Set Theory and the Real Number System

Set Notation and Operations is about describing groups of objects clearly and using symbols to compare or combine those groups. In maths, a set is a collection of items, called elements. Sets help us organize numbers and ideas in a precise way.

This lesson will teach you how to write sets, how to read common symbols, and how to perform basic set operations such as union, intersection, and complement. These ideas are important in algebra, probability, and many other parts of mathematics.

1. What is a set?

A set is a well-defined collection of elements. “Well-defined” means we can tell clearly whether something belongs in the set or not.

For example, the set of even numbers less than 10 is a set because we know exactly which numbers belong:

$$\{2,4,6,8\}$$

The curly brackets \(\{\}\) show that we are writing a set.

2. Basic symbols used with sets

• \(\{ \}\): curly brackets used to write a set
• \(\in\): “is an element of”
• \(\notin\): “is not an element of”
• \(=\): two sets are exactly the same
• \(\subseteq\): is a subset of
• \(U\): the universal set, which contains all items being considered
• \(\varnothing\) or \(\{\}\): the empty set, a set with no elements

Examples:

• \(3 \in \{1,2,3,4\}\)
• \(7 \notin \{1,2,3,4\}\)
• If \(A=\{1,2\}\) and \(B=\{1,2\}\), then \(A=B\)

3. Ways to write sets

There are two common ways to write a set: roster notation and set-builder notation.

Roster notation lists all the elements of the set inside curly brackets.

Example:

$$A=\{1,2,3,4,5\}$$

This means the set \(A\) contains the numbers 1, 2, 3, 4, and 5.

Set-builder notation describes a rule that tells which elements belong to the set.

Example:

$$A=\{x\mid x \text{ is a whole number and } 1\le x\le 5\}$$

This is read as: “the set of all \(x\) such that \(x\) is a whole number and \(1\le x\le 5\).”

Both notations describe the same set.

4. Subsets

A set \(A\) is a subset of set \(B\) if every element of \(A\) is also an element of \(B\).

For example, if

$$A=\{2,4\}, \qquad B=\{1,2,3,4,5\}$$

then \(A\subseteq B\) because every number in \(A\) is also in \(B\).

If even one element of \(A\) is not in \(B\), then \(A\) is not a subset of \(B\).

5. The universal set and complement

The universal set, often written as \(U\), is the set of all elements we are discussing in a problem. The universal set matters because the complement of a set depends on it.

The complement of a set \(A\), written as \(A'\) or sometimes \(A^c\), is the set of all elements in the universal set that are not in \(A\).

If

$$U=\{1,2,3,4,5,6,7,8,9,10\}$$

and

$$A=\{2,4,6,8,10\}$$

then the complement of \(A\) is

$$A'=\{1,3,5,7,9\}$$

6. Union of sets

The union of two sets combines all elements from both sets. The symbol for union is \(\cup\).

For sets \(A\) and \(B\),

$$A\cup B$$

means all elements that are in \(A\), in \(B\), or in both.

Example:

$$A=\{1,2,3,4\}, \qquad B=\{3,4,5,6\}$$

Then

$$A\cup B=\{1,2,3,4,5,6\}$$

Notice that elements are not repeated in a set. Even though 3 and 4 are in both sets, they are written only once.

7. Intersection of sets

The intersection of two sets contains only the elements that are in both sets. The symbol for intersection is \(\cap\).

Using the same sets:

$$A=\{1,2,3,4\}, \qquad B=\{3,4,5,6\}$$

Then

$$A\cap B=\{3,4\}$$

These are the elements shared by both sets.

If two sets have no elements in common, their intersection is the empty set:

$$\{1,2\}\cap\{3,4\}=\varnothing$$

8. Reading and comparing operations

It is important to understand the difference between union, intersection, and complement.

• Union \(A\cup B\): everything in either set
• Intersection \(A\cap B\): only what the sets share
• Complement \(A'\): everything in the universal set that is not in \(A\)

A quick way to remember:

• Union joins sets together.
• Intersection finds overlap.
• Complement finds what is left out.

Worked Example 1: Writing a set in two ways

Write the set of odd numbers less than 10 in roster notation and set-builder notation.

Step 1: List the odd numbers less than 10.

They are \(1,3,5,7,9\).

Roster notation:

$$\{1,3,5,7,9\}$$

Set-builder notation:

$$\{x\mid x \text{ is an odd whole number and } x<10\}$$

Answer: The set can be written as \(\{1,3,5,7,9\}\) or \(\{x\mid x \text{ is an odd whole number and } x<10\}\).

Worked Example 2: Finding union and intersection

Let

$$A=\{2,4,6,8\}, \qquad B=\{4,6,8,10\}$$

Find \(A\cup B\) and \(A\cap B\).

Step 1: Find the union.

Take all elements from both sets without repeating any:

$$A\cup B=\{2,4,6,8,10\}$$

Step 2: Find the intersection.

Look for elements that appear in both sets:

$$A\cap B=\{4,6,8\}$$

Answer:

$$A\cup B=\{2,4,6,8,10\}$$

$$A\cap B=\{4,6,8\}$$

Worked Example 3: Finding a complement

Let

$$U=\{1,2,3,4,5,6,7,8,9,10,11,12\}$$

and

$$C=\{3,6,9,12\}$$

Find \(C'\).

Step 1: Identify the universal set.

The universal set contains the numbers 1 through 12.

Step 2: Remove the elements in \(C\).

