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Rational and Irrational Numbers
Rational and Irrational Numbers

Rational and Irrational Numbers
Rational and Irrational Numbers

Here - Math 9
Here - Math 9

... To add fractions whose denominators are not the same, first find the Lowest Common Denominator , LCD for the fractions. This is the least common multiple, LCM, of the given denominators. Then for each fraction, divide the LCD by that fraction’s denominator, and multiply both terms of the fraction by ...
Exact Computer Calculations With Infinitely Repeating Decimals
Exact Computer Calculations With Infinitely Repeating Decimals

A pragmatic dialogic interpretation of bi
A pragmatic dialogic interpretation of bi

ramanujan
ramanujan

... and possibly liver infection. He left behind several notebooks containing densely packed formulae written by hand. Box 1: Taxi-cab numbers There is a famous story about Ramanujan. When he was ill, Hardy arrived at his residence to visit him. Being generally awkward, Hardy did not know what to say an ...
The Fibonacci Sequence
The Fibonacci Sequence

... The Fibonacci numbers first appeared in the 6th century AD with the Indian mathematician Virahanka’s analysis of metres with long and short syllables. In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202). The Fibonacci numbers are the product ...
Introduction to Logic
Introduction to Logic

... to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate “building blocks”. One has to identify the basic terms, their ki ...
Coinductive Definitions and Real Numbers
Coinductive Definitions and Real Numbers

Syllogistic Logic with Complements
Syllogistic Logic with Complements

The Foundations
The Foundations

Die Grundlagen der Arithmetik §§82–83
Die Grundlagen der Arithmetik §§82–83

... two sections would have contained a remarkably large gap that was never filled by any argument found in Grundgesetze. In any case, it is certain that Frege did not know of this proof. We begin by discussing §§70–81. In §70, Frege begins the definition of equinumerosity by explaining the notion of a ...
Number Theory Begins - Princeton University Press
Number Theory Begins - Princeton University Press

The generalized order-k Fibonacci–Pell sequence by matrix methods
The generalized order-k Fibonacci–Pell sequence by matrix methods

The set of real numbers is made up of two distinctly differe
The set of real numbers is made up of two distinctly differe

Intensified Algebra Standards
Intensified Algebra Standards

Strong Completeness and Limited Canonicity for PDL
Strong Completeness and Limited Canonicity for PDL

UNIT_10
UNIT_10

Reading 2 - UConn Logic Group
Reading 2 - UConn Logic Group

Algebra I Notes
Algebra I Notes

Lesson 13 - UnboundEd
Lesson 13 - UnboundEd

... added 9 ones with the 1 they made from 6 tenths and 4 tenths to get 10 ones and 13 hundredths.  The third solution shows converting tenths to hundredths in one step. Then, they decomposed the hundredths to make 1 from 60 hundredths and 40 hundredths. 6 ones and 4 ones is 10 ones with 13 hundredths. ...
notes on rational and real numbers
notes on rational and real numbers

Solved and unsolved problems in elementary number theory
Solved and unsolved problems in elementary number theory

Lecture2-1
Lecture2-1

CfE AH LI and SC Booklet - Aberdeen Grammar School
CfE AH LI and SC Booklet - Aberdeen Grammar School

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Foundations of mathematics

Foundations of mathematics is the study of the logical and philosophical basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (number, geometrical figure, set, function, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.The foundations of mathematics as a whole does not aim to contain the foundations of every mathematical topic.Generally, the foundations of a field of study refers to a more-or-less systematic analysis of its most basic or fundamental concepts, its conceptual unity and its natural ordering or hierarchy of concepts, which may help to connect it with the rest of human knowledge. The development, emergence and clarification of the foundations can come late in the history of a field, and may not be viewed by everyone as its most interesting part.Mathematics always played a special role in scientific thought, serving since ancient times as a model of truth and rigor for rational inquiry, and giving tools or even a foundation for other sciences (especially physics). Mathematics' many developments towards higher abstractions in the 19th century brought new challenges and paradoxes, urging for a deeper and more systematic examination of the nature and criteria of mathematical truth, as well as a unification of the diverse branches of mathematics into a coherent whole.The systematic search for the foundations of mathematics started at the end of the 19th century and formed a new mathematical discipline called mathematical logic, with strong links to theoretical computer science.It went through a series of crises with paradoxical results, until the discoveries stabilized during the 20th century as a large and coherent body of mathematical knowledge with several aspects or components (set theory, model theory, proof theory, etc.), whose detailed properties and possible variants are still an active research field.Its high level of technical sophistication inspired many philosophers to conjecture that it can serve as a model or pattern for the foundations of other sciences.
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