Write what it says. Then check it. Maybe 2-3 times.

KISS -- What's the easiest way to solve it?

Anton is twice as old as his wife, Marie.

8 years ago, Anton was three years younger than three times Maire's age. How old is Marie now?

A = 2M

(A-8)+3 = 3(M-8) (write A-8 and M-8, cause that's what we're dealing with, then add the 'stuff'.)

Just write down what it says.

Go over it a couple times to make sure you have it right.

Then you have 2 equations and 2 unknowns, and I know you can do that.

Ed: The next thing to do is try to solve it in your head.

2M -8 +3 = 3M -24 You can do that in your head.

Then swap sides, and signs, and your answer is pretty easy. not 24 not 3.

Then check to make sure your numbers work. They do.

Always use letters that tell you what they represent.

Never use x and y. A is Anton, M is Marie. That makes your thinking about it easier.

Always substitute the easy equation into the hard equation.

Be careful with the Anton was 3 years younger than.

You have to add the 3 to make the equation correct. That's an easy mistake to make.

Sometimes, but not this time, you can add the equations together. EG

-4j+4 = 5k-2

4j-4 = 7k-22 we can either add or subtract them. this time it's easier to add.

0j+0 = 12k -24 What you want is for one unknown to cancel out.

To subtract, you change ALL the signs on one of the equations, then add.

Sometimes you can multiply all the factors of one equation by 2 or 3, or divide by 2 or 3,

or whatever it takes, to get one of the unknowns to cancel out when you add or subtract.

That's legal.