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Integers Essays

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This fun-ducational essay is based on material deleted from the article Innumeracy. Feel free to help fill in any gaps.

Depending on how recently you've taken a math course, you may or may not remember that there are different types of numbers, and depending on high you get into math, there are a lot of types of numbers, but, there are a few sort of basic ones.

Natural Numbers[edit]

The first type that is hard to argue about are called Natural Numbers, which are whole numbers (not a decimal or fraction) that are positive (on the right side of a 0). Sometimes 0 is considered a natural number, and sometimes it is not, but for now let's act like it is. It is hard, probably impossible, to really deny or not understand the concept of a natural number or 0. However, there is a wide ranging myth that the concept of 0 was invented (discovered?) by the Maya. This is a myth in the sense that it was also invented (discovered?) by the Indians at about the same time, and a few other societies had at least a placeholder for 0 in their counting systems[1].

Integers[edit]

Integers simply adds the idea of negative numbers to natural numbers. If a number is a Natural Number, it is also an Integer. You would think that the idea of negative numbers is fairly straight forward, but it sadly isn't. Everyone's favorite mathematician insists that negative numbers are evil [2], or something. There is also always a bit of confusion over just how to do operations with negative numbers. For example, for multiplication and division of negative numbers, the result is only negative if one number is negative and the other is positive and if a negative number is raised to an exponent, it is positive if the exponent is even. Knowing how negatives work is pretty important because they are used for a lot of things, not just knowing that in electronics a negative flow means it is heading in the opposite direction, but also in understanding how far in debt someone is.

Confusion over negative numbers can first appear in elementary school, mainly because teachers constrain math to counting numbers, and insist that is the whole scope of mathematics. In some cases, there is a legitimate reason to exclude negative numbers (i.e. you're phrasing questions in terms of counting apples, and there's obviously no way to have -3 apples).

Consider the following simple math problem "5 - 8 = ?". While the correct answer is -3, confusion has resulted in the following:

  • A simple answer stating it can't be done. However, teachers using this method generally aren't upfront about restricting to counting numbers, and treat this as a trick question.
  • Switching around the 5 and 8 to make the problem "8 - 5 = ?". If you follow this logic, "Bobby Tables has 5 apples, and after you take away 8, he is left with 3."

And that's why math is so hard for some people.

And sometimes, people just don't understand that -1 is larger then -2[3]

Rational Numbers, Real Numbers, and Irrational Numbers[edit]

There isn't much problem with Rational and Real Numbers, but they should be explained because they are how Irrational Numbers are defined. Basically, a Rational Number is one that can be expressed as a fraction with an integer numerator and a non-zero Natural Number denominator (remember, numerator is the top of a fraction and denominator is the bottom). [4] So, 0.333... can be expressed as "1/3", so its a rational number. An issue can be had in figuring out if a number is rational or not if it looks like this: 0.30769230769..., which is 12/39. Basically, a number can be a whole lot of garbage, but if it can be expressed as a fraction without zero in the bottom, its rational.

If you have a number that cannot be expressed as a fraction, it is an irrational number. Simple enough. If I have 123.456, I can make that a fraction by putting it over 1, and so can .3845784758944132475. But, because √2 produces a number that does not end, it cannot be expressed as a fraction, so it is not rational. It is irrational. Remember, with an irrational number: the decimal goes on forever without repeating.[5]

A Real Number is basically every number that falls into one of those categories. A number that is not real is called an Imaginary Number, and the imaginary number i is created by √-1

There are a few key Irrational Numbers, now that we know what they are. The first is the Greek letter ϕ, or Phi if you don't speak Greek. Next up is 'e', which is a mathematical constant used in, among other things, finding compounding interest. So, yes, that big pile of garbage I just made you read is important for you to read if you want to understand how the interest in your bank account and on your mortgage piles up. Some non-technical places to learn about e can be found here[6][7]

Finally is Pi. There isn't so much innumeracy with Pi, but there is some hilarious things about it. For example, the Bible makes a reference to what would require Pi to find, and has it that Pi is 3. A brief description of this can be found here, and a more indepth discussion here[8]. For a response from everyone's favorite fundie, see hereAnd once upon a time, someone tried (thankfully they failed) to define Pi Biblically, as exactly 3[9]. For a real headspin, there is this guy, proving that the value of Pi is 4[10]

And now for something completely different!

