by R. M. Stainforth
(1) In arithmetic a “square” is simply the result obtained by multiplying any number by itself.
25 is the square of 5 ( = 5 x 5 ) : 4 is the square of 2 ( = 2 x 2 )
and so on.
A “square root” is the opposite. It is the number which was multiplied by itself to obtain a square. In the examples above,
5 is the square root of 25 : 2 is the square root of 4
Likewise 10 is the square root of 100, 25 of 625, and so on. To save a lot of writing, mathematicians use a lot of ‘symbols’ to express mathematical ideas. The symbol for a square root is like this:
You can see for yourself that to write:
is a lot shorter than: “the square root of 100 is 10”
(2) In the examples above the square roots are whole numbers. The corresponding squares are sometimes called “perfect squares” because they have whole-number roots.
But every number has a square root and you can find it by a fairly simple calculation. But in most cases the roots are fractions which are called “irrational” because the formula for calculating them results in long decimals which could go on and on without ever coming to an end. For practical purposes we only use the first few numbers of the decimal. For instance, * You can check that with a calculator, but you will also find that to use only two places of decimals (1.73) or, at most, three places (1.732) is accurate enough for all ordinary use.
Some calculators will give you the square root of any number if you press the symbol, or you can buy tables of them.
The calculations below can be done on a calculator to check on and
For normal practical purposes, 1.99…=2, 2.99…=3, and so on.
(3) “Squares” in geometry.
A square, as you know, is a rectangle whose sides are all the same length. If you divide each side into equal segments, then join all the opposite points, you get a family of small, equal squares. For instance:
4 equal segments on each side,
16 small squares
So you see the connection. Four ‘squared’ becomes 16, and similarly for any number you choose. That explains the use of the geometrical word ‘square’ in arithmetic.
(4) Algebra and squares.
Don’t be frightened of the word algebra. Actually it’s from Arabic, like numerous other English words beginning with “al-”, which simply means “the”: alcohol, alchemy, alembic, and you can find others in a dictionary.
Algebra is really arithmetic, but it deals with numbers in a general sense. It uses a symbol, very often the letter x, which can stand for any number at all until you limit it in some way.
As a very simple example, let’s say that
We know immediately that x is the square root of z
but that’s all we know until we have more information.
Let’s now say that
x must be 5: there is no alternative (unless we go into negative numbers, but that’s a whole new field).
Try another simple case:
you are told that
(the unknown number y is twice the unknown x)
and at the same time
(the unknown x and y total 6 when added together)
Fiddle around in your head and you’ll quickly see (I hope!) that the only solution is that
Note that there are any number of solutions (answers) when you only have one of the equations, but only a single solution that satisfies both of them.
I hope this is enough to demonstrate to you that algebra is a shorthand way of expressing mathematical ideas and relationships. You just dealt with a pair of what are called ‘simultaneous equations’ and in algebra all we needed to write was:
Now try to express the concept in English without using symbols. The shortest I can find is:
“Find two numbers: they add up to 6 and one is twice the other.”
This requires 50 letters/numbers whereas the algebraic way needed only 10 letters/numbers/symbols—and this is a very simple case.
(1) René Descartes was a French philosopher and mathematician born 400 years ago.
His name is preserved in the phrase ‘Cartesian coordinates.’ They are a way of converting mathematical problems, equations, formulæ, etc., into drawings which are much easier to understand than a jumble of numbers and symbols.
You can begin with graph paper, of which the simplest type shows a grid of small equal squares. Usually (but not necessarily) the bottom line and the left-hand margin are thicker than the lines of the grid, and they are marked off with equal divisions. Usually these divisions are marked with successive numbers, from left to right across the bottom and upwards on the left-hand margin.
Now, suppose you want an easy way to find half of any number you think of, without making an arithmetical calculation. You begin with a diagram like ‘Figure 1.’ Mark off some points (as done in red) where you know the answer. For instance, start off at 10 on the bottom row then move up to the level of half-of-ten, = 5, on the side line and mark the point where your pencil is. Do the same for 4/2, 8/4, 24/12 points (as in red) or any others you like of the same family.
Now you can’t help but notice that all these known points lie on a straight line.
Next you realize that if you choose any number you like, find its place on the bottom line, move straight up to the sloping line, and read across to the side line, that gives you half of your first number.
