A equals B

Weather icon: temperature equal

Weather icon: temperature equal (Photo credit: Wikipedia)

The whole universe is full of inequality. No two galaxies are exactly alike, no two planets are exactly alike, no two grains of sand are exactly alike, no two atoms of silicon are exactly alike. Wait a minute, is that last one correct?

Well, in one sense each atom of silicon is alike. Every silicon atom has 14 protons in its nucleus, and, usually, 14 neutrons. However it could have one or two neutrons extra if it is a stable atom, or even more if it is a radioactive atom. Alternatively it could have less neutrons and again it would be radioactive.

Monocrystalline silicon ingot grown by the Czo...

Monocrystalline silicon ingot grown by the Czochralski process (Photo credit: Wikipedia)

So two silicon atoms with the same number of neutrons in the nucleus are “equal” right? Well, of course a single atom by itself is seldom if ever found in nature, and two isolated similar atoms are very unlikely. But suppose.

An atom of silicon is said to have electron shells with 14 electrons in them. Without going into unnecessary details these electrons can be in a base (lowest) state or in an excited state. With multiple excitation levels and multiple electrons the probability of two isolated atoms of silicon with all electrons in the same excitation state is extremely low.

Atom Structure

Atom Structure (Photo credit: Wikipedia)

In practise of course, you would not find isolated atoms of silicon at all. You would find masses of silicon atoms, perhaps in a random conformation, or maybe in organised rows and columns. One of the tricks of semi-conductors is that the silicon atoms are organised into an array, with an occasional atom of another element interspersed.

Atoms according cubical atom model

Atoms according cubical atom model (Photo credit: Wikipedia)

This has the effect of either providing an extra electron or one fewer in parts of the array. Under certain conditions this allows the silicon atoms and the doping element to pass the extra electron, or the lack of an electron (known as a hole) along the array in an organised manner, a phenomenon known in the macroscopic world as an electric current.

English: Drawing of a 4 He + -ion, with labell...

English: Drawing of a 4 He + -ion, with labelled electron hole. (Photo credit: Wikipedia)

So, while two atoms of silicon may in some theoretical physical and chemical sense be equal, in practice, they will be in different states, in different situations. What can be said about two silicon atoms is that fit an ideal pattern of a silicon atom, in that the nucleus of the atom has 14 protons. Some of the properties and states of the two atoms will be different.

At the very least the two atoms will be in different locations, moving with different velocities and with different amounts of energy. They can never be “equal as such. The best that you could probably say is that two atoms of the same isotope of silicon have the same number of neutrons and protons in their nuclei.

Periodic table with elements colored according...

Periodic table with elements colored according to the half-life of their most stable isotope. Stable elements. Radioactive elements with half-lives of over four million years. Half-lives between 800 and 34,000 years. Half-lives between 1 day and 103 years. Half-lives ranging between a minute and 1 day. Half-lives less than a minute. (Photo credit: Wikipedia)

When we talk about numbers we stray into the field of mathematics, and in maths “equal” has several shades of meaning. When we say that one integer equals another integer we are essentially saying that they are the same thing. So 2 + 1 = 3 is a bit more than a simple equality and in fact that expression can be referred to as an identity.

Algebraic proofs are all about changing the left hand side of an expression or the right hand side of the expression or both and still retaining that identity between the two sides.

Mnemosyne with a mathematical formula.

Mnemosyne with a mathematical formula. (Photo credit: Wikipedia)

In the real world we use mathematics to calculate things, such a velocities, masses, energy levels, in fact anything that can be calculated. Issues arise because we cannot measure real distances and times with absolute accuracy. We measure the length of something and we know that the length that we measure is not the same as the actual length of the object that we are measuring.

Lengths are conceptually not represented by integers but by ‘real numbers’. Real numbers are represented by two strings of digits separated by a period or full stop. Both strings can be infinite in length though the both strings are usually represented as being finite in length.

1 Infinite Loop, Cupertino, California. Home o...

1 Infinite Loop, Cupertino, California. Home of Apple Inc. and one of Silicon Valley’s best known streets. (Photo credit: Wikipedia)

If we measure a distance with a ruler or tape measure, the real distance will usually fall between two marks on the ruler or measure. So we can say that the length is, say, between 1.13 and 1.14 units of measurement. If use a micrometer we might squeeze and extra couple of decimal places, and say that the length is between 1.1324 and 1.1325. With a laser measuring tool we can estimate the length more accurately still.

You can see what is happening, I hope. The more accurately we measure a distance, the more decimal places we need. To measure something with absolute accuracy we would need an infinite number of decimal places. So when we say that the distance from A to B equals 1.345 miles, we are not being exact, but are approximating to the level of accuracy that we need. Hence A is not really equal to B.

Aurora during a geomagnetic storm that was mos...

Aurora during a geomagnetic storm that was most likely caused by a coronal mass ejection from the Sun on 24 May 2010. Taken from the ISS. (Photo credit: Wikipedia)

A particularly interesting case of A not being equal to B is in the mathematical case where one is trying to determine the roots of an equation. There are various method of doing this and there is a class of methods which can be designated as iterative.

One first makes a guess as to the correct value, puts that into the equation which generates a new value which is, if the iterative method chosen is appropriate, closer to the correct value. This process is repeated getting ever closer to the correct answer.

Plot of x^3 - 2x + 2, including tangent lines ...

Plot of x^3 – 2x + 2, including tangent lines at x = 0 and x = 1. Illustrates why Newton’s method doesn’t always converge for this function. (Photo credit: Wikipedia)

Of course this process never finishes, so we specify some rule to terminate the process, possibly some number of decimal places, at which to stop. More technically this is called a limit.

To prove convergence, in other words to prove that the process will generate the root if the process is taken to infinity, has proved mathematically difficult. I’m not going to attempt the proof here, but after several attempts from the time of Isaac Newton, this was achieved last century, with the introduction of the concept of limits.

English: A comparison of gradient descent (gre...

English: A comparison of gradient descent (green) and Newton’s method (red) for minimizing a function (with small step sizes). Newton’s method uses curvature information to take a more direct route. Polski: Porównanie metody najszybszego spadku(linia zielona) z metodą Newtona (linia czerwona). Na rysunku widać linie poszukiwań minimum dla zadanej funkcji celu. Metoda Newtona używa informacji o krzywiźnie w celu zoptymalizowania ścieżki poszukiwań. (Photo credit: Wikipedia)

One can then say, roughly, that the end result of an infinite sequence of steps in a process (A) is equal to a required value (B), even though the result no particular step is actually equal to B. You have to creep up on it, as it were.

I’ll briefly mention equality in computer programs and social equality/inequality, if only to say that I might come back to those topics some time.

English: Income inequality in the United State...

English: Income inequality in the United States, 1979-2007 (Photo credit: Wikipedia)

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