I try to illustrate the formation and derivation of algebraic numbers and transcendental numbers. I found that both categories of numbers can be made by continuous summation or division/fraction method. Are there some distinguishable features between algebraic and non-algebraic numbers, if you make them via summation or continued fraction method?
I thought I found an easy way to explain the difference between rational and irrational numbers by a continued fraction, which is a sort of precision made with a limit of infinity. But now this method doesn't seem to explain how the next categories of numbers, namely the imaginary, the transcendental and the complex numbers differ from each other. At least transcendental numbers can be expressed by a summation too.
I guess the same question can be asked if summation / continued fraction method can reveal if the number is algebraic?
Can we pinpoint the transcendentality of the number from the continued fraction or the summation notation in mathematics?