Paul P. Mealing

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Tuesday, 25 June 2013

Algebra - the language of mathematics

I know I’m doing things back-to-front – arse-about - as we say in Oz (and possibly elsewhere) but, considering all the esoteric mathematics I produce on this blog, I thought I should try and explain some basics.

As I mentioned earlier this year in a post on ‘analogy’, mathematics is a cumulative endeavour and you can’t understand calculus, for example, if you don’t know algebra. I’ve come across more than a few highly intelligent people, of both sexes, who struggle with maths (or math as Americans call it) and the sight of an equation stops them in their tracks.

Mathematics is one of those topics where the gap, between what you are expected to know and what you actually learn, can grow as you progress through school, mainly because you were stumped by algebra. You know: the day you were suddenly faced with numbers being replaced by letters; and things like counting, adding, subtracting, dividing, multiplying, fractions and even decimals suddenly seemed irrelevant. In other words, everything you’d learned about mathematics, which was firmly grounded in numbers – something you’d learned almost as soon as you could talk – suddenly seemed useless. Even Carl Jung, according to his autobiography, stopped understanding maths the day he had to deal with ‘x’. In fact, his wife, Emma, had a better understanding of physics than Jung did.

But for those who jump this hurdle, seemingly effortlessly, ‘x’ is a liberator in the same way that the imaginary number i is perceived by those who appreciate its multi-purposefulness. In both cases, we can do a lot more than we could before, and that is why algebra is a stepping-stone to higher mathematics.

Fundamentally, mathematics is not so much about numbers as the relationship between numbers, and algebra allows us to see the relationships without the numbers, and that’s the conceptual hurdle one has to overcome.

I’ll give a very simple example that everyone should know: Pythagoras’s triangle.

I don’t even have to draw it, I only have to state it: a2 + b2 = c2; and you should know what I’m talking about. But a picture is worth innumerable words.

The point is that we can use actual integers, called Pythagorean triples, that obey this relationship; the smallest being 52 = 42 + 32. Do the math as you Americans like to say.

But the truth is that this relationship applies to all Pythagorean triangles, irrespective of their size, length of sides and units of measurement. The only criteria being that the triangle is ‘flat’, or Euclidean (is not on a curved surface) and contains one right angle (90o).

By using letters, we have stated a mathematical truth, a universal law that applies right across the universe. Pythagoras’s triangle was discovered well before Pythagoras (circa 500BC) by the Egyptians, Babylonians and the Chinese, and possibly other cultures as well.

Most of the mathematics, that I do, involves the manipulation of algebraic equations, including a lot of the stuff I describe on this blog. If you know how to manipulate equations, you can do a lot of mathematics, but if you don’t, you can’t do any.

A lot of people are taught BIDMAS, which gives the priority of working out an equation: Brackets, Indices, Division, Multiplication, Addition and Subtraction. To be honest, I’ve never come across a mathematician who uses it.

On the other hand, a lot of maths books talk about the commutative law, the associative law and the distributive law as the fundaments of algebra.

There is a commutative law for addition and a commutative law for multiplication, which are both simple and basic.

A + B = B + A  and  A x B = B x A (that’s it)

Obviously there is no commutative law for subtraction or division.

A – B B – A  and  A/B B/A (pretty obvious)

There are some areas of mathematics where this rule doesn’t apply, like matrices, but we won’t go there.

The associative law also applies to addition and multiplication.

So A + (B + C) = (A + B) + C  and  A x (B x C) = (A x B) x C

It effectively says that it doesn’t matter what order you perform these operations you’ll get the same result, and, obviously, you can extend this to any length of numbers, because any addition or multiplication creates a new number that can then be added or multiplied to any other number or string of numbers.

But the most important rule to understand is the distributive law because it combines addition and multiplication and can be extended to include subtraction and division (if you know what you're doing). The distributive law lies at the heart of algebra.

A(B + C) = AB + AC  and  A(B + C) ≠ AB + C (where AB = A x B)

And this is where brackets come in under BIDMAS. In other words, if you do what’s in the brackets first you’ll be okay. But you can also eliminate the brackets and get the same answer if you follow the distributive rule.

But we can extend this: 1/A(B - C) = B/A - C/A (where B/A = B ÷ A)

And  -A(B – C) = CA – BA  because (-1)2 = 1, so a minus times a minus equals a plus.

If 1/A(B + C) = B/A + C/A then (B + C)/A = B/A + C/A

And  A/C + B/D = (DA + BC)/DC

To appreciate this do the converse:

(DA + BC)/DC = DA/DC + BC/DC = A/C + B/D

But the most important technique one can learn is how to change the subject of an equation. If we go back to Pythagoras’s equation:

a2 + b2 = c2  what’s b = ?

The very simple rule is that whatever you do to one side of an equation you must do to the other side. So if you take something away from one side you must take it away from the other side and if you multiply or divide one side by something you must do the same on the other side.

So, given the above example, the first thing we want to do is isolate b2. Which means we take a2 from the LHS and also the RHS (left hand side and right hand side).

So b2 = c2 – a2

And to get b from b2 we take the square root of b2, which means we take the square root of the RHS.

So b = (c2 – a2)

Note b ca  because (c2 – a2) c2 - a2

In the same way that (a + b)2 a2 + b2

In fact (a + b)2 = (a + b)(a + b)

And applying the distributive law: (a + b)(a + b) = a(a + b) + b(a + b)

Which expands to  a2 + ab + ba + b2 = a2 + 2ab + b2

But (a + b)(a – b) = a2 – b2  (work it out for yourself)

An equation by definition (and by name) means that something equals something. To maintain the equality whatever you do on one side must be done on the other side, and that’s basically the most important rule of all. So if you take the square root or a logarithm or whatever of a single quantity on one side you must take the square root or logarithm or whatever of everything on the other side. Which means you put brackets around everything first and apply the distributive law if possible, and, if not, leave it in brackets like I did with the example of Pythagoras’s equation.

Final Example:  A/B = C + D    What’s B = ?

Invert both sides:  B/A = 1/(C + D)

Multiply both sides by A:   B = A/(C + D)   (Easy)

Note: A/(C + D) A/C + A/D

To understand why, take the inverse:  (C + D)/A = C/A + D/A

And take the inverse again:  A/(C + D) = 1/(C/A + D/A) A/C + A/D


Eli Horowitz said...

"Obviously there is no commutative law for subtraction or division."

Plausibly that's because subtraction and division aren't true operations. Really, subtraction is just the addition of a negative number and division is just multiplication by a number between 0 and 1.

Paul P. Mealing said...

Hi Eli,

Yes, and, basically that's the way it is in algebra: a number can be positive or negative and it can be inverted or not, but it's important to keep track of those attributes.

In fact, if I was teaching this, I'd make that point about the associative law: you can do them in any order you want as long as the negative numbers stay negative and the inverted ones stay inverted.

I think algebra is a conceptual hurdle for a lot of people and I don't think BIDMAS helps at all. It assumes you're ignorant and, if you depend on it, it will keep you ignorant.

Regards, Paul.