The Spoken Reals in Pictures and Text

(Mathematicians call the real numbers the reals.)


The point ... IS ... that much mathematics and number crunching can be derived from the seven first principles which are represented by the oblong pieces. The cubical pieces represent needed concepts. The scale and the "hour glass" egg timer just happened to be on the counter, and I decided to use them.

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The first concept block, labelled C1 in the picture, represents sets and ordered pairs. Sets are just collections of things, and an ordered pair is a set of two things with a first element and a second element. Look down at a pair of dice and you will see a set of two things that is not ordered. Look at a person's two feet and you will see an ordered pair.

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The second concept block represents the empty set. I noticed that the block did not move the scale, and this seemed helpful. Like an empty box the empty set is something but contains nothing.

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Now the empty set has been moved down to the bottom of the picture where it will be needed later.

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For this first pass through the presentation, we can ignore the next two concept blocks. We consider the real number system to consist of a set of objects, called the real numbers, together with the two functions of addition and multiplication. Once the seven first principals are stated, everything else, like subtraction and division, is derived. In this first pass, I would like to show that these first principles are things that almost everyone already knows.

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The first five First Principles are called the Field Properties of the real number system - so the abbreviation FP is quite convenient!   FP1 says that the sum and product of any two real numbers does not depend on the order. Two times three equals three times two, and six plus one equals one plus six etc.   FP2 says that any three real numbers have a unique sum and product.   FP3 says that multiplying a sum of two real numbers by a third gives the same result as multiplying the two numbers in the sum by the third number and then adding.   FP4 says that zero and one exist. Zero is the "neutral element" for addition - add zero and there is no change. One is the "neutral element" for multiplication - multiply by one and there is no change.   FP5 says the following: (1) Every real number has an additive inverse, and, when the two are added, the result is zero. (2) Every real number except zero has a multiplicative inverse, and, when the two are multiplied, the result is one.

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I will skip the first magenta concept block, which is needed to give a precise definition of the second concept block - which represents subsets. For this first pass through the presentation it is sufficient to say that if all the elements of one set are in another, then the first is a subset of the second. The oblong piece here stands for the order property of the real numbers: Some are larger than others. Specifically, the principal says that there exists a subset of the real numbers such that (1) If two real numbers are in the set, then so is their sum and product, and (2) every real number is either in the set, equal to zero, or its additive inverse is in the set. The subset is the set of "positive" real numbers, which are the ones that are greater than zero.

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The first green concept block stands for the concept of an upper bound for a set of real numbers. If a given number is greater than or equal to every member of a set of real numbers, then the given number is said to be an upper bound on the set. If all other upper bounds are greater than or equal to the given number, then you have the least upper bound. Now recall that we moved the empty set down here. The last first principal says that every non-empty set of real numbers that is bounded from above has a least upper bound. It is easier to say "non-empty set" than to say "set which contains at least one element".

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Summary:   Again, the whole point of what we are doing here is that a small number of ideas can be used to derive all the things that give so many people so much trouble in number crunching and mathematics. On the one hand it is difficult because it is very abstract, but on the other hand it is a long lever that can move a world of trouble out of the way. Other essential long levers are to be found in the computational tools at the next two links. Calculators are inadequate because we need to do symbolic mathematics. Please find MapleSoft and Wolfram. I recommend that you get at least the free players from these two vendors. Use the contact links if you can not find the download links. With the players, you will be able to view extensively the capabilities of modern computational software. These programs are to ordinary calculators - even advanced scientific or financial ones - as the star ship Enterprise would be to the Wright Brother's first airplane.