What is a suitable name for numbers like $a + b\sqrt{c}$

Question

The motivation for this is to find a succinct name for a data type in a Python module.

Suppose I choose an integer $c$ and I want to talk about the set of numbers of the form $a + b\sqrt{c}$, where $a$ and $b$ are rational numbers. If $c = -1$, for example, then this set is called the set of Gaussian Rationals. Of course, we would usually think only of values of $c$ such that $|c|$ is non-square. If $|c|$ is a square, for example if $c = 4$, then we obtain either the set of rational numbers (if $c$ is nonnegative) or the set of Gaussian Rationals (otherwise).

Suppose, for example, that $c = 2$. What would the set of numbers of the form $a + b\sqrt{2}$ ($a$, $b$ rational) be called?

An obvious generalisation would be to cube and higher roots, for example numbers of the form $a + b \times 2^{1/3} + c \times 2^{2/3}$ where $a$, $b$ and $c$ are rational.

Answer 1

The standard notation for $\{a+b\sqrt{2}:a,b \in \mathbb{Q}\}$ is $\mathbb{Q}[\sqrt{2}]$. We use the square brackets when we're thinking about it as a ring, with addition and multiplication defined in the usual way.

As you know $\mathbb{Q}[x]$ is the ring of polynomials in $x$ with rational coefficients. The strict definition of $\mathbb{Q}[\sqrt{2}]$ is $\{p(\sqrt{2}) : p \in \mathbb{Q}[x]\}$. It just so happens that things like $a_0 + a_1\sqrt{2} + a_2(\sqrt{2})^2 + \cdots + a_n(\sqrt{2})^n$ all boil down to $a + b\sqrt{2}$ when you're done simplifying.

The field $\mathbb{Q}(\sqrt{2})$ is defined to be $\{p/q : p,q \in \mathbb{Q}[\sqrt{2}]\}$. As a set we have $\mathbb{Q}[\sqrt{2}] = \mathbb{Q}(\sqrt{2})$.

In the case of the generalisation $a + b\times 2^{1/3} + c\times 2^{2/3}$, this would be $\mathbb{Q}[\sqrt[3]{2}]$. The formal definition of $\mathbb{Q}[\sqrt[3]{2}]$ is $\{p(\sqrt[3]{2}) : p \in \mathbb{Q}[x]\}$. Again, things like $a_0 + a_1\sqrt[3]{2} + a_2(\sqrt[3]{2})^2 + \cdots + a_n(\sqrt[3]{2})^n$ will look like $a+b\sqrt[3]{2} + c(\sqrt[3]{2})^2$ when you're done.

You can add other stuff, e.g. \begin{array} 1\mathbb{Q}[\sqrt{2},\sqrt{3}] &=& \mathbb{Q}[\sqrt{2}][\sqrt{3}] = \{ a+b\sqrt{3} : a,b \in \mathbb{Q}[\sqrt{2}]\} \\ &=& \{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} : a,b,c,d \in \mathbb{Q}\} \end{array}

Any generalisation you can come up with will be of he form $\mathbb{Q}[\alpha_1,\ldots,\alpha_k]$, where $\alpha_i \in \mathbb{C}$. Of course, there's no point choosing $\alpha_i \in \mathbb{Q}$. Once you have $\operatorname{i}$ involved, there's no point having any other $\alpha_i \in \mathbb{Q}[\operatorname{i}]$ (the Gaussian rationals).

Answer 2

Since $\Bbb Q(\sqrt c)=\{a+b\sqrt c\colon a,b\in \Bbb Q\}$, you could call the elements of this set $\sqrt c\text{-rationals}$.

Answer 3

When c = -1 , we usually call that set as complex numbers.

When c = $\sqrt 2$, we can the set as surds (of type $\sqrt 2$) or simply surds because it is the simplest.

Answer 4

You might want to explore the concept of a number field. There are so many different number fields (finite degree field extensions of the rational numbers) that there are only standard names for elements of very few of them. Elements of a number field would do. All the numbers you are describing are algebraic numbers. It is unclear to me why you want to have a name for the specific numbers you are talking about - but you may be interested also in the concept of an algebraic integer which looks a strange idea at first sight and out of context, but turns out to be the right generalisation of integer in the number field context and also turns out to be very useful.

In this question, like in your follow-up one, you would get better answers if you gave more explanation as to why you are interested in these numbers.

If you are using a special class of numbers in some essay or paper you are writing, you can always define a name for the purpose of the paper.