The numbers not in \(C\) are:

$$1,2,4,5,7,8,10,11$$

Answer:

$$C'=\{1,2,4,5,7,8,10,11\}$$

Worked Example 4: Using all three ideas together

Let

$$U=\{1,2,3,4,5,6,7,8,9,10\}$$

$$A=\{1,2,3,4,5\}$$

$$B=\{4,5,6,7\}$$

Find:

1. \(A\cup B\)
2. \(A\cap B\)
3. \(A'\)

1. Union

Combine all elements in \(A\) and \(B\):

$$A\cup B=\{1,2,3,4,5,6,7\}$$

2. Intersection

Find the elements in both sets:

$$A\cap B=\{4,5\}$$

3. Complement of \(A\)

Take everything in \(U\) that is not in \(A\):

$$A'=\{6,7,8,9,10\}$$

9. Common mistakes to avoid

• Repeating elements: In a set, each element is written only once.
• Mixing up union and intersection: Union means combined; intersection means shared.
• Forgetting the universal set: You need the universal set to find a complement.
• Listing elements out of the rule: In set-builder notation, make sure the rule matches the set exactly.

10. Quick practice ideas

Try these on your own:

• Write the set of multiples of 5 less than 25 in both roster and set-builder notation.
• If \(A=\{1,3,5,7\}\) and \(B=\{5,6,7,8\}\), find \(A\cup B\) and \(A\cap B\).
• If \(U=\{1,2,3,4,5,6,7,8\}\) and \(D=\{2,4,6,8\}\), find \(D'\).

Summary

A set is a collection of elements, and it can be written in roster notation or set-builder notation. The symbol \(\cup\) means union, which combines elements from sets. The symbol \(\cap\) means intersection, which gives only the shared elements.

The complement of a set gives the elements in the universal set that are not in that set. Learning these symbols and operations helps you describe number groups clearly and solve problems in a logical way.

Real Number System Hierarchy

In math, numbers are grouped into sets based on their properties. The real number system hierarchy shows how these sets fit inside one another.

Learning this hierarchy helps you classify numbers correctly and understand why some numbers belong to more than one set. It also builds a strong foundation for algebra, graphing, and later math topics.

The main sets we will study are:

• Natural numbers
• Whole numbers
• Integers
• Rational numbers
• Irrational numbers
• Real numbers

These sets form a hierarchy, which means some sets are contained inside larger sets.

Here is the basic structure:

\[ \text{Natural} \subset \text{Whole} \subset \text{Integers} \subset \text{Rational} \subset \text{Real} \]

Irrational numbers are also part of the real numbers, but they are not rational.

So the real numbers are made of two big parts:

\[ \text{Real Numbers} = \text{Rational Numbers} \cup \text{Irrational Numbers} \]

And rational and irrational numbers do not overlap:

\[ \text{Rational Numbers} \cap \text{Irrational Numbers} = \varnothing \]

1. Natural Numbers

Natural numbers are the counting numbers:

\(1, 2, 3, 4, 5, \dots\)

These are the numbers we use when counting objects. In many 9th grade courses, natural numbers start at 1.

2. Whole Numbers

Whole numbers include all natural numbers and also 0:

\(0, 1, 2, 3, 4, \dots\)

This means every natural number is a whole number, but 0 is whole and not natural.

3. Integers

Integers include negative whole numbers, 0, and positive whole numbers:

\(\dots, -3, -2, -1, 0, 1, 2, 3, \dots\)

Integers do not include fractions or decimals unless the decimal ends with .0, like \(5.0\), which is equal to 5.

4. Rational Numbers

A rational number is any number that can be written as a fraction of two integers:

\[ \frac{a}{b} \quad \text{where } a \text{ and } b \text{ are integers and } b \ne 0 \]

Examples of rational numbers include:

• Integers, like \(4\), because \(4 = \frac{4}{1}\)
• Fractions, like \(\frac{3}{5}\)
• Terminating decimals, like \(0.75\)
• Repeating decimals, like \(0.333\dots\)

A decimal is terminating if it ends, such as \(2.4\) or \(0.125\).

A decimal is repeating if a digit or group of digits repeats forever, such as \(0.777\dots\) or \(1.232323\dots\).

All integers are rational because they can be written over 1. For example:

\[ -7 = \frac{-7}{1} \]

5. Irrational Numbers

Irrational numbers are real numbers that cannot be written as a fraction of two integers.

Their decimal forms do not end and do not repeat.

Common examples are:

• \(\pi\)
• \(\sqrt{2}\)
• \(\sqrt{3}\)
• \(\sqrt{5}\)

Not every square root is irrational. For example, \(\sqrt{9} = 3\), which is rational. A square root is irrational when the number inside is not a perfect square.

6. Real Numbers

Real numbers include all rational and irrational numbers. Any number you can place on a number line is a real number.

Examples of real numbers:

• \(-4\)
• \(0\)
• \(\frac{2}{3}\)
• \(1.8\)
• \(\sqrt{7}\)
• \(\pi\)

Understanding Subset Relations

When one set is completely contained inside another set, we say it is a subset.

For example, every natural number is also a whole number, so:

\[ \text{Natural} \subset \text{Whole} \]

Similarly:

• Every whole number is an integer.
• Every integer is a rational number.
• Every rational number is a real number.

So if a number is natural, it is automatically also whole, integer, rational, and real.

For example, the number \(5\) belongs to many sets:

• Natural
• Whole
• Integer
• Rational
• Real

But the number \(-2\) is not natural or whole. It is:

• Integer
• Rational
• Real

A Helpful Way to Picture the Hierarchy

You can imagine the sets like nested boxes:

• Natural numbers are inside whole numbers.
• Whole numbers are inside integers.
• Integers are inside rational numbers.
• Rational numbers and irrational numbers are both inside real numbers.

