How it works : Mathematics


Mathematics is often thought of as dealing with numbers, and associating numbers with quantities, but it is much more than that .It is a language in its own right, which has been evolved to define exactly the relationships between physical objects, and to simplify discussion of them, just as any other language is used to talk about humans' thoughts and feelings. It has many of the qualities of a natural language, and involves more than computing or arithmetic.

In branches such as geometry, numbers play no real part (although it is possible to re-express the results in terms of numbers); in others such as calculus and algebra the language deals with the most general case, and numbers do not occur until the final stage when they are substituted for letters to study special cases. Today, electronic computers and calculators do all the manipulation of numbers, where this is needed at all.

It is tempting, but unwise, to classify branches of mathematics according to their practical value. The important application of mathematics over the last century have been of results found earlier, which seemed at the time quite useless.

Arithmetic began with the natural numbers, 1, 2, 3... (in mathematics, the symbol '. . .' means 'and so on', without specifying the actual numbers), and the operations of addition and multiplication. The natural numbers are inconvenient because subtraction is not always possible: we can take 3 from 5 but not 5 from 3. Accordingly the integers or whole numbers (positive and negative) are defined to always allow subtraction, by the rule that since 3 from 5 is 2, 5 from 3 is to be written a. This can be shown as follows:

The same problem arises with division, and is again answered by inventing new numbers. 24 divided by 4 gives 6, but divided by 5 does not give a natural number. The fractions, or rational numbers, were invented with the result that dividing 24 by 5 is written 24/5. The rational numbers are dense in the sense that between any two there lies another (their average, for example), so there is no 'next' one.

The early Greek geometers thought that the rationals were all the numbers there were and set up their theory of similar figures on that basis. The idea was to represent numbers by suitable shapes, which is why we talk today of a number multiplied by itself as being its square, while a number multiplied by itself twice is the cube of the number:

By this means Pythagoras, or maybe his collaborators, were able around 500 ad to prove the theorem that the square of the side opposite the right angle, the hypotenuse, in a right angled triangle, is the sum of the squares on the other two sides. (A theorem is any proposition which is not obvious, but which can nevertheless be proved).

Pythagoras' theorem gives a good insight into the way mathematics relates to the physical world: the relationship a2 = b2 + c2, which holds true for any right angled triangle ever drawn, shows how to turn shapes into mathematics. It can be used to find unknown quantities-given any two sides, the third can be calculated-and so has immediate practical uses.

Applied to a square which has sides one unit long, the theorem says that the square of the diagonal must be 2 units; but no rational number can be squared to give the answer 2.

The proof that the square root of 2 (Ö2) cannot be represented by a rational number was discovered by Pythagoras. A rational number is a number that can be represented as a fraction-one integer (whole number) divided by another. In this way, the rational number 6.2 can be represented as 31/5.

To prove the irrationality of Ö2 take two integers m and n such that m/n cannot be reduced to a fraction containing smaller integers-unlike, for example, 62/10, which can be reduced to 31/5, leaving its value unchanged, by dividing the top and bottom by 2.

Therefore m and n cannot both be even numbers, or they would both be divisible by 2.

Now if m/n= 2, then m2/n2= 2, and m2 = 2n2. But twice any integer is an even number; therefore m2 is even, which means that m is also even (an even number multiplied by an even number gives another even number).

Thus, there must be another integer which is half of m. Call it p. m = 2p, so m2= 4p2. Substituting 4p2 for m2 in the original statement m2/n2 = 2 gives 4p2/n2=2. It therefore follows that n=2p2.

By the same argument, however, n2 must therefore be even, and so must n. m has already been shown to be even.

But m and n cannot both be even (see above), so there can be no rational number of the form m/n such that m /n= 2. Therefore

Ö2 is irrational.

This result gave rise to the first of the great crises to the foundations of mathematics, which was solved by inventing vet more numbers-the irrationals. These and the rationals together form the real numbers all numbers expressed by (finite and infinite) decimals. The number whose square is 2 (the square root of 2) is now 1.41421.......

The square of 1.41421 differs from 2 by 0.0000100759; no matter how many more figures are taken the square cannot quite be 2. But it can approach 2 are nearly as desired. The real numbers are the final step in constructing bigger number systems so long as we restrict attention to numbers which are in some order (so that for any unequal pair we can determine which is the greater). But it became necessary from the sixteenth century onwards to extend the number system one stage further, by giving up the requirement of ordering.

This need arose in the solution of equations, and because of the 'double negative' situation. In mathematics, as in any language, two negatives multiplied together produce a positive (for example, 'to not not do something' means 'to do something'). If we ask what numbers when squared give the answer 1, the answer is two-fold: 1and - 1. If we ask what numbers when squared give zero, there is but one, zero. But if we ask what numbers when squared give - 1, there are no such real numbers. It was possible to invent complex numbers, defined in terms of reals and one other unit, called i by mathematicians and now j by electrical engineers, whose square is - 1.

