Mathematics History

Equations Mathematics

In the algebra section, we looked at how we can use letters as numbers and how to find their values. There are times, however, when life gets a little harder:

x + y = 25

We cannot say what x and y are in the equation above because there are an infinite number of possibilities:

x = 1, y = 24
x = 21, y = 4 etc.

In general, a single equation with two unknowns (we don't know x and y in this case) cannot be solved. What we need is a second equation involving at least one of the unknown variables.

Simultaneous Equations

Suppose that we were given the following equations:

x + y = 25
3x = 15

We have two different equations with two unknowns - this is enough information for us to solve them both (that is, find the values of x and y). We call these simultaneous equations because we use them at the same time.

In the second equations, we can solve immediately for x (see the algebra pages if you're not happy with this):

3x = 15 so x = 5

Now we put this value of x (called substitution) into the first equation:

5 + y = 25 so y = 20

So the only correct solution for the two equations given at the top of the page is x = 5 and y = 20.

So far so good! Let's look at a harder example:

2a + b = 21
a + b = 15

In this case we cannot just solve for a or b individually, as both variables appear in each equation. Instead we must rearrange an equation to find an unknown in terms of the other:

b = 15 - a
[This is the second equation re-stated]

Now we can substitute this statement for b into the first equation:

2a + b = 21
2a + (15 - a) = 21

This can now be tidied up, giving a correct answer for a:

2a - a = 21 - 15
a = 6

Finally we can use any of the first 3 equations to find b:

b = 15 - 6
b = 9

Testing Simultaneous Equations

Now would be a good time for you to test what you have learned. Work out the value of each variable in the examples below:

2a + b = 21 a + 2b = 24 5p + q = 100 2p - q = -30 m + n = 0 6m - n = 49

Hints: Rearrange one equation to give you one variable. Substitute it, then solve. Be careful with minus signs!

Let's see how you got on:

1. 2a + b = 21
   a + 2b = 24

   Rearrange one of the equations: b = 21 - 2a
   Next substitute in the other: a + 2(21 - 2a) = 24
   Solve: a + 42 - 4a = 24
   Rearrange: 42 - 24 = 3a
   Answer: a = 6
   Finally substitute: b = 21 - 2×6
   Answer: b = 9
   

2. 5p + q = 100
   2p - q = -30

   Rearrange one of the equations: q = 100 - 5p
   Next substitute in the other: 2p - (100 - 5p) = -30
   Solve: 2p - 100 + 5p = -30
   Rearrange: 7p = 70
   Answer: p = 10
   Finally substitute: q = 100 - 5×10
   Answer: q = 50
   

3. m + n = 0
   6m - n = 49

   Rearrange one of the equations: m = -n
   Next substitute in the other: 6(-n) - n = 49
   Solve: -7n = 49
   Answer: n = -7
   Finally substitute: m = -(-7)
   Answer: m = 7

Quadratic Equations Mathematics

Brace yourself! These are probably the most complicated equations you'll meet at GCSE maths. That means two things:

1. You'll have to work a little harder to crack them
2. You'll get many more marks in exams when you do!

Don't worry why they're called quadratic - it basically means "involving squared powers".

What's a Quadratic?

Quadratic equations take the following form:
ax² + bx + c = 0

Where x is the only variable and a, b and c are just numbers (constants, that may also be zero!)

If a=0 then the equation is not quadratic: bx + c = 0

However, if b=0 then it can be: ax² + c = 0

Whilst if c=0 then it's: ax² + bx = 0

It is all much less confusing with numbers!

Quadratics With Numbers

Normally, of course, equations like ax² + bx + c = 0 are not written with a, b and c: they're usually just numbers.

e.g. 4x² - 3x + 5 = 0

It's your job normally to find the values of x for which the equation works - nightmare!

Let's start with equations of the form: ax² + c = 0

Solving Quadratic Equations

Solving equations like ax² + c = 0 can be quite straightforward.
e.g. x² - 25 = 0

From your work on algebra, you should be able to rearrange the equation to: x² = 25

By taking the square-root of both sides, we end up with:
x = 5

That wasn't too bad, was it? Another solution is x = -5, but we'll look at that another time.

Here's one for you. Find the solution to the equation: x² - 121 = 0.
Once you've worked it out, click here.

Modal Temperatures Mathematics

Sean asked the following:

We have a record of temperatures and we are asked to find the MODAL TEMPERATURES. Any help would be great.

Thanks for this, Sean! The word MODAL refers to the mode, which is a type of average of a set of numbers.

So, for instance, you could have a set of peak temperatures for a month, and what you're asked for is the mode (or the modal temperature). The mode, as you can see from our tutorial, is the most common of a set of numbers.

There can be more than one mode, so if 20°C was the peak temperature on 5 days, and 22°C was the peak for another 5 days, then you have what's called a bimodal set of numbers. More than 2 modes and it's a multimodal set.

Click on the tutorials link below to learn more about averages. Good luck!

Trigonometry Mathematics

The word trigonometry is very old. In fact the Ancient Greeks drew all this up. Was it important? Well, for one thing it allowed them to estimate the distance to the Sun!

It all starts with a triangle, specifically a right-angled one:

A right-angle is 90°, rather like the corner of a square. The hypotenuse is the longest side of a right-angle triangle, found always opposite the right angle.