Just as an example, I chose 12 1/2 (see purple lines) and the diagram (‘graph’) shows that half of that is 6 1/4.†
(2) Simply to save space and a lot of writing, we usually call the bottom line of a graph the “X-axis” and use it for plotting the numbers we know or choose to start with. They are often called “x.” We call the left-hand margin the “Y-axis” and use it for plotting the numbers derived from the “x” numbers. Quite often we call these derived numbers “y.”
In Figure 1 the Y-number for any point on the red line corresponds to half the X-number immediately below.
You can see that it needed two lines of writing to convey that idea, but in algebra we shorten it to simply
Let’s try another graph. See Figure 2. This time we plot the squares (remember them?) of the numbers on the bottom line (the X-axis). In algebra we say,
We agreed that Figure 2 would be the graph of the equation . The method of construction is just the same as on Figure 1, which plotted but this time you can see that the result is not a straight line. It is a curve (called a parabola). You can use it to get the square root or the square of a number. For example, you can check that 4 1/2 squared = 20, (there is some approximation, partly depending on how accurately the curve is plotted):
(3) A general point about graphs of algebraic equations. You might reasonably wonder why (or if) the red lines plotted on Figure 1 and Figure 2 stop where the X-axis and Y-axis meet. (This is called the zero-point.) The answer is NO. In the complete picture the red line would continue to the left BUT when it passes the zero-point it gets into the field of negative numbers, and that is a complication that we don’t need to discuss today.
You will meet the whole picture in the next section, when we get to the circle, but we’ll just take it for granted without comment.
(4) The circle in Cartesian coordinates
See Figure 3. It’s easy, with a compass, to draw a circle and then it seems natural (symmetrical) to make its centre the zero-point and to put the X-axis and the Y-axis through it, as shown. We’ll call its radius “r”. But how do we find an equation to match the circle???
The secret is that long ago a Greek mathematician named Pythagoras made a discovery in geometry. It was about triangles in which one of the three angles is a right-angle (90˚). The apparent shape can vary quite a lot as sketched below:
if we keep the long side the same length (which we’ll call “r”, as in the circle above), the combined area of the two squares drawn on its short sides is equal to the area of the square drawn on the long side—no matter whether it’s a “fat” or a “skinny” triangle.
That’s quite a mouthful to understand straight away, but if you look at the examples in Figure 4 you can see for yourself that it’s true.
Now look at Figure 3 again. You could draw any number of triangles like the two shown, with one of the small angles at the zero point and the other on the circle itself. In each one the longest side would be the same, = radius of the circle (= r), and the angle opposite the long side would be a right-angle.
So, treating this drawing as an algebraic graph and applying Pythagoras’ Theorem, you can see that the equation for the circle is , where r is the symbol we have chosen for the radius of the circle.
Pi, always written as “π” is a letter in the Greek alphabet corresponding to our P and it is the accepted symbol for the ratio of the area of a circle to the area of a square which exactly contains the circle.†
The area of the circle is πr2, by definition. The area of the square, as you can see, is 4r2 and the area of the “inside square is 2r2 (half as big). So obviously π is smaller than 4 but bigger than 2. Our problem is to find an exact figure for it.
Areas in general
If your diagram, a piece of land, the floor space in a building, or any other space, is bounded by straight lines, its area is quite easy to calculate. You can divide it into triangles and rectangles and the solution is straightforward.
But when the boundaries are curved the straightforward methods are no use.
We have to go to a branch of mathematics called calculus. It was invented by the famous Sir Isaac Newton about 300 years ago. His basic idea was to think of the diagram (graph) as if made up of a lot of parallel strips, all of the same width, each consisting of a narrow rectangle plus a small triangle caused by the slope of the curved graph. Depending on the algebraic equation, he calculated the area of the rectangular strips. Then he visualized the strips being made narrower and narrower (but more and more of them), so that the little triangles all became smaller and smaller until their total area could be dismissed as ZERO. I’m sorry I can’t be more precise, but that was his basic idea… and it worked!
As regards a circle, we decided that its formula is .
The radius can be as long or as short as we like, so we are going to say that it is one—one foot, one inch, one yard, one meter, any unit you like. In algebraic terminology, r=1 .*
So the equation for the circle becomes
which can be rearranged as
Now take the square root of each side and it becomes
By the square-root calculation mentioned earlier, we can find the square root of . It is a long series of fractions which could be calculated one after another without ever coming to an end, but they get progressively smaller until there’s no practical value in continuing. You have to accept my word for it, the answer is:
Now, back to areas.
Isaac Newton used special symbols for his calculus. One was a funny tall ‘S’ which we call the integral of a mathematical formula. Loosely speaking, if you have an equation stating that y = some formula with an x in it, then the integral of y, written , tells you the area under the curve on your graph.