General results about numbers are the domain of algebra, using symbols (usually letters) to represent quantities. An example of this has already occurred, for the statement a2 = b2 + c2 is an algebraic one referring to the right angled triangle, in which numbers do not arise until we need a specific result.

Algebra depends on equalities, and interprets the relationships between quantities in terms of equations. By making a number of statements about a situation, each one an equation, it is then possible to draw inferences from the statements using simple rules, and arrive at an answer for an unknown quantity or quantities.

A simple algebraic equation is, for example, y = 9/5x +32. This is the conversion of a temperature of x° Centigrade into its equivalent in Fahrenheit, y°F. A graph of y against x looks like:

This gives a straight line graph, and the equation y = 9/5x + 32 is therefore called a linear equation. It has the general form y = ax + b, where a and b are constants, which can be replaced by any numbers, depending on the application; x is called the independent variable and y the dependent variable, since it depends on x.

Many quantities, however, do not vary linearly but involve squares, cubes or higher powers of numbers. Power in this case means the number of times the number is multiplied by itself -2 to the power 4, Or '2 to the fourth' is 24= 2 ´ 2 ´ 2 ´ 2 = 16.

An example of a non-linear equation is y = 2x2 which looks like:

Non-linear equations have a curved graph, this one being the curve called a parabola. Because the square of a negative number is positive, the values of y are always positive and the curve has two upward arms. This gives the result that when, say, y= 8, x has two values (called the roots of the equation):
either -2 or -2. An equation of this form is called a quadratic, with the general form y = ax2 +bx+c.

Real progress in algebra had to wait till the sixteenth century when the detailed solution of equations was attempted. For quadratic equations the rule had been known since Babylonian times; but if the cube of the unknown enters the problem (cubic equations), the solution (found about 1500) involves first solving a certain quadratic. Half a century later equations involving the fourth power of the unknown (biquadratics) could be solved, the process involving first solving a subsidiary cubic. But equations of fifth degree (quintics) when tackled similarly relied for their solution on an equation of sixth degree. It was not till the nineteenth century that Abel showed that no such method could be found for the quintic.

Geometry is the study of relationships in space and was the principal component of Greek mathematics. Among its most widely known results are the properties of the circle:

Basic properties of a circle:

diameter d = 2r where r is the radius. An arc whose length is equal to the radius, r, always subtends the same angle, q, at the centre of the circle. q = 57.3 and this angle is called one radian. The ratio of the circumference, C, of a circle, to its diameter is a constant number given the Greek symbol p (pi).

p =c/2r = 3.1416 ........... The circumference c is equal to 2pr because there are 2p radians around a circle (360o = 2p radians). Area of a circle = pr2.

As well as straight lines and circles in the plane (that is, flat), the Greeks studied three dimensional geometry and the shapes made by cutting circular cones with planes. These shapes are called the conic sections-the elipses, parabolas and hyperbolas.

Plane sections of the cone. Plane 1 cuts the cone twice, so producing the two branches of the hyperbola, plane 2 cuts in an ellipse, plane 3 is parallel to the generators and so produces the parabola.

But for us the importance of these curves is in celestial mechanics -the motions of planets, satellites, and so on. Kepler found that the orbits of the planets round the Sun were ellipses; the orbit of an artificial satellite round the Earth is an ellipse; and the limiting case of this, when we throw a ball in the air, is a parabola.

We do not now derive these properties by the methods of the Greeks, but two things from Greek geometry remain valuable to us. One is the axiomatic system-that everything is supposed to be derived from axioms or 'self-evident truths'. Greek or Euclidean geometry is confined to flat planes. But this is not the only sort possible, and it was by questioning the absolute truth of these axioms that Euclidean geometry was supplemented by non-Euclidean geometries. These were used by Einstein in 1915 to replace Newton's theory of Gravitation by General Relativity.

The second important aspect of Greek geometry was the invention of trigonometry by Hipparchus (about 125 BC) and Ptolemy (about 150 AD).

If two triangles have the same angles,but different sizes, their corresponding sides are proportional. Hipparchus realized that a table could be made relating the ratio of the sides to the angles in degrees. It proved sufficient to do this for right angled triangles: the ratio of the side opposite one of the smaller angles to the hypotenuse is called the sine (abbreviated to sin) of the angle.

Since any triangle can be divided into two right angled ones by forming a perpendicular from one corner to the opposite side, tables of sines of angles expressed in degrees suffice to calculate the ratios of sides for all triangles.

The Greeks were originally interested in trigonometry for astronomy: it was only much later that it was used for surveying the Earth as in the triangulation method used in map-making. But its importance is not in applications but in the introduction of the idea of a function.

A function is a rule that governs the way one value depends on another. It assigns to every number in a certain set (its domain) exactly one value, the value of the function. The sine function assigns a certain ratio to every angle. The cosine (cos) and tangent (tan) are other examples of functions based on the sides and angles of a right angled triangle. The sine and other trigonometric functions were for a long time the only examples, until the invention of the Logarithm and exponential functions; and they were the most important ones, because they are periodic functions.