Another way to write this is:

\[ \mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \]

Here, the symbols often mean:

• \(\mathbb{N}\): natural numbers
• \(\mathbb{W}\): whole numbers
• \(\mathbb{Z}\): integers
• \(\mathbb{Q}\): rational numbers
• \(\mathbb{R}\): real numbers

How to Classify a Number

When classifying a number, it helps to ask these questions in order:

1. Is it a counting number? Then it is natural.
2. If it is 0, is it whole?
3. Is it a negative or positive whole number? Then it is an integer.
4. Can it be written as a fraction of integers? Then it is rational.
5. If it cannot be written as a fraction and its decimal does not end or repeat, then it is irrational.
6. Either way, if it is rational or irrational, it is real.

Important Notes

• A number can belong to more than one set.
• When asked for the smallest set, choose the most specific category.
• Fractions are not integers unless they simplify to a whole number.
• Terminating decimals and repeating decimals are rational.
• Nonterminating, nonrepeating decimals are irrational.

Worked Example 1: Classify \(8\)

Step 1: \(8\) is a counting number, so it is a natural number.

Step 2: Since all natural numbers are also whole numbers, integers, rational numbers, and real numbers, \(8\) belongs to all of these sets.

Answer: \(8\) is natural, whole, integer, rational, and real.

The smallest set is natural numbers.

Worked Example 2: Classify \(-3\)

\(-3\) is not natural and not whole because it is negative.

It is an integer because integers include negative whole numbers.

It is also rational because:

\[ -3 = \frac{-3}{1} \]

So it is also real.

Answer: \(-3\) is integer, rational, and real.

The smallest set is integers.

Worked Example 3: Classify \(0.625\)

\(0.625\) is a terminating decimal, so it is rational.

We can write it as a fraction:

\[ 0.625 = \frac{625}{1000} = \frac{5}{8} \]

It is not natural, whole, or integer because it is not a counting number or a whole number.

Answer: \(0.625\) is rational and real.

The smallest set is rational numbers.

Worked Example 4: Classify \(\sqrt{11}\)

Since 11 is not a perfect square, \(\sqrt{11}\) cannot be written as a whole number or integer.

Its decimal goes on forever without repeating, so it is irrational.

Because all irrational numbers are real numbers, \(\sqrt{11}\) is also real.

Answer: \(\sqrt{11}\) is irrational and real.

The smallest set is irrational numbers.

Comparing Rational and Irrational Numbers

• \(\frac{1}{2} = 0.5\) is rational because it ends.
• \(0.121212\dots\) is rational because it repeats.
• \(\pi = 3.14159\dots\) is irrational because it does not end or repeat.
• \(\sqrt{16} = 4\) is rational because 16 is a perfect square.
• \(\sqrt{10}\) is irrational because 10 is not a perfect square.

Common Mistakes to Avoid

• Mistake: Thinking every decimal is irrational.
  Some decimals are rational, such as terminating or repeating decimals.
• Mistake: Thinking 0 is natural.
  In this lesson, 0 is whole but not natural.
• Mistake: Thinking square roots are always irrational.
  Square roots of perfect squares are rational.
• Mistake: Giving only one category when the number belongs to several.
  A number may fit many sets in the hierarchy.

Quick Practice Ideas

Try classifying these on your own:

• \(0\)
• \(\frac{-4}{7}\)
• \(12\)
• \(\sqrt{49}\)
• \(\sqrt{6}\)
• \(0.444\dots\)

Answers:

• \(0\): whole, integer, rational, real
• \(\frac{-4}{7}\): rational, real
• \(12\): natural, whole, integer, rational, real
• \(\sqrt{49} = 7\): natural, whole, integer, rational, real
• \(\sqrt{6}\): irrational, real
• \(0.444\dots\): rational, real

Summary

The real number system is organized as a hierarchy of sets. Natural numbers are inside whole numbers, whole numbers are inside integers, integers are inside rational numbers, and rational numbers are inside real numbers.

Irrational numbers are also real numbers, but they are separate from rational numbers. To classify a number, decide whether it is counting, whole, integer, rational, or irrational, and remember that numbers in smaller sets also belong to the larger sets around them.

Proof of Irrationality

In the real number system, numbers can be grouped into different sets. One important split is between rational numbers and irrational numbers.

• A rational number can be written as a fraction \\(\frac{a}{b}\\), where \\(a\\) and \\(b\\) are integers and \\(b \ne 0\\).
• An irrational number cannot be written as a fraction of two integers.

Some decimals help us recognize these types of numbers:

• Rational numbers have decimals that end or repeat, such as \\(0.75\\) or \\(0.333\ldots\\).
• Irrational numbers have decimals that go on forever without repeating, such as \\(\sqrt{2}\\) or \\(\pi\\).

But in mathematics, we do not just say a number is irrational because its decimal “looks non-repeating.” We often need a proof.

A proof of irrationality is a logical argument that shows a number cannot be written as a fraction of two integers.

The most famous example is proving that \\(\sqrt{2}\\) is irrational. Learning this proof helps you understand how mathematicians use logic carefully and step by step.

Main Idea: Proof by Contradiction

One common method for proving irrationality is called proof by contradiction.

Here is how it works:

1. Assume the opposite of what you want to prove.
2. Use logical steps to follow that assumption.
3. Reach an impossible result, called a contradiction.
4. Conclude that the original assumption must be false.

For irrationality, we usually start by assuming the number is rational. Then we try to show that this leads to something impossible.

An Important Idea: Simplest Form

If a number is rational, we can write it as a fraction in lowest terms, meaning the numerator and denominator have no common factor other than 1.

