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eljose
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How can you prove in general that given a real number "a" this is rational or irrational?..
BSMSMSTMSPHD said:Well, if I'm given a number - that is, any real number in its standard form (not a series or infinitely continued fraction, etc.) then it should be pretty easy.
NateTG said:What is the "standard form" for a real number?
BSMSMSTMSPHD said:I was thinking of numbers written in decimal or fraction form, using only the 10 digits and no symbols (such as pi or e). I realize that it was an elementary way of looking at things, and perhaps I didn't add much to the discussion. You're certainly right about e^e.
Curious3141 said:Any number that has either a terminating or a nonterminating periodic representation (in any base, including 10 of course) is rational. Irrational numbers are aperiodic in any base.
Office_Shredder said:What about base pi?
True; though, I have made a simple system for rational non-natural bases (greater than one of course!) that comply with the rules (coefficient selection included) for natural bases.Curious3141 said:Generally, only natural numbers greater than one are used as bases.
Proving a scientific theory involves conducting experiments, collecting data, and analyzing the results to support the hypothesis. This process is known as the scientific method and it involves making observations, forming a hypothesis, testing the hypothesis, and drawing conclusions based on the evidence. The more evidence and data that supports the hypothesis, the stronger the theory becomes.
In science, the existence of something is proven through empirical evidence. This means that there needs to be observable and measurable data that supports the existence of the thing in question. Scientists use various methods such as experiments, observations, and mathematical models to gather this evidence and support their claims.
Proving a cause and effect relationship involves identifying an independent variable (the cause) and a dependent variable (the effect) and then manipulating the independent variable to see how it affects the dependent variable. By controlling other variables and repeating the experiment, scientists can determine if there is a cause and effect relationship between the two variables.
In science, nothing can be proven with 100% certainty. Instead, scientists use the weight of evidence to support their claims and theories. This involves conducting experiments, analyzing data, and replicating results to strengthen the evidence and increase confidence in the truthfulness of the claim.
To prove something is reliable, it needs to be tested multiple times under different conditions and produce consistent results. This is known as reliability testing and it is an essential part of the scientific process. If an experiment can be replicated and produces similar results each time, it is considered reliable and can be used to support scientific claims.