Hypotheses


 * checked by Carlos Torrealba **

**I. Hypotheses**

When hypothesizing you are giving a possible solution to a problem or situation. Please visit the following link so that you can learn how to write hypotheses and when to use them. [|http://www.accessexcellence.org/LC/TL/filson/writhypo.php]


 * As you could see in the link above, hypotheses are written using modal verbs, like may, could, should. would, and if conditional structures. They can also be written using expresions (__key words__) as probably, possibly, and verbs such as: think, assume, hypothesize, imagine, suppose, guess, believe, among others. When reading a text, the indicators of hypotheses are the previously mentioned grammatical structures and key words. **


 * Read the following information extracted from the web page**: [] on Dec 27th, 2008
 * Hypotheses and mathematics**

So where does mathematics enter into this picture? In many ways, both obvious and subtle: Very often, the situation under analysis will appear to be complicated and unclear. Part of the mathematics of the task will be to impose a clear structure on the problem. The clarity of thought required will actively be developed through more abstract mathematical study. Those without sufficient general mathematical skill will be unable to perform an appropriate logical analysis. (Taken from [] on Dec 27th, 2008)
 * A good hypothesis needs to be clear, precisely stated and testable in some way. Creation of these clear hypotheses requires clear general mathematical thinking.
 * The data from experiments must be carefully analysed in relation to the original hypothesis. This requires the data to be structured, operated upon, prepared and displayed in appropriate ways. The levels of this process can range from simple to exceedingly complex.

There is often confusion between the ideas surrounding proof, which is mathematics, and making and testing an experimental hypothesis, which is science. The difference is rather simple: Of course, to be good at science, you need to be good at deductive reasoning, although experts at deductive reasoning need not be mathematicians. Detectives, such as Sherlock Holmes and Hercule Poirot, are such experts: they collect evidence from a crime scene and then draw logical conclusions from the evidence to support the hypothesis that, for example, Person M. committed the crime. They use this evidence to create sufficiently compelling deductions to support their hypotheses //beyond reasonable doubt//. The key word here is 'reasonable'. There is always the possibility of creating an exceedingly outlandish scenario to explain away any hypothesis of a detective or prosecution lawyer, but judges and juries in courts eventually make the decision that the probability of such eventualities are 'small' and the chance of the hypothesis being correct 'high'. (Taken from [] on Dec 27th, 2008)
 * Using deductive reasoning in hypothesis testing**
 * Mathematics is based on //deductive reasoning// : a proof is a logical deduction from a set of clear inputs.
 * Science is based on //inductive reasoning// : hypotheses are strengthened or rejected based on an accumulation of experimental evidence.

**II. Assignment** 1. Check the following links and explain what deductive reasoning is and inductive reasoning is. [] []


 * //Deductive Reasoning//**


 * Also called deductive logic, it consists in evaluating deductive arguments in order to find the truth. A deductive argument is based on logical assumptions. The reasoning consists in gathering information and formulating a premise. The deductive argument is valid as long as the conclusion that follows is true.**


 * Inductive Reasoning.**


 * Inductive reasoning is a logical reasoning that implies a truth, yet it's not entirely certain. It basically gives a probability of something being true. It's susceptible to change if new information is gathered.**

2. Please visit the following page and read the text **"Geometrical proportions of the Egyptian Pyramids"** then find and extract the hypotheses in it. There are 6 hypotheses in the text extract 5 and explain how you found them.
 * [|Details]
 * [[file:englishformath2/Geometrical proportions of the Egyptian Pyramids.doc|Download]]
 * 80 KB


 * //Namely if the height of triangle XEK is equal 318 cubits then height of triangle AEK approximately 314 cubits provided that the height of human growth is little bit more than 4 cubits. //
 * //It is possible to speak that magnitudes of the Egyptian Pyramids have fixed sizes of measurements which allow to understand structure of world around, and allow to apply "Great Egyptian Measures" to designing environmental space and for an arrangement of the objects of the human world created by people. //
 * //If exact geometrical calculations are not required, then it is possible to count that approximately the cubit is equal to the side of a correct diheptagon which is entered within the framework of a correct circle. //
 * //If to project the given measuring rod on the diheptagonal network of lines then the rod is equal to line AT, and also it is approximately equal to length of the side of a heptagon. //
 * //For construction of pyramids it is enough to know proportions of a heptagon and to use ratio of lines which exist in the geometrical figure of a heptagon //


 * The basic structure of a hypothesis is**


 * "If (premise) ... Then ... (Conclusion)"**


 * Then, a hypothesis consist on giving a premise, a truth, or an assumption. The conclusion it's basically what would confirm the premise, it can also have other conditions.**


 * For example: "If I go swimming on an shark infested beach. Then I'd probably get eaten by a shark."**


 * Most hypothesis follow that grammatical structure, but there are exceptions.**

3. Look for any mathematical hypothesis and put it in your wiki. Please make sure you cite the source properly so that you do not commit plagiarism. Explain whether the hypothesis you are explaining is deductive or inductive and give reasons to your explanation.

In mathematics, the **Riemann hypothesis**, proposed by [|Bernhard Riemann] ( [|1859] ), is a [|conjecture] about the location of the [|nontrivial zeros] of the [|Riemann zeta function] which states that all [|non-trivial] zeros (as defined below) have real part 1/2. The name is also used for some closely related analogues, such as the [|Riemann hypothesis for curves over finite fields]. The Riemann hypothesis implies results about the distribution of [|prime numbers] that are in some ways as good as possible. Along with suitable generalizations, it is considered by some mathematicians to be the most important unresolved problem in [|pure mathematics] ( [|Bombieri 2000] ). The Riemann hypothesis is part of [|Problem 8], along with the <span style="background-color: transparent; color: #0b0080; font-family: sans-serif; text-decoration: none;">[|Goldbach conjecture] , in <span style="background-color: transparent; color: #0b0080; font-family: sans-serif; text-decoration: none;">[|Hilbert] 's list of <span style="background-color: transparent; color: #0b0080; font-family: sans-serif; text-decoration: none;">[|23 unsolved problems] , and is also one of the <span style="background-color: transparent; color: #0b0080; font-family: sans-serif; text-decoration: none;">[|Clay Mathematics Institute] [|Millennium Prize Problems]. Since it was formulated, it has withstood efforts of many outstanding mathematicians but remains unsolved.

The Riemann zeta function ζ(//s//) is defined for all <span style="background-color: transparent; color: #0b0080; font-family: sans-serif; text-decoration: none;">[|complex numbers] //s// ≠ 1. It has zeros at the negative even integers (i.e. at //s// = −2, −4, −6, ...). These are called the **trivial zeros**. The Riemann hypothesis is concerned with the non-trivial zeros, and states that: "The real part of any non-trivial zero of the Riemann zeta function is 1/2.Thus the non-trivial zeros should lie on the **critical line**, 1/2 + //i t//, where //t// is a <span style="background-color: transparent; color: #0b0080; font-family: sans-serif; text-decoration: none;">[|real number] and //i// is the <span style="background-color: transparent; color: #0b0080; font-family: sans-serif; text-decoration: none;">[|imaginary unit] ."

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 * This hypothesis is Inductive Reasoning by concept. It's called the Riemann Hypothesis which is a conjecture.**
 * By definition, a conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Since inductive reasoning is **
 * a logical reasoning that implies a truth, yet it's not entirely certain; this hypothesis has not been disproved, so it can still be susceptible to change if new information is gathered, and therefore it remains true until is proven wrong.**