# Category Archives: Rational Numbers

## Fractions first then definition of length

Suppose we have defined fractions algebraically and worked out their rules including for adding fractions with different denominators.  Now we are ready to apply this algebraic construct to length.   We consider a function from point sets on the fraction number … Continue reading

## Euler’s Elements of Algebra Gaps in Logic

http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ElementsAlgebra.html Paragraph 19 is where Euler defines the natural numbers as adding one starting from zero. http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ElementsAlgebra.html#tth_sEc1.1 19.   In the same manner, therefore, as positive numbers are incontestably greater than nothing, negative numbers are less than nothing. Now, we obtain positive … Continue reading

## Why people justify rational numbers with geometry

The tendency is to justify rational numbers by a resort to geometry.  This is what Euler did in 1765. If we start from the Peano Axioms, we define natural numbers and the successor function through the axioms.  We then build … Continue reading

## #mtt2k 2012 math wars procedural v conceptual v object oriented

In programming languages like C++, a class or object has its own data and procedures or methods that use its own data. Natural numbers, signed integers as a sign and a natural number, rational numbers as a pair of signed … Continue reading