For example:

• \\(\frac{6}{8} = \frac{3}{4}\\), and \\(\frac{3}{4}\\) is in lowest terms.
• \\(\frac{10}{15} = \frac{2}{3}\\), and \\(\frac{2}{3}\\) is in lowest terms.

This matters because many irrationality proofs show that the numerator and denominator would both have to be even, or both divisible by the same number. That would mean the fraction was not in lowest terms, which creates the contradiction.

Key Fact About Even Numbers

We will use this fact often:

• If \\(n\\) is even, then \\(n = 2k\\) for some integer \\(k\\).
• If \\(n^2\\) is even, then \\(n\\) must also be even.

Why is the second fact true? Because if \\(n\\) were odd, then \\(n = 2k+1\\), and

which is odd. So an odd number squared stays odd. Therefore, if \\(n^2\\) is even, \\(n\\) cannot be odd, so it must be even.

Worked Example 1: Prove that \\(\sqrt{2}\\) is irrational

We want to prove that \\(\sqrt{2}\\) cannot be written as a fraction of integers.

Step 1: Assume the opposite.

Assume \\(\sqrt{2}\\) is rational. Then we can write

where \\(a\\) and \\(b\\) are integers, \\(b \ne 0\\), and \\(\frac{a}{b}\\) is in lowest terms.

Step 2: Square both sides.

Multiply both sides by \\(b^2\\):

Step 3: Use the equation to learn about \\(a\\).

Since \\(a^2 = 2b^2\\), \\(a^2\\) is even. That means \\(a\\) is even.

So let \\(a = 2k\\) for some integer \\(k\\).

Step 4: Substitute back.

Divide both sides by 2:

This shows \\(b^2\\) is even, so \\(b\\) is even.

Step 5: Find the contradiction.

We have shown that both \\(a\\) and \\(b\\) are even. That means both are divisible by 2.

But this is impossible because we said \\(\frac{a}{b}\\) was in lowest terms. If both are even, the fraction can be simplified.

This contradiction means our original assumption was false.

Conclusion: \\(\sqrt{2}\\) is irrational.

Why This Proof Works

The contradiction happens because the assumption that \\(\sqrt{2}\\) is rational forces the fraction to be both:

• in lowest terms, and
• not in lowest terms.

That cannot happen at the same time, so the assumption must be wrong.

Worked Example 2: Prove that \\(\sqrt{3}\\) is irrational

This proof follows a very similar pattern.

Step 1: Assume \\(\sqrt{3}\\) is rational.

where \\(\frac{a}{b}\\) is in lowest terms.

Step 2: Square both sides.

So

This means \\(a^2\\) is divisible by 3. That tells us \\(a\\) is divisible by 3.

So let \\(a = 3k\\).

Step 3: Substitute back.

Divide by 3:

So \\(b^2\\) is divisible by 3, which means \\(b\\) is divisible by 3.

Step 4: Contradiction.

Both \\(a\\) and \\(b\\) are divisible by 3, so \\(\frac{a}{b}\\) is not in lowest terms.

This contradicts our starting assumption.

Conclusion: \\(\sqrt{3}\\) is irrational.

A Useful Pattern

The proofs for \\(\sqrt{2}\\) and \\(\sqrt{3}\\) suggest a pattern:

• If a square root can be simplified to a whole number, then it is rational. For example, \\(\sqrt{16} = 4\\).
• If the number under the square root is not a perfect square, the square root is often irrational.

For 9th Grade, an important result to remember is:

The square root of a whole number that is not a perfect square is irrational.

Examples:

• \\(\sqrt{4} = 2\\) is rational.
• \\(\sqrt{9} = 3\\) is rational.
• \\(\sqrt{5}\\), \\(\sqrt{7}\\), and \\(\sqrt{10}\\) are irrational.

Worked Example 3: Decide whether each number is rational or irrational

1. \\(\sqrt{25}\\)
2. \\(\sqrt{11}\\)
3. \\(\sqrt{49}\\)
4. \\(\sqrt{18}\\)

1. \\(\sqrt{25}\\)

Since 25 is a perfect square,

So it is rational.

2. \\(\sqrt{11}\\)

11 is not a perfect square, so \\(\sqrt{11}\\) is irrational.

3. \\(\sqrt{49}\\)

Since 49 is a perfect square,

So it is rational.

4. \\(\sqrt{18}\\)

18 is not a perfect square, so \\(\sqrt{18}\\) is irrational.

Even though \\(\sqrt{18}\\) can be simplified, it is still irrational:

Since \\(\sqrt{2}\\) is irrational, \\(3\sqrt{2}\\) is also irrational.

Worked Example 4: Explain why \\(2 + \sqrt{2}\\) is irrational

This example uses an important idea: adding a rational number and an irrational number gives an irrational number.

We know:

• 2 is rational.
• \\(\sqrt{2}\\) is irrational.

Suppose \\(2 + \sqrt{2}\\) were rational.

Then subtract 2 from both sides. Since subtracting rational numbers keeps the result rational, \\(\sqrt{2}\\) would have to be rational.

But that is false, because we already proved \\(\sqrt{2}\\) is irrational.

So our assumption is wrong.

Conclusion: \\(2 + \sqrt{2}\\) is irrational.

Common Mistakes to Avoid

• Mistake 1: Saying a number is irrational just because its decimal looks long. A long decimal is not enough; you need a mathematical reason.
• Mistake 2: Forgetting to say the fraction is in lowest terms at the start of the proof.
• Mistake 3: Saying “if \\(a^2\\) is even, then \\(a\\) is even” without understanding why. This fact is important and must be used carefully.
• Mistake 4: Thinking every square root is irrational. Square roots of perfect squares are rational.

