Is $x+y -\pi$ an algebraic expression or not?

Question

I came across this Wikipedia definition of an algebraic expression:

"In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, $3x^2 − 2xy + c$ is an algebraic expression.

It talks about integer constants in its definition, hence if I involve $\pi$ in my expression, $$x+y-\pi$$ will this be regarded as algebraic expression?

Answer 1

For saying an expression is algebraic or not, specifying the underlying field of constants is necessary. If no underlying field of constants is specified, $\mathbb{Q}$ is meant. And this is the case in the cited definition in the Wikipedia article.

$x+y-\pi$ is an algebraic expression (means algebraic over $\mathbb{Q}$) regarding $x$, $y$ and $\pi$. The cause is that $x+y-\pi$ is generated from rational numbers, $x$, $y$ and $\pi$ only by only algebraic operations. But the expression is not algebraic (means non-algebraic over $\mathbb{Q}$) regarding $x$ and $y$ - because $x+y-\pi$ is generated from rational numbers, $x$ and $y$ with help of the number/expression $\pi$ which is not algebraic (means non-algebraic over $\mathbb{Q})$.

Answer 2

$\pi$ is not a variable, an integer constant nor an operation, so no.

Answer 3

It depends on the definition of "algebraic expression", and especially the meaning of "algebraic" you take to use in your context. If you talk from a high-school perspective, then yes, $x+y-\pi$ is an algebraic expression, because... well, it looks like algebra and has algebraic-looking characters. But if you want to take a higher-level definition, especially the one used in Wikipedia (which is common in abstract algebra), then no, it is not an algebraic expression. In particular, it is not algebraic over $\mathbb Q$. The reason is precisely those outlined by the definition: $\pi$ is not rational, so it cannot be "built" in an easy way (using only the four operations) from the set of integers. The manner in which you might analytically define $\pi$, for example, might be as a limit working in $\mathbb R$, which itself takes some effort to define starting only from $\mathbb Z$. By contrast, from $\mathbb Z$ you essentially get $\mathbb Q$ for free, just by considering all ratios over the field.