Euler, in Complete introduction to algebra (1771) wrote in the introduction:. No definition was really thought necessary, and in fact the mathematics was considered the science of magnitudes. Real numbers became very much associated with magnitudes. This leads into the study of infinite series but without the necessary machinery to prove that these infinite series converged to a limit, he was never going to be able to progress much further in studying real numbers. He considered approximations by continued fractions, and also approximations by taking successive square roots. For Wallis there were a variety of ways that one might achieve this approximation, so coming as close as one pleased. Now, as for other incommensurable quantities, though this proportion cannot be accurately expressed in absolute numbers, yet by continued approximation it may so as to approach nearer to it than any difference assignable. such proportion is not to be expressed in the commonly received ways of notation: particularly that for the circles quadrature. However, Wallis understood that there were proportions which did not fall within this definition of number, such as those associated with the area and circumference of a circle:. He still only considers finite decimal expansions and realises that with these one can approximate numbers (which for him are constructed from positive integers by addition, subtraction, multiplication, division and taking nth roots ) as closely as one wishes. Details of the earlier contributions are examined in some detail in our article: The real numbers: Pythagoras to Stevin If we move forward almost exactly 100 years to the publication of A treatise of Algebra by Wallis in 1684 we find that he accepts, without any great enthusiasm, the use of Stevin's decimals. By the time Stevin proposed the use of decimal fractions in 1585, the concept of a number had developed little from that of Euclid's Elements.
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