How to Write a Proof of Irrationality

When you are asked to prove a number like \\(\sqrt{2}\\) is irrational, you can follow this structure:

1. Assume the number is rational.
2. Write it as \\(\frac{a}{b}\\) in lowest terms.
3. Use algebra to create an equation.
4. Show that both \\(a\\) and \\(b\\) must be divisible by the same number.
5. This contradicts the fact that the fraction was in lowest terms.
6. Conclude that the number is irrational.

Quick Check for Understanding

• What does it mean for a fraction to be in lowest terms?
• Why does proving both numerator and denominator are even create a contradiction?
• Is \\(\sqrt{36}\\) rational or irrational? Why?
• Is \\(\sqrt{6}\\) rational or irrational? Why?

Brief Summary

A proof of irrationality shows that a number cannot be written as a fraction of two integers.

The most common method is proof by contradiction: assume the number is rational, write it as a fraction in lowest terms, and then show that the numerator and denominator must still share a common factor. That contradiction proves the number is irrational.

You should especially remember that \\(\sqrt{2}\\) is irrational, and more generally, the square root of a whole number that is not a perfect square is irrational.

Real Number Line Density is the idea that the real number line has no gaps. If you pick any two different real numbers, no matter how close they are, there is always another real number between them.

Even more surprising, there are not just one or two numbers between them. There are infinitely many numbers between any two real numbers. In fact, there are infinitely many rational numbers and infinitely many irrational numbers between them.

This lesson will explain what density means, how to find numbers between two given numbers, and why this property is so important in the real number system.

First, recall the types of numbers in the real number system:

• Rational numbers: numbers that can be written as a fraction of integers, such as \(\frac{1}{2}\), \(-3\), and \(0.75\).
• Irrational numbers: numbers that cannot be written as a fraction of integers, such as \(\sqrt{2}\), \(\pi\), and \(\sqrt{3}\).
• Real numbers: all rational and irrational numbers together.

On a number line, rational and irrational numbers are mixed together everywhere. There is no section that has only one type.

What does “dense” mean?

A set of numbers is called dense on the number line if between any two different numbers in that set, you can always find another number from the set.

For real number line density, we focus on these important facts:

• Between any two different real numbers, there is another real number.
• Between any two different real numbers, there is a rational number.
• Between any two different real numbers, there is an irrational number.

This means the number line is packed full of numbers.

A simple way to find a number between two numbers

If you have two different real numbers, say \(a\) and \(b\), one easy way to find a number between them is to take their average:

This number is exactly halfway between \(a\) and \(b\).

If \(a < b\), then

So there is always at least one real number between any two real numbers.

But density says much more: once you find one number between them, you can repeat the process again and again to get infinitely many more.

For example, between \(1\) and \(2\), the average is \(1.5\). Then between \(1\) and \(1.5\), the average is \(1.25\). Then between \(1\) and \(1.25\), the average is \(1.125\), and this can continue forever.

Why are there infinitely many rational numbers between any two real numbers?

Suppose you have two different real numbers \(a\) and \(b\), with \(a < b\). Because decimals can be made more and more precise, you can always find fractions or terminating/repeating decimals between them.

For example, between \(0.2\) and \(0.3\), you can find:

• \(0.25\)
• \(0.21\)
• \(0.201\)
• \(0.299\)

All of these are rational numbers, and there is no end to how many you can make.

Another useful idea is that if one rational number lies between two numbers, then you can average again to create more rational numbers between them. Since the average of two rational numbers is rational, this process can continue forever.

Why are there infinitely many irrational numbers between any two real numbers?

This may seem less obvious, but it is also true. Between any two real numbers, there is always an irrational number.

One way to think about this is to start with a rational number between the two numbers, then add or subtract a very small irrational number so that the result stays in the interval.

For example, suppose a rational number \(r\) is between \(a\) and \(b\). If we choose a very small irrational amount, such as \(\frac{\sqrt{2}}{1000}\), then

is irrational. If that number stays between \(a\) and \(b\), then we have found an irrational number in the interval. By choosing smaller and smaller irrational amounts, we can create infinitely many irrational numbers between \(a\) and \(b\).

Important idea: having decimals that “go on forever” does not automatically mean a number is irrational. A repeating decimal is rational. An irrational decimal goes on forever without repeating in a pattern.

Worked Example 1: Find a real number between two numbers

Find a real number between \(4\) and \(10\).

Step 1: Take the average.

Answer: \(7\) is a real number between \(4\) and \(10\).

Why it works: \(4 < 7 < 10\).

Worked Example 2: Find rational numbers between two decimals

Find three rational numbers between \(1.2\) and \(1.3\).

We can choose any terminating decimals in that interval.

• \(1.21\)
• \(1.25\)
• \(1.29\)

Each of these is rational because each can be written as a fraction:

Answer: Three rational numbers between \(1.2\) and \(1.3\) are \(1.21\), \(1.25\), and \(1.29\).

Worked Example 3: Find an irrational number between two numbers

Find an irrational number between \(2\) and \(3\).

A well-known irrational number in this interval is \(\sqrt{5}\).

Check its size:

Since \(5\) is between \(4\) and \(9\), its square root is between \(2\) and \(3\):

Also, \(\sqrt{5}\) is irrational because \(5\) is not a perfect square.

Answer: \(\sqrt{5}\) is an irrational number between \(2\) and \(3\).

Worked Example 4: Show that there are infinitely many numbers between two numbers

Explain why there are infinitely many real numbers between \(0\) and \(1\).

