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Happy Valentine's Day, one day late! We hope you all had a wonderful day yesterday, no matter how (or if) you celebrated the day. For all of our online friends, I have the only Valentine's Day card you'll ever need for your mathematician friends.

It references counting principles in combinatoric mathematics, and in case you're not familiar with the notation, please allow me to explain.

Suppose you went to a pizza restaurant, and when you ordered your pizza, they told you, "You can choose from any of the following toppings: pepperoni, olives, hamburg, mushrooms, onions, ham, pineapple, or sausage. Of those, you must choose three toppings." (I know, what restaurant demands that you choose exactly three toppings?)

So how do you determine the number of possible pizzas? Well, the reasoning goes something like this:

  1. I have eight ways to choose the first topping (because there are eight toppings available).
  2. Once I've done that, I now have seven ways to choose the second topping (because I've already chosen one).
  3. Now for my third topping, I have six choices.
  4. That gives me 8 x 7 x 6 = 336 possibilities.
  5. BUT WAIT A MINUTE! If I chose "pepperoni, olives and ham" - is that really any different from choosing "olives, ham, and pepperoni"?? No, it's not; it's the same pizza no matter what order I pick the toppings in. So I have to eliminate the repetitions.
  6. It turns out the number of repetitions is 6. Why? Because you could put pepperoni in 3 positions, leaving 2 positions for olives, and then 1 position for ham. 3 x 2 x 1 = 6.
  7. So we take the total number of possibilities (336) and, recognizing that every different pizza has been counted six times, we divide by 6: 336/6 = 56. There are 56 different ways of choosing your pizza.

There's a nice way of representing the fact that we've got 8 things to choose from, aned we're choosing 3 of them, and it looks like this:

If you crank out those factorials (in case you're not familiar with the notation, 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1), you'll see that it works out to 56.

But how do we read that funny notation with the parentheses and two numbers inside of it? Since there are eight things, and we are choosing three, we read it as follows: "Eight choose three."

So, in honor of Valentine's Day, be sure to tell your favorite mathematician: 

"I choose you."
 

One of my Pre-Cal students asked me a few days ago if the Law of Sines was hard to prove. I told him I'd write up a proof here. I also made my Pre-Cal class work through the proof themselves. I'll share their train of reasoning here.

First, we stated the Law of Sines:

in triangle ABC, a/sinA = b/sinB = c/sincC.

They quickly recognized that we need a diagram to work from. So we drew something like the diagram to the right.

 

Then one of the students said, "Maybe we should draw in an altitude." Thus we had the following diagram. Their choice to label that new point F was somewhat random, I think, but I liked it, because F can stand for "foot" of the altitude.

After puzzling over their diagram for a bit, someone suggested setting up an equation involving the sine function and triangle ABF, and after a bit more thinking, a second equation involving the sine function and triangle CBF:

sinA = h/c; sinC = h/a.

They then did one of their favorite algebraic steps: multiplying both sides of an equation by a denominator in order to get rid of fractions (they don't like fractions, so they're always happy to get rid of them):

csinA = h; asinC = h.

A little substitution gives csinA = asinC. They then chose to divide both sides of this equation by sinAsinC, which gives:

c/sinC = a/sinA.

At this point, someone said, "Well we can just do this all over again with another side." Sure enough, if you drew an altitude to a different side, you'd get b/sinB = c/sinC (or b/sinB = a/sinA, depending on which altitude you drew).

Thus, a/sinA = b/sinB = c/sinC, and the Law of Sines is proved. Nice teamwork, class!

This question comes from one of my Pre-Cal students, who wants to know if there's a nice rule like the "sum and product" factoring rule, that works when the roots are complex numbers.

The sum-and-product method he's referring to is this:

If we have a quadratic x2 + bx + c, and we can find two numbers (let's call them m and n) that add to b and multiply to c, then the quadratic factors into (x + m)(x + n). Here's an example:

Factor x2 + 13x + 36.

In order to factor this, I look for two numbers that add to 13, and multiply to 36. Can you find them? I usually start by asking, "What are two numbers that multiply to 36?" The first pair I think of is 6 x 6. But 6 + 6 isn't 13, so that doesn't work. Next I think of 9 x 4. Aha! 9 + 4 is 13. So, since 9 x 4 = 36 and 9 x 4 = 13, I conclude that:

x2 + 13x + 36 = (x + 4)(x + 9).

Okay, so now the question is, what if the quadratic factors with complex numbers? Is there a nice way of doing this factoring rule?

Let's start with an example:

x2 - 4x + 5

If I knew that the factorization of this resulted in complex numbers, I would remember another rule, that complex roots come in pairs: if one root is a + bi, then the other is a - bi.