Mathematical Explanation

"Mathematical Explanation:
Problems and Prospects"
THURSDAY, MARCH 1, 2001 - 4:00 PM; **ROOM 1116, IPST BLDG.

Paolo Mancosu
Department of Philosophy, University of California, Berkeley

In this talk I have three major aims. The first is to introduce the topic of mathematical explanation by listing a number of problems followed by a reflection on the status of research and prospects for further development. The general discussion in the first part motivates the specific contributions to be presented in the remaining two parts of the talk. In the second part I will draw attention to an important tradition in philosophy of mathematics for which explanation is a concern. Here I discuss Mill, Russell, Godel, Lakatos and other philosophers of mathematics on mathematical explanation. The last part of the talk will present a case study of a development in mathematical practice that originates from explanatory concerns, i.e. Alfred Pringsheim's "explanatory" approach to the foundations of complex analysis.

Paolo Mancosu is Associate Professor of Philosophy at U.C. Berkeley. His main interests are in mathematical logic and the history and philosophy of mathematics. He is the author of "Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century" (OUP 1996) and "From Brouwer to Hilbert."(OUP 1998).

If you have questions, contact the CHPS Office at (301) 405-5691 or by e-mail at hp26@umail.umd.edu. Information about colloquia is also available on-line through the CHPSCOL LISTSERV and on the WWW at http://carnap.umd.edu/chps or write to:

Probability and Pascal's Triangle Mathematics

Date: 28 Feb 1995 14:19:21 -0500
From: Dr. Ken
Subject: Re: Probability and Pascal's Triangle

Hello there!

Well, since I'm not quite sure how much you know about Pascal's Triangle
(for instance, you seem to know that it is connected to the study of
probability), I'll start pretty much from the beginning.

The way I see it, Pascal's Triangle is kind of a collection of neat things
in mathematics.  The way you construct it follows:

                                 1
                               1   1
                             1   2   1
                           1   3   3   1
                         1   4   6   4   1
                       1   5  10   10  5   1
                     1   6  15  20   15  6   1
                   1   7  21  35   35  21  7   1
                                 .
                                 .
                                 .

You start out with the top two rows: 1, and 1 1.  Then to construct each
entry in the next row, you look at the two entries above it (i.e. the one
above it and to the right, and the one above it and to the left).  At the
beginning and the end of each row, when there's only one number above,
put a 1.  You might even think of this rule (for placing the 1's) as included
in the first rule: for instance, to get the first 1 in any line, you add up the
number above and to the left (since there is no number there, pretend it's
zero) and the number above and to the right (1), and get a sum of 1.

When people talk about an entry in Pascal's Triangle, they usually give a
row number and a place in that row, beginning with row zero and place
zero.  For instance, the number 20 appears in row 6, place 3.

That's how you construct Pascal's Triangle.  An interactive version where
you can specify the number of rows you want to see can be found at

 http://mathforum.org/dr.cgi/pascal.cgi  

But Pascal's Triangle is more than just a big triangle of numbers. There
are two huge areas where Pascal's Triangle rears its head, in Algebra
and in Probability/Combinatorics. 

First let's look at the Algebra version.

Let's say you have the polynomial 1+x, and you want to raise it to some
powers, like 1,2,3,4,5,....  If you make a chart of what you get when you
do these power-raisings, you'll get something like this:

(x+1)^0   =                           1
(x+1)^1   =                      1    +    x
(x+1)^2   =                 1    +   2x    +    x^2
(x+1)^3   =             1   +   3x    +   3x^2  +    x^3
(x+1)^4   =         1   +  4x    +   6x^2  +   4x^3  +    x^4
(x+1)^5   =     1   +  5x   +  10x^2  +  10x^3  +   5x^4  +    x^5  .....


If you just look at the coefficients of the polynomials that you get, voila!
Pascal's Triangle!  Because of this connection, the entries in Pascal's
Triangle are called the _binomial_coefficients_.  They are usually written
in parentheses, with one number on top of the other, for instance

20 =  (6)   <--- note: that should be one big set of
     (3)              parentheses, not two small ones.

I don't think it's standard notation, but when I write binomial coefficients
in a text document like this, I usually write them [6:3].

The other main area where Pascal's Triangle shows up is in Probability.
Let's say you have five hats on a rack, and you want to know how many
different ways you can pick two of them to wear.  It doesn't matter
to you which hat is on top, it just matters which two hats you pick.
So this problem amounts to the question "how many different ways can you
choose two objects from a set of five objects?" The answer? It's the
number in the second place in the fifth row, i.e. 10. (Remember that the
first number is in place zero.)

Because of this choosing property, the binomial coefficient [6:3] is
usually read "six choose three."  If you want to find out the probability
of choosing one particular combination of two hats, that probability is 1/10.

There's a pretty simple formula for figuring out the binomial coefficients.
It's
           n!
[n:k] = --------
       k! (n-k)!
                      6 * 5 * 4 * 3 * 2 * 1
For example, [6:3] =  ------------------------  =  20.
                      3 * 2 * 1 * 3 * 2 * 1


That's a basic introduction to Pascal's Triangle.  It certainly also shows
up in lots of other places (for example, the triangular numbers are in
there, if you know what those are), but I think it would be too much for
me to go into those right now.  Thanks for the question!

-Ken "Dr." Math

Mathematics History