Start by finding one number between them:

Now find a number between \(0\) and \(0.5\):

Now find a number between \(0\) and \(0.25\):

This process never has to stop. We can keep halving the interval and getting new numbers:

So there are infinitely many real numbers between \(0\) and \(1\).

The same idea works for any two different real numbers.

Common mistakes to avoid

• Mistake 1: Thinking consecutive whole numbers have no numbers between them. For example, there are many numbers between \(2\) and \(3\), such as \(2.1\), \(2.5\), and \(\sqrt{5}\).
• Mistake 2: Thinking only decimals count as numbers between two values. Fractions and square roots can also lie between two numbers.
• Mistake 3: Thinking “infinitely many” means “very large but countable by listing.” Infinity means the process never ends.
• Mistake 4: Thinking irrational numbers are rare. They appear all throughout the number line, just like rational numbers do.

Why this matters in algebra and the real number system

The density of the real numbers helps explain why graphs look continuous and why intervals such as \((1,2)\) contain far more than just a few points.

It also supports estimation. For example, if you know a solution is between \(1.4\) and \(1.5\), density tells you there are many numbers in between, so you can keep narrowing the answer.

This property is one reason the real number system is so powerful for algebra, graphing, and solving equations.

Summary

• The real number line is dense: between any two different real numbers, there is another real number.
• Between any two different real numbers, there are infinitely many rational numbers.
• Between any two different real numbers, there are infinitely many irrational numbers.
• A simple way to find a number between two numbers is to use the average, \(\frac{a+b}{2}\).
• Because you can repeat this process forever, there is no “next” real number after a given real number.

Whenever two real numbers are different, no matter how close they seem, the number line always has infinitely many more numbers waiting between them.

Absolute value tells us how far a number is from zero on the number line.

This is why absolute value is connected to distance. Distance is always nonnegative, so absolute value is always zero or positive.

For example, both 5 and 5 are 5 units away from 0. So they have the same absolute value:

The absolute value symbols are the vertical bars around a number: \(|x|\).

Key idea: \(|x|\) means the distance between \(x\) and 0 on the number line.

We can picture this on a number line:

• If \(x=3\), then \(|3|=3\) because 3 is 3 units from 0.
• If \(x=-3\), then \(|-3|=3\) because 3 is also 3 units from 0.
• If \(x=0\), then \(|0|=0\) because 0 is 0 units from 0.

This leads to an important rule:

• If the number is positive, its absolute value is the same number.
• If the number is negative, its absolute value is its positive distance from 0.

So:

Absolute value does not tell whether a number is left or right of 0. It only tells how far away the number is.

That is why two different numbers can have the same absolute value. For example, \(4\) and \(-4\) both have absolute value 4.

We can also use absolute value to describe the distance between two numbers, not just a number and zero.

The distance between two numbers \(a\) and \(b\) is:

This works because subtracting tells us how far apart the numbers are, and the absolute value makes the answer positive.

For example, the distance between 2 and 7 is:

The distance between 7 and 2 is also:

Distance is the same either way, which makes sense.

Worked Example 1

Find each absolute value: \(|6|\), \(|-9|\), and \(|0|\).

Step 1: Think about distance from 0.

• 6 is 6 units from 0, so \(|6|=6\).
• 9 is 9 units from 0, so \(|-9|=9\).
• 0 is 0 units from 0, so \(|0|=0\).

Answer:

Worked Example 2

Order these numbers from least to greatest: \(-5\), \(3\), \(-2\), \(0\).

Step 1: Place them on a number line in your mind.

Numbers farther left are smaller. The order is:

Step 2: Compare their absolute values if needed.

• \(|-5|=5\)
• \(|3|=3\)
• \(|-2|=2\)
• \(|0|=0\)

This shows an important idea: a larger absolute value does not always mean a larger number.

For example, \(-5\) has a greater absolute value than 3, but \(-5<3\).

Worked Example 3

Find the distance between \(-4\) and \(3\).

Method 1: Count on the number line.

From 4 to 0 is 4 units, and from 0 to 3 is 3 units. Total distance:

Method 2: Use absolute value.

You could also write:

Answer: The distance between \(-4\) and \(3\) is \(7\).

Worked Example 4

Solve: \(|x|=5\)

This means the distance from \(x\) to 0 is 5.

There are two numbers that are 5 units from 0:

• \(5\)
• \(-5\)

So the solutions are:

This is a very important pattern:

• If \(|x|=a\) where \(a>0\), then \(x=a\) or \(x=-a\).
• If \(|x|=0\), then \(x=0\).
• If \(|x|=a\) where \(a<0\), there is no solution, because distance cannot be negative.

For example, \(|x|=-3\) has no solution.

Common mistakes to avoid

• Mistake 1: Thinking absolute value makes a number "bigger." It does not. It gives the distance from 0.
• Mistake 2: Saying \(|-7|=-7\). This is wrong because distance cannot be negative. The correct answer is \(|-7|=7\).
• Mistake 3: Confusing a number with its absolute value. For example, \(-4\) and \(|-4|\) are not the same. One is \(-4\), and the other is \(4\).
• Mistake 4: Forgetting that equations like \(|x|=6\) usually have two solutions: \(6\) and \(-6\).

Main ideas to remember

1. \(|x|\) means the distance from \(x\) to 0.
2. Absolute value is always greater than or equal to 0.
3. Positive and negative numbers the same distance from 0 have the same absolute value.
4. The distance between two numbers \(a\) and \(b\) is \(|a-b|\).
5. If \(|x|=a\) and \(a>0\), then \(x=\pm a\).