So, we look for two numbers such that:

(a + bi) + (a - bi) = -4
(a + bi)(a - bi) = 5

Simplifying these gives:

2a = -4, or a = -2
a2 + b2 = 5
4 + b2 = 5
b2 = 1
b = +/- 1

Therefore, this factors as follows:

x2 - 4x + 5 = (x - 2 - i)(x - 2 + i) Multiply this out, and sure enough, it's true.

Let's try generalizing this a bit. Suppose the quadratic was x2 + mx + n, and we wanted to find complex roots with the sum and product method.

(a + bi) + (a - bi) = m
(a + bi)(a - bi) = n

2a = m
a2 + b= n

a = m/2

m2/4 + b2 = n
b = SQR(n - m2/4)

So let's take another example, and put our rule into practice.

x2 + 6x + 13

a is half of 6, or 3.

b is SQR{13 - 62/4) = SQR{13 - 9} = 2.

Therefore x2 + 6x + 13 = (x +3 + 2i)(x + 3 - 2i).

This isn't too bad, but it presumes that the leading coefficient of the quadratic is 1; things might get a bit dicier if it wasn't. And, of course, if you're paying close attention, the result we obtain is actually just a slightly re-arranged version of the quadratic formula as it would appear in the special case of the leading coefficient being one!

Everett, from Illinois, writes, "I was born in 1937 in Southern Illinois, Franklin County. We were not taught PEMDAS in school. When did they start teaching this order of operations. Is it taught all over the state now?"

Hi Everett, since the beginning of people writing down mathematical ideas, mathematicians have had to develop conventions for indicating the meaning of expressions on paper. Those rules have not always been consistent. For instance, as a precursor to parentheses, some mathematicians used to write expressions with a horizontal bar over the top to indicate that things should be grouped.

Consider the expressions 7 x 2 + 3. This looks like it should be 14 + 3 = 17. But if the author of a text wanted the addition to be done first, he would write:

    _____
7 x 2 + 3

Typographically, this was cumbersome, and eventually led to the idea that parentheses could be used, and would be simpler to write. This was first suggested in the early 1600s, so it was in vogue as a mathematical ordering notation long before you were born.

Exponents were first used much earlier than the 1600s, but there was no "official" way to write them. Some used Roman Numerals for exponents, some wrote a number after a variable, but not raised (thus, 3x meant 3 times x, but x3 meant x to the third power). It was probably Rene Descartes who made the raised notation popular, and it became the standard. Descartes was doing this in the mid 1600s (not too long after the parentheses started becoming popular). And because of the way it was written, outside the normal baseline of the equation, it had an implicit ordering to it. You would never see the following expression:

23 + 1

and interpret it as though the 3 and the 1 should be combined first, and then 2 raised to the 4th power.

So there you have the first two elements of PEMDAS - parentheses and exponents - that have been implicitly understood since the 1600s.

As for multiplication and division coming before addition and subtraction, I'm not clear on exactly when these became part of the standard ordering notation. However, if you took an Algebra class in high school, you used this ordering of operations, even if no one taught you a name for what you were doing. Consider the following examples:

Example One: Multiplication before Addition
5 + 3x

If multiplication was not evaluated before addition, this would be evaluated (as some of my Algebra students like to do) as:

5 + 3x = 5 + 3 · x = (5 + 3)x = 8x.

The ordering of multiplication and addition was implictly given in the rules for how we can manipulate terms in an expression.

Example Two: Division before Addition
5 + 1/x 

If division was not evaluated before addition, this would be evaluated as (5 + 1)/x or 6/x. Again, this order of operations was inherent in the rules for manipulating terms of an algebraic expression.

Which brings me to the answer to your question. First, I do think it's possible that you were never taught PEMDAS as a specific notational convention, but all the rules of the PEMDAS mnemonic were in existence long before you were born. Secondly, even if you weren't taught it specifically, you still used it when you did Algebra, and if you passed your algebra class, then you were doing it correctly! Finally, yes, PEMDAS (or BODMAS*) is taught consistently worldwide. There are still arguments that arise about some ambiguous notations (you may have read one or more posts about those), but PEMDAS is the standard convention for interpreting expressions worldwide.

* BODMAS is used (I believe) in Britain, and perhaps other places as well. B = Brackets, O = Order. Brackets are parentheses, of course, and Order is the word used for Exponents, so even though it looks different, it's the same convention.

S. Ventar writes, "I saw video that explains divisibility for 19. Multiply the last digit by 2 and add to left over digits. repeat until result two digits multiple of 19. Why does method work?"