In the present instance , so we want the integral of . To find this, we use the series of fractions above and apply a basic formula which says (you just have to believe me!)
The result (remembering that x = 1) is a new series of fractions, namely:
To make this series usable, we convert all the common fractions to decimals and add them up:
and then we subtract them from the ‘1’ at the beginning:
and this number (0.79101…) is the numerical value of . This is the area of only a quarter-circle (the one on Figure 3 with the purple triangles in it). So the answer we are looking for is
This is supposed to be π but you know from your books that the true value of π is 3.14159…, so our answer is too high. The questions that arise are:
(a) What went wrong? (b) Can we fix it?
The answers are:
(a) The fractions in the series we used got progressively smaller, but even so we stopped calculating after only a few terms. If we had continued, the remaining fractions, even though smaller and smaller, would have added a little bit to the addition-sum above. If they only added 0.006 (a quite tiny amount), our first number would become 0.78501…, and 4 times that = 3.14064…, which is very close to the true value of π.
(b) To cure our difficulty, we need to find a series of fractions which get smaller, one after the other, much more quickly than in the first series we used. The secret is to find a comparable series, but one which uses “1/2” instead of “1” for the top line of the fractions
whereas 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 always = 1
—no matter how many fractions you include in your series.
quickly becomes a very small amount and after only a few fractions of the series are calculated, the rest become so very tiny that you don’t need to bother with them. Once a mathematician gets the idea of using a formula that gives rapidly-shrinking series of fractions, there are several different ways to do it. My own favourite uses the diagram shown on Figure 5. Please look at it…
Again the circle is the graph of , or but now we consider only the coloured area, not the whole quarter-circle.
In calculus the formula for this area is an integral expressed as
On the diagram it divides into two natural parts. The red part is a segment of a circle, like a wedge of pie. You can see that twelve such segments would make up the whole circle and you know that the area of the circle is π, so the area of the red part has to be .
The green part is half an equilateral triangle (which is one with all of its sides and angles equal). From Pythagoras’ Theorem, we know that its height is half the square-root of three , so the green area
SO… from the geometrical diagram of Figure 5 we arrive at the algebraic equation:
(the whole coloured area) (the red area) (the green area)
By rearranging, we get:
so π = 12 x the difference between those two numbers
SO… to find π we need to calculate two irrational numbers:
The first one, , is another series of fractions, related to the one above but modified to shrink away quickly, namely:
which simplifies to:
As before, we convert the common fractions to decimals and add them up, noting how quickly they shrink to almost zero.
and now subtract the total from the “1/2” at the beginning:
For the second one, we have to calculate the square root of three to as many places of decimals as we like, let’s say seven. When you know how, this is quite straightforward, and the answer is:
So to get the number we want,
Now we’re ready. If we subtract the second decimal from the first one, the answer should be one-twelfth of π. Let’s see!
E U R E K A!!!!! That agrees with the accepted value of π to the fifth place of decimals, which is more accurate than is ever normally necessary.
So here we are! We set out looking for an accurate value for π and we found it. In ancient Egypt, when the measurement of land (= geo + metry) was a serious matter for Euclid and his colleagues, they used the approximation π = 3. We can see now that their usage was reasonably close but not precise.
* The line of dots means that the decimal goes on and on, but we choose to stop here.
† The numbers used for finding each point are called its coordinates. In geography the location of anywhere in the world can be identified by its coordinates of longitude and latitude. For instance, Ottawa is roughly 45° North, 76° West. You may find similar use of the word in a more local sense, on maps and atlases.
* Here is a new symbol. That little 2 indicates that you multiply the number by itself . If you multiplied three, or four, or five, or as many as you like of a number together, you would write it as x3, x4, x5, and so on. (You will see some examples before we get to π.)
When you talk about a number multiplied by itself several times, all except for twice and for three times, you use the phrase “x to the 4th power,” “x to the 5th power,” and so on. Very often the word “power” is omitted (understood) and “x to the fourth” is enough.
In the two exceptions, x2 ( = x multiplied by itself) gives the square of x, as we have already discussed. In speaking, x2 is called “x squared.” Then if you do it once again, , we call the result the cube of the number. = 4-cubed = the cube of 4. (The reason for using a geometric word ties in with the use of “square” for second-power numbers.)
† Also the ratio of the circumference to the diameter = π
* The reason is that in the formulas which you will shortly see, to multiply or divide by one makes no differences.