The Imaginary Number and Complex Numbers[edit]

“”The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and not being.

—Gottfried Wilhelm Leibniz

A problem mathematicians had for awhile is, how do you deal with ? The solution eventually decided upon was to just invent a number, specifically the Imaginary Number i. In teaching the concept if i, some prefer to say "i is a number, deal with it", while others say it is simply a short hand version of saying "square root negative one". Ultimately, as long as the concept is grasped, it doesn't make much difference. Complex numbers are numbers comprising a "real" part and an "imaginary" part. While this may seem silly to anyone who hasn't encountered imaginary numbers before, it is a very powerful branch of mathematics, and just because something is imaginary, it doesn't mean that it doesn't exist.

There are some pretty amusing videos on youtube of people loosing their freaking minds over how evil the concept of i is. Here is a twoparter that gets pretty funny towards the end. Andy, of course, agrees that Complex Numbers are the debbah

[edit]

Math Songs: Integers

Integers Rap

Yeah…Integers are whole numbers
That are negative, positive, or zero.
If you master this simple trick,
They’ll all call you a hero.

Two opposite signs you take away,
And give the sign of the bigger one;
Two similar signs always add up
And keep that sign and then you’re done.

Opposite signs when multiplying
Just follow the following rule:
Just check the number of negatives
And you won’t become a fool.

It’s about the number
Of negatives that you multiply.
Even is positive, odd is negative;
That’s how to simplify.

Integers are useful
In comparing a direction.
Mr. Chatterjee, I don’t understand
Can you please repeat the question?

I said…Integers are whole numbers
That are negative, positive, or zero.
If you master this simple trick,
They’ll all call you a hero.

For adding and subtracting
You can use the number lines
Or you can just follow the rule
And it will be just fine.

Integer in Latin means
Untouched integrity.
They are all whole numbers
That go to infinity.

Integers can be neither decimals nor fractions.
They have a positive negative interaction.
Negatives and positives, that’s the trick.
So get your textbook and grab it quick.

I can travel across the number line
In rise or trough
Or I can pretend it’s sea water
Below and above.

I said…Integers are whole numbers
That are negative, positive, or zero.
If you master this simple trick,
They’ll all call you a hero.

Don’t Be Negative about Negative Signs

Chorus
Don’t be negative about negative signs
Just use your brain that you’ll train, yeah you’re gonna be fine
So don’t be negative about negative signs
You’ll be dancing with the answers to the questions in time

Verse I
When you’re adding two numbers and the signs are all the same
You add the absolute value and the sign doesn’t change
Let’s say you’re adding (−5) with (−5)
The answer’s (−10) with the negative sign
This time we’ll mix it up and we’ll use one negative
We’ll add integers with signs opposite
Subtract the smaller from the larger
Keep the larger number’s sign, so 8 + (−17) is (−9)

Chorus

Verse II
To subtract you change the sign on the number being subtracted
So sit back, relax and just let the math just happen
You’ll convert what you’re subtracting to the opposite sign
And add the two together—yeah the answer you will find
If you’re confused, listen up as the examples start to flow
Yeah, we got 10 − (−4)
4 becomes positive, we convert it you see
Now add them up and get 14

Chorus

Verse II
When you’re multiplying numbers there are tricks to keep in mind
Like positive times positive is a positive sign
And negative times negative is positive too
But negative times positive is negative, yo
Lets go on to dividing, the rules are just the same
The only change is that you’re in division’s domain
So the only time you get a sign that isn’t positive
Is when you divide two numbers with signs opposite

Chorus

Bridge
When negative numbers find their way in the equation
Sometimes it might feel like an alien invasion
I’m sayin’—it’s wacky, you know, but with some patience
You’re gonna succeed, yeah it’s success you’ll be tasting

 

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