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9. The definitions of rational numbers are somewhat confusing for me. The definition of rational numbers on wikipedia and most other sites is: In mathematics, a rational number is any number that can be expressed as the quotient or fraction p q of two integers, with the denominator q not equal to zero. All integers can be expressed as the ...
2. In high school, in definition of rational number we used to say, "A number, which can be written in the form p q p q, where p, q(≠ 0) ∈Z p, q (≠ 0) ∈ Z is called rational number." But now I am realising that this is not a definition but it is a characterisation of rational numbers. Because the above definition starts with "a number ...
Wikipedia correctly states: In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers. In a way, it's not enough to say that any number that is not rational is irrational, because most complex numbers (like i i) are neither rational nor irrational. Wikipedia is wrong (not surprising). Irrational is ...
Very often I find this definition of rational numbers in my textbooks: A rational number is a number determined by the ratio of some integer p to some nonzero natural number q. But numbers $$\frac{-1}{-2};\frac{-2}{-4};...$$ are surely rational. Why the denominator has to be a natural number?
$\begingroup$ A positive integer is a non-zero natural number (or a natural number, if you definition of that term does not include zero.) Most mathematics starts from the natural numbers, with some assumed axioms (usually Peano's and 'second order logic') and then defines all the other sorts of numbers from there. @user258254 $\endgroup$
Usually "rational integer" and "rational prime" is terminology thrown around in algebraic number theory to mean integer of Q Q and prime of (the ring of integers of) Q Q, i.e. elements of Z Z or primes in Z Z. This terminology is used to differentiate between the term algebraic integer, which will often be said without the algebraic preceding ...
41. It means that between any two reals there is a rational number. The integers, for example, are not dense in the reals because one can find two reals with no integers between them. That definition works well when the set is linearly ordered, but one may also say that the set of rational points, i.e. points with rational coordinates, in the ...
My understanding of rational numbers is that they are generated by integers (I may already be wrong here). The definition I was given years ago in school was as follows: "a rational number is a fraction of two integers." This was fine and good, but after giving it some thought, I ran into a logical hurdle. I take "a fraction of two integers" to ...
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$\begingroup$ Since $0$ is rational but neither positive nor negative, both definitions might need extending $\endgroup$ – Henry Commented Apr 7, 2022 at 15:24
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