Number Systems and Real Numbers – Detailed Study Notes

Natural numbers and the foundations of number systems

Let's start with the basic building blocks of numbers. Natural numbers are our counting numbers (1, 2, 3…) and also include zero (0). These are like the sizes of simple collections of items. We then expand to:

Integers ( $\mathbb{Z}$ ): which include all natural numbers, plus their negative counterparts ( $…-2, -1, 0, 1, 2…$
Rational numbers ( $\mathbb{Q}$ ): which are any numbers that can be written as a fraction of two integers (like $\frac{1}{2}$ or $\frac{3}{4}$ ).
Real numbers ( $\mathbb{R}$ ): which include all rational numbers, but also numbers that can't be written as simple fractions (like $\sqrt{2}$ or $\pi$ ).

Real numbers are super important for a field called Calculus because they form a solid, unbroken line of values, meaning there are no gaps. Sometimes, we add $\pm\infty$ (positive and negative infinity) to create an extended real line for convenience in certain calculations.

The real numbers and their geometric background

Think of real numbers as every single point on an infinitely long number line. This line is completely filled, holding all kinds of numbers:

Rational numbers: like $\frac{2}{3}$ , which can be perfectly placed on the line as a simple fraction.
Irrational numbers: like $\sqrt{2}$ , $\pi$ , or $\mathrm{e}$ , which also have a specific spot on the line but can't be written as a simple fraction.

Even though rationals exist, most numbers on the real line are actually irrational! We use the symbol $\mathbb{R}$ for real numbers and $\overline{\mathbb{R}}$ for the extended real line (which includes $\pm\infty$ ).

Extended real line and arithmetic with infinity

Infinity ( $\infty$ ) and negative infinity ( $!-\infty$ ) aren't regular numbers you can count to. Instead, they are concepts or tools we use in calculus to describe things that are endlessly large or small.

Here are some simple rules for using them:

Any real number is always less than or equal to $\infty$ , and greater than or equal to $!-\infty$ .
If you add any finite number $x$ to $\infty$ , you still get $\infty$ ( $\infty + x = \infty$ ). Same for $!-\infty$ .
When multiplying, watch the signs: if $a$ is positive, $a \cdot \infty = \infty$ ; if $a$ is negative, $a \cdot \infty = !-\infty$ .

However, some combinations are tricky and undefined, like $\infty - \infty$ or $0 \times \infty$ . These don't have a single, definite value and must be handled with care. (In very specific situations, like calculating areas, $0 \times \infty$ might be interpreted as $0$ , but this is an exception).

Sets, unions, intersections, and set operations

A set is simply a collection of distinct items, like $\text{{apple, banana, orange}}$ or $\text{{1, 2, 3}}$ . The order of items doesn't matter, and each item appears only once. We have some basic ways to combine or compare sets:

Union ( $A \cup B$ ): This combines all unique items from set A and set B. Think of it as "everything in A, or everything in B, or both."
Intersection ( $A \cap B$ ): This finds items that are common to both set A and set B. If there are no common items, the intersection is an empty set ( $\emptyset$ ).
Difference ( $A \setminus B$ or $A - B$ ): This takes all items that are in set A but not in set B.

Cartesian products and ordered pairs

An ordered pair is like a coordinate $(x,y)$ , where the order of the items matters. For example, $(1,2)$ is different from $(2,1)$ .

The Cartesian product ( $A \times B$ ) of two sets A and B creates a new set of all possible ordered pairs where the first item comes from A and the second from B.

For example, if $A = \text{{1, 2}}$ and $B = \text{{x, y}}$ , then $A \times B = \text{{(1,x), (1,y), (2,x), (2,y)}}$

If we take the Cartesian product of the real numbers with themselves ( $\mathbb{R} \times \mathbb{R}$ ), we get the familiar Cartesian plane – our two-dimensional graph paper where points are represented by $(x,y)$ coordinates.

Upper bounds, lower bounds, supremum, and infimum

Imagine a group of numbers on the real line (a set of numbers).

An upper bound is any number that is greater than or equal to every number in your set. For example, for the set $\text{{1, 3, 5}}$ , numbers like 5, 6, 10, or $\infty$ are all upper bounds.
The supremum ( $\sup S$ ) is the smallest (or least) of all these upper bounds. For $\text{{1, 3, 5}}$ , the supremum is 5.

Similarly:

A lower bound is any number that is less than or equal to every number in your set.
The infimum ( $\inf S$ ) is the largest (or greatest) of all these lower bounds.

Completeness of the real numbers

This is a super important idea for real numbers: they are complete. Imagine drawing the number line without lifting your pencil – there are absolutely no holes or gaps.

What this means mathematically is that if you have any set of real numbers that are all 'stuck' below a certain value (it has an upper bound), then there will always be a smallest possible upper bound (a supremum) that exists right there on the real line. This "no gaps" property is crucial for calculus to work smoothly.

Functions, relations, and representations

A relation describes how items from one set are linked to items in another set (like a list of ordered pairs).

A function is a special kind of relation with a very strict rule: for every single input you give it, you get exactly one output. Think of it like a machine: you put something in (an input from the domain), and it always gives you one specific thing out (an output which is part of the range).

We often represent functions in different ways:

As a table of inputs and outputs.
As a graph on the Cartesian plane.
As a formula (like $f(x) = x^2$ ).

Intervals, neighborhoods, and limits

Intervals are just continuous sections of the number line. For example, $(a,b)$ means all numbers between $a$ and $b$ (but not including $a$ or $b$ themselves).

A neighborhood of a point is simply a small open interval around that point. We also have neighborhoods of infinity, which mean all numbers beyond a very large (or very small) value.

Limits are about understanding what value a function's output approaches as its input gets closer and closer to a specific point. This is important because the function might not even be defined at that exact point (e.g., there might be a hole in the graph). We can look at how the function behaves when approaching from the left ( $x \to a^-$ ) or from the right ( $x \to a^+$ ).

The practical details and caveats of infinity

Remember, $\infty$ and $!-\infty$ are conceptual tools in calculus, not actual numbers. While they help us understand extreme behaviors (like very large or very small values), their arithmetic isn't always like regular numbers. You have to be careful with expressions that are undefined, such as $\infty - \infty$ , $\infty + (!-\infty)$ or $0 \times \infty$ , as these don't lead to a single clear value. These conventions are designed to make calculus work, but they have specific rules and exceptions.

Practical calculus concepts and closing notes

To summarize, we've covered the foundational ideas for calculus:

The family of numbers: from natural numbers to integers, then rationals, and finally real numbers (which are complete and continuous).
How we extend this to include infinity for certain calculations.
Ways to describe collections of numbers using sets, unions, intersections, and differences.
How to create ordered pairs and the Cartesian plane ( $\mathbb{R} \times \mathbb{R}$ ).
Understanding upper/lower bounds, supremum, and infimum to define the 'edges' of a set of numbers.
The key idea of functions (where each input provides one distinct output) and how we represent them.
Using intervals, neighborhoods, and the concept of limits to study how functions behave around specific points.

These fundamental concepts are the bedrock for understanding calculus and real analysis. (Just a quick correction: a number like $\frac{2}{3}$ is a rational number, not irrational.)