The sine function
The sine curve is a fabulous example that concretely solidifies the scientific basis of thousands of natural phenomena. The function that generates the curve is simple:
f(t) = A . sin(ωt + θ),
where A is the amplitude of the wave, ω the angular frequency and θ the phase angle. The behavior of the curve itself bears a close relationship with the circle: draw a circle on a sheet of paper and mark its center as the origin of a Cartesian system. Now, if you were to trace the locus of the figure and project the position of your hand on the x-axis, you will observe that for every rotation that you make a point on the projection will first move away from the center, travel to the axial extreme (a length which is equal to the radius), come back to the center, and then move on to the other. If you were to plot the distance of that point from the center of the circle, you will get the sine curve – of course, you’ve to keep in mind the sign changes.
Because of its convoluted relationship with the circle, the sine curve has basis in its applications as a periodic waveform – a typical wave that repeats itself in design over a specific time period. Remember the path traced by a pendulum in your grandfather clock? Same. Also goes with undamped oscillations of a block suspended from a rigid ceiling by a spring, the propagation of heat waves, the vibration of strings on a guitar, the intonations in human speech, the signal processing that is required in most electronic gadgets, Heisenberg’s inequality and proofs of quadratic reciprocity.
2. Cosine
The cosine function
The cosine function differs from the sine function only in that the phase angle θ is displaced by 90 degrees. In other words, the cosine waveform is the sine waveform that’s got itself a small headstart. Such a wave becomes useful when one studies the propagation and interaction of multiple sine waves that give birth to interesting interference patterns like the one shown.
An interference pattern
3. Exponential
The exponential function
The graph is generated by the following function:
x(t) = α . βt/τ
Suppose that a gambler plays a slot machine with a one in n probability and plays it n times. Then, for large n (such as a million) the probability that the gambler will win nothing at all is (approximately) 1/e. And so Napier’s constant raises its fiery head.
At first, nobody saw nothing peculiar about the value of e – not until Jacob Bernoulli began to study the gambler’s issue and began to take note of a particular value that could be approximated to the value of another famous limit. Nestled between and 2 and 3 on the integer scale, the exponential growth factor manifests itself in hundreds and hundreds of mathematical problems, all of which have direct impact on business strategies and economic development. You will know the chessboard problem, wherein an Indian king was once gifted a beautifully crafted chessboard by a courtier. In return, the courtier asked for this: one grain of rice on the first square of the board, two grains on the second, four grains on the third, and so on. The king obliged, only to find that for 64 squares to be filled, 18,446,744,073,709,551,615 grains of rice would be needed – something that would weigh 461,168,602,000,000 kilograms!
Applications? Unsurprisingly many because of the way the function tends to grow with respect to time. If you observe the graph, you can see that for small increments of ‘x’ after some lower values, the value of the function skyrockets. This is similar to, if not the same as, the many growth factors manifested, for example, in the following:
- The Malthusian catastrophe
- Population of microorganisms in a culture
- Loudness of sound
- Nuclear chain reactions
- Ponzi schemes
- Moore’s law
- Compound interest
- The water lily problem
4. Logarithm
The logarithmic function
Draw a line passing through the origin and having a slope of 1, and using this line as a mirror, reflect the exponential curve. The image is the logarithmic curve, the inverse of the exponent. You must be reminded of having used the log tables, a small booklet with those long lists of extremely tiny numbers with which you could solve complex math problems. That’s the log function for you: it is the mathematical equivalent of scaling. The only difference is that, unlike in practical situations where it is constant, the scale in the universe of numbers is part of a sequence characterised by a growth ratio.
And as the exponential series represents growth, the log function represents a pattern on which to base that growth. The applications of both are the same since one is the inverse of the other, but it is worth detailing both because of the same reasons.
5. Asymptotes
Asymptotes
Asymptotes are like porn movies you find on the net for free: they tease and tease and tease, and then they end abruptly leaving everything else to your imagination. The one shown above is a rectangular hyperbola. The sections of the hyperbola that extend towards infinity along either axes are the asymptotes – and only because they exhibit a tendency towards touching the axes but never do. Put an other way, the first curve (+x-axis asymptote) gets closer to the second (+x-axis) as it gets farther away from the origin.
(x – h)(y – k) = m
m = x . y
The real application lies in the field of asymptotic analysis, which in turn is a key tool for exploring the ordinary and partial differential equations in the mathematical modeling of fluid flow through the Navier-Stokes equations.
6. Modulus
The modulus function
The modulus function is the noble gentleman amongst functions: you feed it with negative values but all it returns is their positive cunterpart. That is probably why it is also known as the absolute function.
Introduced by Jean-Robert Argand, and later conferred a denotation by Karl Weierstrass, the applications of this function pertain to the concepts of complex numbers, quaternions, ordered rings, fields and norms – which in turn are used in the modeling of real-world phenomena.
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Have you ever wondered how 10 digits that were introduced a few millenia ago gave birth to so many different and varied functions and behaviorisms? The need that gave rise to them in the first place was that of quantification: simple and abstracted notations that each stood for a particular value that was a multiple of one. The second step lay in the classifiability of these numbers into similar-seeming sets, each of which was deigned to behave the same way. After this categorization arose the applications, where real world objects were compared to the numbers in terms of their respictive classifiabilities. Next in line was dimensions: the number of objects in a particular direction. Ultimately, there came modeling which represented the interface between understanding what was already there and what we could do in order to mimic it. The computer and the engine, two machines that completely changed the way the world understood and functioned, are both conclusively based on the functions shown above. The computer uses the functions to generate higher values via (complex variations of) Boolean and set logic, while the engine uses them to magnify inputs to result in larger outputs. However, their true importance lay in the fact that each one of them represented hundreds of us. One computer or one engine did what a thousand humans could have together in a single day. They saved time; rather, they brought in extra time, time that was devoted to other purposes, time that quickened the process of civilisation.
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