If an arm rotates, so as to sweep out an ever increasing angle, it will have made a complete rotation at 360°; for larger angles than that, the sine repeats itself.

Mathematicians rarely describe angles in degrees, but instead use radians, where 360°= 2p radians. This arises because a radian is defined as the angle between the centre of a circle and an arc the length of the radius. Since the circumference is 2pr, there are 2p radians around a circle, in 360° .

Equation of circle is x2 + y2 = r2 (from Pythagoras' theorem). As q increases, the radial arm sweeps around the circle. When it has travelled 360° (2p radians) it has completed one circle. But sin q = y/r so adding 2p to q does not change the value of sin q . sin q = sin (2p + q) = sin (4p + q) . . sin q is therefore a periodic function.

So for the sine function there is a certain number, the period, here 2p, which is such that adding it to any element of the domain leaves the value of the function unchanged. Periodic phenomena are very common in practice (see Sine Wave). It was realized in the eighteenth century that the behaviour of almost anything which varies periodically can be reproduced by adding a sufficient number of sines or cosines of different sizes ---called Fourier analysis.

The motion of a heavy stretched string or the oscillation of a particle along a line attracted to a fixed point on the line are examples of two problems in which sines enter although there is no connection with angles. In the same way, logarithms enter in problems of natural growth when the original use for multiplication does not come in at all.

The later investigations of functions all hinge on the calculus, which arose in the seventeenth century in mechanics, the study of the forces on objects and the motion they produce

(see Dynamics and Statics). The key notation is the 'derivative function of a function'.

It is easy enough, for example, to calculate the distance, x, travelled by a particle moving with a steady velocity, v, in a time, t, using the formula x = v ´ t. In reverse, we can calculate the average velocity by dividing the distance travelled by the time taken, in other words v = x/t In many problems, however, the velocity itself will be changing, such as that of a stone accelerating towards the ground:

Trajectory or body thrown off cliff. Because of earth's gravity its vertical velocity increases (from zero), while its horizontal velocity remains unchanged. The body therefore falls in a curved path.

ab is a small part of the body's trajectory. When the distance ab is made small enough the trajectory can be considered a straight line. Around time t = 6.5 sees the instantaneous velocity is distance ah divided by time taken to travel ab-as ab approaches zero.

It is then important to know how the velocity changes with respect to time. This introduces the idea of instantaneous velocity, that is, velocity at an instant. This can be found by considering very small intervals of time, and the correspondingly short distances travelled. Over very short intervals, the velocity can be said to be practically constant. This can be done at each instant of the motion and so the velocity is, in turn, a function of time, the derived function or derivative of distance with respect to time. This process is called differentiation.

The mathematical process of differentiation, which can be applied to any desired function (nor just velocity), is a matter of manipulation of the function according to standard rules.

During each minute time interval, the distance travelled by the falling stone is the instantaneous velocity multiplied by the time interval. Consequently, the total distance travelled by the stone will be the sum of all these small distances. This process is called integration, and is the reverse of differentiation.

The falling stone example is a simple case of the most important problems in calculus especially where applications are concerned, the solution of differential equations.

Infinite series
In many problems encountered in science and engineering an exact answer cannot be found. Instead an approximate answer is determined which contains a series of terms where each succeeding term is of less importance than the one before it. Only with an infinitely large number of such terms can the exact answer be found, but an engineer is not concerned with such accuracy-he only needs to know how many terms to include to give a sufficiently precise value for his purposes. Infinite series are, however, very important.

As already mentioned, Fourier analysis of a generally periodic function (of f periods per second, that is, a fundamental frequency f) indicates that it is composed of an infinite series of sine waveforms (sinusoids) of frequency f; 2f, 3f. . . . But if each of the terms (sinusoids) of this infinite series were of roughly the same magnitude, the combined effect would be an infinitely large function. Yet we know that this is not so the original periodic function is finite and measurable. Fourier analysis also shows that each succeeding term is of less importance than the last-that is, the higher frequency components in the series are of diminishing magnitude.

The theory of convergence worked out in the nineteenth century, particularly by Cauchy, provided the criterion of whether an infinite series has a finite sum or not. For example, the series 1 + 1/2 + 1/4 + 1/8 +1/16+... is convergent and has a sum (in the limit, its sum is 2).

Not all decreasing series converge, however; the most famous one that does not is called the harmonic series 1 + 1/2 +1/3 + 1/4 + 1/5 +...... It is easy to see this, because 1/3 + 1/4 is greater than 2/4 = 1/2; 1/5 + 1/6 + 1/7 + 1/8 is greater than 4/8 = 1/2 and so on. So the sum,if the series were convergent, would be greater than 1 + 1/2 + 1/2 +1/2 +..... Actually the sum of n terms of the harmonic series differs by an almost constant amount from the natural logarithm (that is, to the base e) of the number n; this constant, whose value is roughly 0.577... ,is called Euler's constant. It is rather a mysterious number. No one knows yet whether it is the solution of an algebraic equation.

Reproduced from HOW IT WORKS p1472