Summary

Absolute value is best understood as distance on the number line. It tells how far a number is from 0, not whether the number is positive or negative. This is why \(|5|\) and \(|-5|\) are both 5. We can also use absolute value to find the distance between any two numbers using \(|a-b|\).

Terminating and Repeating Decimals are two important types of decimal numbers in the real number system. Understanding them helps you move easily between fractions and decimals, and it also helps you recognize which numbers are rational.

A rational number is any number that can be written as a fraction of two integers, such as \(\frac{3}{4}\), \(\frac{-7}{2}\), or \(\frac{11}{9}\), where the denominator is not zero. When rational numbers are written as decimals, their decimal form will either end or repeat in a pattern.

In this lesson, you will learn:

• what terminating decimals are,
• what repeating decimals are,
• how to tell whether a fraction will terminate or repeat,
• and how to convert repeating decimals into fractions using algebra.

1. Terminating Decimals

A terminating decimal is a decimal that stops after a finite number of digits.

Examples:

• \(0.5\)
• \(2.75\)
• \(-3.125\)

These decimals end, so they are terminating.

Many terminating decimals come from fractions. For example,

2. Repeating Decimals

A repeating decimal is a decimal in which one digit or a group of digits repeats forever.

Examples:

• \(0.3333\ldots = 0.\overline{3}\)
• \(0.121212\ldots = 0.\overline{12}\)
• \(2.1666\ldots = 2.1\overline{6}\)

The bar shows the repeating part. For example, in \(0.\overline{12}\), the block \(12\) repeats forever.

Repeating decimals are also rational numbers because they can always be written as fractions.

3. How Fractions Connect to Decimals

Every rational number has a decimal form that is either:

• terminating, or
• repeating.

For example:

• \(\frac{1}{4} = 0.25\) terminates
• \(\frac{1}{3} = 0.\overline{3}\) repeats
• \(\frac{5}{6} = 0.8\overline{3}\) repeats

This means that if a decimal neither ends nor repeats, it is not rational. Numbers like \(\pi\) and \(\sqrt{2}\) are examples of decimals that go on forever without repeating. Those are called irrational numbers.

4. When Does a Fraction Terminate?

To decide whether a fraction will become a terminating decimal, first write the fraction in simplest form. Then look at the prime factorization of the denominator.

A fraction in simplest form will have a terminating decimal if the denominator has only the prime factors \(2\) and/or \(5\).

Why \(2\) and \(5\)? Because decimal place values are based on powers of \(10\), and

So denominators that can fit into powers of 10 will terminate.

Examples of denominators that give terminating decimals:

• \(2 = 2\)
• \(4 = 2^2\)
• \(5 = 5\)
• \(8 = 2^3\)
• \(20 = 2^2 \times 5\)
• \(25 = 5^2\)

Examples of denominators that give repeating decimals:

• \(3\)
• \(6 = 2 \times 3\)
• \(7\)
• \(12 = 2^2 \times 3\)

These repeat because they include prime factors other than \(2\) or \(5\).

Important: You must simplify the fraction first. For example,

The denominator in simplest form is \(5\), so the decimal terminates:

Worked Example 1: Decide if a fraction terminates or repeats

Determine whether \(\frac{7}{40}\) is terminating or repeating.

Step 1: Check whether the fraction is in simplest form. It is.

Step 2: Factor the denominator.

The denominator has only factors of \(2\) and \(5\), so the decimal terminates.

In fact,

Worked Example 2: Decide if a fraction terminates or repeats

Determine whether \(\frac{5}{18}\) is terminating or repeating.

Step 1: The fraction is already in simplest form.

Step 2: Factor the denominator.

Since the denominator contains a factor of \(3\), not just \(2\) and \(5\), the decimal repeats.

Its decimal form is

5. Converting a Repeating Decimal to a Fraction

You can convert repeating decimals into fractions using algebra. The key idea is to let the decimal equal a variable, then line up the repeating digits and subtract.

Worked Example 3: Convert a simple repeating decimal to a fraction

Convert \(0.\overline{3}\) to a fraction.

Let

Because one digit repeats, multiply both sides by \(10\):

Now subtract the original equation from this new equation:

Solve for \(x\):

So,

Worked Example 4: Convert a repeating decimal with two repeating digits to a fraction

Convert \(0.\overline{27}\) to a fraction.

Let

Two digits repeat, so multiply by \(100\):

Subtract the original equation:

Solve for \(x\):

Simplify:

So,

6. Repeating Decimals That Start After Some Non-Repeating Digits

Sometimes a decimal has a part that does not repeat at first, and then a repeating part begins.

Example: \(0.1\overline{6} = 0.1666\ldots\)

You can still convert it to a fraction. The idea is to use two multiplications so the repeating parts line up.

Worked Example 5: Convert \(0.1\overline{6}\) to a fraction

Let

First, move the decimal so the repeating part begins right after the decimal point. Since one non-repeating digit comes first, multiply by \(10\):

Now one digit repeats, so multiply this equation by \(10\) again, or multiply the original by \(100\):

Subtract:

Solve for \(x\):

So,

7. A Quick Pattern to Remember

• If a decimal ends, it is terminating.
• If a decimal has a repeating pattern, it is repeating.
• Every terminating or repeating decimal is rational.
• A fraction in simplest form terminates only if the denominator’s prime factors are just \(2\) and/or \(5\).

8. Common Mistakes to Avoid

• Forgetting to simplify first: Always reduce the fraction before checking the denominator.
• Thinking all long decimals repeat: Some decimals never repeat. Those are irrational numbers.
• Using the wrong power of 10: If one digit repeats, multiply by \(10\). If two digits repeat, multiply by \(100\). If three digits repeat, multiply by \(1000\), and so on.
• Not simplifying the final fraction: After converting, always reduce your answer.

