Table of Contents
Numbers are one of the oldest and most central ideas in mathematics, but even something that feels as familiar as “numbers” has a structure and a history. This chapter introduces the general landscape of numbers and number systems, without going into the detailed properties of each particular type of number (those appear in the subchapters that follow).
Why there are different kinds of numbers
It might seem that “numbers are numbers,” but as soon as you try to answer different kinds of questions, you discover that one single kind of number is not enough.
Consider these questions:
- “How many apples are in the basket?”
- “What is your bank balance if you owe money?”
- “If you cut a pizza into 4 equal slices and eat 1, what part is left?”
- “What is the exact length of the diagonal of a square with side 1 unit?”
- “Where is this point on a number line?”
Each of these questions leads to a slightly different need:
- Only counting whole objects: needing $1,2,3,\dots$
- Representing losses or debts: needing negative numbers like $-3$
- Representing parts of a whole: needing fractions like $\frac{1}{4}$
- Representing lengths that cannot be written as exact fractions: needing numbers like $\sqrt{2}$
- Combining all of these into a single system where we can compare and measure: needing a complete “number line”
Different number systems were created to meet these needs, and over time mathematicians have organized them carefully.
Sets of numbers as building blocks
In mathematics, it is common to group numbers into sets, each with its own symbol. You will meet these sets in detail in the subchapters, but it is useful here to see the overall structure.
Common sets of numbers include:
- The natural numbers (for counting)
- The integers (allowing negatives)
- The rational numbers (fractions)
- The irrational numbers (non-fraction decimals)
- The real numbers (all points on the number line)
These sets are related in a nested way: simpler sets sit inside more general ones. The later subchapters will give precise definitions; here we focus on how they fit together and why they exist.
Number systems as solutions to limitations
Each larger system was developed to fix a limitation of an earlier one.
From counting to whole-number arithmetic
Counting begins with numbers like $1,2,3,\dots$ (and often $0$). These are enough for:
- Counting objects
- Comparing “how many”
- Doing basic addition and multiplication, as long as answers stay positive
But subtraction causes a problem: what is $3-5$? The result is not a counting number.
Allowing subtraction fully: integers
To make subtraction always possible (for example, $3-5=-2$), we introduce negative numbers. The set of all whole numbers, both positive and negative, plus zero, is called the integers.
Now:
- Addition, subtraction, and multiplication always stay inside this set.
- Division still has limitations: $3 \div 2$ is not an integer.
Allowing division (except by zero): rational numbers
To handle division like $3 \div 2$, we create fractions, or ratios of integers. These are the rational numbers. With them:
- Any number that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$, is included.
- Many familiar decimals (like $0.5$ or $1.25$) are rational.
However, certain lengths and quantities (such as the diagonal of a unit square, or the circumference-to-diameter ratio of a circle) cannot be captured exactly by any fraction.
Beyond fractions: irrational numbers
Some numbers cannot be written as a fraction of two integers at all. These are irrational numbers. They include:
- Square roots of many whole numbers, like $\sqrt{2}$
- Certain famous constants, like $\pi$
Their decimal expansions never end and never repeat in a simple pattern. They fill in “gaps” left by rational numbers on the number line.
Completing the number line: real numbers
If we take:
- all rational numbers (fractions), and
- all irrational numbers (non-fraction decimals)
and put them together, we get the real numbers. This system is designed so that:
- Every point on the usual horizontal number line corresponds to exactly one real number.
- We can measure distances and place quantities continuously, not just at discrete fractional steps.
In everyday arithmetic and most school mathematics, “number” usually means “real number,” unless stated otherwise.
Visual picture: nesting of number sets
It is helpful to imagine the main sets of numbers as nested inside each other:
- Start with natural numbers (counting numbers).
- Enlarge to integers by including negatives.
- Enlarge to rational numbers by allowing fractions.
- Enlarge to real numbers by adding irrationals.
Symbolically, this nesting is often written in the form:
$$
\text{Natural numbers} \subset \text{Integers} \subset \text{Rational numbers} \subset \text{Real numbers}.
$$
Each step adds new numbers to solve new kinds of problems, but the old numbers are still included and still behave in familiar ways within the larger system.
Different ways to write numbers
A “number system” can also refer to how we write numbers, not just which numbers exist. For example:
- The familiar system using digits $0$ to $9$ and place value is called the decimal (base $10$) system.
- Other possible systems include binary (base $2$), often used in computers, and others like base $8$ or base $16$.
In this course, the main focus is on the kinds of numbers themselves and their properties. When alternative notations or bases matter (especially later in more advanced topics), they will be introduced in context.
Numbers and the number line
A unifying idea behind modern number systems is the number line:
- A horizontal line extending endlessly in both directions.
- A marked point for $0$.
- Unit steps marking $1$, $2$, $3$, and so on to the right, and $-1$, $-2$, $-3$, and so on to the left.
Different kinds of numbers locate themselves differently on this line:
- Natural numbers: the counting points $1,2,3,\dots$ to the right of $0$.
- Integers: all whole steps left and right, including $0$.
- Rational numbers: all points that can be reached by fractions (they form a dense pattern).
- Irrational numbers: the remaining points that fill all the gaps between rationals.
- Real numbers: every single point on the line.
Thinking in terms of the number line helps clarify why we keep enlarging our number systems: we aim for a system where “every position” that might represent a measurement, length, or quantity is included.
Operations and number systems
The choice of number system affects which operations are always possible:
- In naturals, subtraction and division might not give a natural result.
- In integers, subtraction is always possible, but division might not be.
- In rationals, all four basic operations (addition, subtraction, multiplication, division by a nonzero number) stay within the system.
- In reals, the same four operations are available, and we also gain a better framework for limits, continuity, and measurement.
Later chapters (especially in Algebra and Calculus) rely heavily on using real numbers precisely because their system of operations and their connection to the number line work so well together.
Looking ahead
The rest of this part of the course takes the broad picture from this chapter and focuses in on each major kind of number:
- Natural numbers
- Integers
- Rational numbers
- Irrational numbers
- Real numbers
Each subchapter will:
- Give a clear definition.
- Show typical examples.
- Explain how that kind of number fits into the number line and into the larger system.
Together, these ideas form the foundation for almost everything else you will do in mathematics.