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Common Sets in Algebra (Part 1)

Sets are collections of elements. If an element $x$ belongs to a set $A$, we write $x \in A$.

There are several sets that you might commonly encounter when solving problems, below is a list of some of those sets.

Set of natural numbers

Often denoted as $\mathbb{N}$ or $\mathbb{Z_+}$.

$\mathbb{N}$ or $\mathbb{Z_+}$ is the set of positive integers, i.e. $\{1, 2, 3, 4, ...\}$. For example, $39991 \in \mathbb{N}$, but $2.5, -2 \not\in \mathbb{N}$.

Set of integers

Often denoted as $\mathbb{Z}$.

$\mathbb{Z}$ is the set of integers, i.e. $\{0, \pm 1, \pm 2, ...\}$. It's the union of $\mathbb{N}$, $\{0\}$, and the set of negative integers.

For example, $2014, -3 \in \mathbb{Z}$, but $\frac{3}{4} \not\in \mathbb{Z}$.

Set of rational numbers

Often denoted as $\mathbb{Q}$.

$\mathbb{Q}$ is the set of rational numbers. What is a rational number, some might ask. A rational number is a number that can be written as the quotient of two integers with the denominator not equal to $0$. For example, $\frac{3}{4}, \frac{2}{3}$ are all rational numbers. (and thus $\in \mathbb{Q}$)

One might think that all numbers are rational, since even $0.333.... = \frac{1}{3}$ is rational. However, not all numbers are rational. An example is $\sqrt{2}$, which can be proved that it is not a rational number. In fact, there is a "larger" set of values of values on the "number line" that includes $\mathbb{Q}$. We call it the set of real numbers $\mathbb{R}$.

Set of real numbers

One can visualize this intuitively as the entire "continuous" set of points on the number line. Real numbers can be compared, e.g., we know that $\sqrt{2} < 2$, but $\sqrt{5} > 2$.

Now, suppose one day you want to find the square root of $\sqrt{2}$. No problem, it's still a real number. However, there will be problems when we square root a negative number. After all, what number multiplied itself will produce a negative number, i.e. find $x$ such that $x^2 < 0$. One can easily see that this is clearly impossible in the set of real numbers, but......

Set of complex numbers

If we introduce a new element $i$, defined as the "square root of $-1$", i.e. $i^2 = -1$. Including $i$ among the real numbers give the set of complex numbers $\mathbb{C}$, defined formally as :

$C := \{a + bi : a, b \in \mathbb{R}\}$.

So, why do we need complex numbers? It's because it possesses the important property of algebraic closure. This means that any finite combination of additions, subtractions, multiplications, divisions (execpt by zero), extractions of roots, when applied to a complex number, will always produce a complex number. None of the other sets described above possesses this property.

There is one more set I'm going to cover in the next part of this article. Stay tuned!

Part 2 of the article can now be found here.

Just as there are special numbers like $\pi$ and $e$, there are special sets that show up all the time that have been given special names.

The integers (written $\mathbb{Z}$) that can be written without decimals or fractions. They include $2$, $0$, $-5$, $23$, etc.

$$\mathbb{Z}=\{\ldots,-5,-4,-3,-2,-1,0,1,2,3,4,5,\ldots\}$$

The rationals (written $\mathbb{Q}$) are all numbers that can be written as fractions, both proper and improper. This includes repeating decimals. All rationals can be written as an integer divided by another integer. The rationals contain the integers since $n=\frac{n}{1}$ and $n$ and $1$ are integers.

The rationals contain numbers such as $0.3333\ldots, \frac{3}{2}, 67, -23\!\!\frac{1}{2},$ etc.

The reals (written $\mathbb{R}$) are all the numbers that correspond to points on a number line. These correspond to any number that can be written as a finite or terminating string of decimals. Cool related thing: Computable numbers (there are some numbers which have decimal expansions, but the digits are impossible to determine).

The reals include all the rationals and hence all the integers, as well as numbers like $e$ and $\pi$ without terminating decimal expansions.

There are other special sets, but these are the three most important ones.