Hi S! That's a great question. Let's do an example first, to make sure everyone understands the method. Let's take the number 65968 and do the process on it. Ready?

65968: 6596 + 2 x 8 = 6612
6612: 661 + 2 x 2 = 665
665: 66 + 2 x 5 = 76
76: 7 + 2 x 6 = 19

Since the result of this repeated process is 19, the number is divisible by 19. Note that you can stop the process anywhere you happen to know that a number is a multiple of 19. For instance, if you knew that 76 was a multiple of 19, you wouldn't have to do the last step. Similarly, if you knew that 76 wasn't a multiple of 19, you could stop there.

For completeness, let's do one that isn't a multiple of 19, and then we'll explore your questions of why this works. Let's do the number 6121.

6121: 612 + 2 x 1 = 614
614: 61 + 2 x 4 = 69
69: 6 + 2 x 9 = 24

And I know that 24 is not divisible by 19, so I conclude 6121 is not divisible by 19 either.

Before I go any further, I want to introduce you (in case you're unfamiliar) with a branch of mathematics we call "modular arithmetic." In modular arithmetic, we aren't dealing with equations; instead, we're dealing with something called "congruences." A congruence looks like this:

a ≡ b mod c

It looks like an equation, except that instead of an equals sign, the symbol in the middle has three bars. We read this congruence like this:

"a is congruent to b mod c."

It's a short-hand way of saying that if you subtract b from a, you get a multiple of c.

Thus, all of the following are true:

7 ≡ 1 mod 3 (7 - 1 = 6, which is a multiple of 3)
10 ≡ 1 mod 3 (10 - 1 = 9, which is a multiple of 3)
15 ≡ 0 mod 5 (15 - 0 = 15, which is a multiple of 5).

There are a couple of nice rules for congruences that will help us see why this divisibility rule works.

Multiplicative Congruence Property
If a ≡ b mod c, then ax ≡ bx mod c.

Additive Congruence Property
If a ≡ b mod c, and d ≡ e mod c, then a + d ≡ b + e mod c.

These should look very familiar to you; they are just like the multiplicative and additive properties for equality, except that they involve congruences (with the same modular base - note that the c stays constant throughout!).

With all of this in mind, let's explore what happens with our 19 divisibility rule. First of all, I'll start by pointing out that any integer can be written in the form 10x + y, with the y representing the last digit, and x representing the number formed by all the other digits. Here are some examples:

145: x = 14, y = 5; 145 = 10(14) + 5
32794: x = 3279, y = 4; 32794 = 10(3279) + 4

Let's suppose we have a number K which is divisible by 19. We can write K = 10x + y, for some integer x, and y representing the last digit of K. Using congruences, we can write:

Congruence 1. 10x + y ≡ 0 mod 19 

Now, just because K is divisible by 19 doesn't mean that 10x is divisible by 19. Let's say that when we divide 10x by 19, we get a remainder of z. This means we can write:

Congruence 2.  10x ≡ z mod 19

Subtracting the second congruence from the first gives us the following:

Congruence 3. y ≡ -z mod 19

If we do our process on the number 10x + y, we obtain the following: x + 2y. (divide 10x by 10, and add twice the last digit). Remember this result, because we're going to return to it later! If we start with 10x + y, the next number in our process is x + 2y. If we can show that x + 2y is congruent to 0 mod 19 whenever 10x + y is congruent to 0 mod 19, then we'll have shown that the divisibility rule works (because each step in the process will recursively be a multiple of 19 as well).

Now it's time to invoke the Multiplicative Congruence Property on Congruences 2 and 3, using the number 2 as our multiplier:

Congruence 4.  20x ≡ 2z mod 19 

Congruence 5.  2y ≡ -2z mod 19 

Now we note that 19x has no remainder when divided by 19, because x is an integer, so 19x must be a multiple of 19. This means:

Congruence 6.  19x ≡ 0 mod 19

Combining 4 and 6, using the Additive Congruence Property (technically, subtractive property, or multiplication by -1 and then addition) gives us:

Congruence 7.  x ≡ 2z mod 19 

Now add together Congruences 5 and 7:

Congruence 8. 2y + x ≡ -2z + 2z mod 19 

Congruence 9. 2y + x ≡ 0 mod 19.

In other words: if you start with a multiple of 19, and do the process described, you'll end up with another multiple of 19. And if you do the process on that number, you'll still have a multiple of 19. And so on, and so on. And that's why the process works.

Incidentally, I should add that as far as I know, no one actually uses this divisibility rule for anything; the more cumbersome a divisibility rule is, the less useful it is compared to doing long division. After all, if a number is divisible by 19, you probably want to know what the other factor is, so you'll end up doing long division anyway!

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