Brief Summary

Terminating decimals stop, while repeating decimals continue forever in a repeating pattern. Both are rational numbers and can be written as fractions.

To tell whether a fraction will terminate, simplify it and factor the denominator. If the denominator has only \(2\) and/or \(5\) as prime factors, the decimal terminates. Otherwise, it repeats.

To convert a repeating decimal to a fraction, use algebra: let the decimal equal a variable, multiply to line up the repeating part, subtract, and solve. This method works for simple repeating decimals and for decimals with a non-repeating part before the repetition starts.

Interval and inequality notation are two different ways to describe the same set of real numbers.

In 9th Grade Maths, you often need to describe numbers that are greater than, less than, or between certain values. Inequalities show this using symbols like \(<\), \(>\), \(\le\), and \(\ge\). Interval notation shows the same idea using parentheses and brackets.

Learning to switch between these forms helps you read graphs, solve equations, and describe solution sets clearly.

1. Inequality notation

An inequality compares numbers or expressions.

• \(x < 5\) means all real numbers less than 5.
• \(x \le 5\) means all real numbers less than or equal to 5.
• \(x > -2\) means all real numbers greater than \(-2\).
• \(x \ge -2\) means all real numbers greater than or equal to \(-2\).
• \(2 < x < 7\) means all real numbers between 2 and 7, not including 2 or 7.

The inequality symbols tell whether an endpoint is included:

• Strict inequality: \(<\) or \(>\) means the endpoint is not included.
• Inclusive inequality: \(\le\) or \(\ge\) means the endpoint is included.

2. Interval notation

Interval notation uses symbols to describe a continuous set of numbers on the number line.

The basic forms are:

• Parentheses \((\, )\) mean the endpoint is not included.
• Brackets \([\, ]\) mean the endpoint is included.

For example:

• \((2, 7)\) means all numbers between 2 and 7, but not 2 or 7.
• \([2, 7]\) means all numbers from 2 to 7, including both 2 and 7.
• \((2, 7]\) means greater than 2 and up to 7, including 7 but not 2.
• \([2, 7)\) means at least 2 and less than 7.

3. Matching inequalities and intervals

Here is how the two notations connect:

• \(x < 5\) corresponds to $$(-\infty, 5)$$
• \(x \le 5\) corresponds to $$(-\infty, 5]$$
• \(x > -2\) corresponds to $$(-2, \infty)$$
• \(x \ge -2\) corresponds to $$[-2, \infty)$$
• \(2 < x < 7\) corresponds to $$(2, 7)$$
• \(2 \le x \le 7\) corresponds to $$[2, 7]$$
• \(2 < x \le 7\) corresponds to $$(2, 7]$$
• \(2 \le x < 7\) corresponds to $$[2, 7)$$

Important: Infinity is never written with a bracket.

We use parentheses with \(\infty\) and \(-\infty\) because infinity is not a real endpoint. It is an idea that numbers continue without end.

• Correct: \((-\infty, 4)\), \([3, \infty)\)
• Not correct: \([-\infty, 4)\), \((3, \infty]\)

4. Reading interval notation

When reading an interval, look at:

1. The left endpoint
2. The right endpoint
3. Whether each side uses a parenthesis or bracket

Example: \([ -1, 6 )\)

• The interval starts at \(-1\)
• The bracket means \(-1\) is included
• The interval goes up to \(6\)
• The parenthesis means \(6\) is not included

So the inequality is:

$$-1 \le x < 6$$

5. Graphing on a number line

Both inequalities and intervals can be shown on a number line.

• An open circle means the endpoint is not included.
• A closed circle means the endpoint is included.
• A shaded line shows all the numbers in the set.

For example:

• \(x < 3\): open circle at 3, shade left
• \(x \ge 3\): closed circle at 3, shade right
• \(-2 < x \le 4\): open circle at \(-2\), closed circle at 4, shade between them

6. Worked examples

Example 1: Write an inequality as interval notation

Write \(x \le 8\) in interval notation.

Step 1: The values include 8 and all numbers less than 8.

Step 2: Since the set goes forever to the left, use \(-\infty\).

Step 3: Since 8 is included, use a bracket at 8.

Answer:

$$(-\infty, 8]$$

Example 2: Write interval notation as an inequality

Write \(( -3, 5 ]\) as an inequality.

Step 1: The parenthesis at \(-3\) means \(-3\) is not included, so use \(>\).

Step 2: The bracket at 5 means 5 is included, so use \(\le\).

Answer:

$$-3 < x \le 5$$

Example 3: Describe a number line in both forms

A graph has a closed circle at 1, an open circle at 6, and shading between them.

Inequality form: Since 1 is included and 6 is not included, we write

$$1 \le x < 6$$

Interval form:

$$[1, 6)$$

Example 4: Use a real-life style description

A temperature is above \(-4\) degrees and can be as high as 2 degrees, including 2.

Step 1: “Above \(-4\)” means

$$x > -4$$

Step 2: “As high as 2, including 2” means

$$x \le 2$$

Step 3: Combine them:

$$-4 < x \le 2$$

Interval notation:

$$(-4, 2]$$

7. Common mistakes to avoid

• Mixing up parentheses and brackets
  Parentheses mean not included. Brackets mean included.
• Using a bracket with infinity
  Always use parentheses with \(\infty\) or \(-\infty\).
• Reversing the endpoints
  Write intervals from smaller number to larger number.
• Forgetting that “between” may or may not include endpoints
  Read carefully to decide whether to use strict or inclusive inequalities.

8. Quick comparison table

• Included endpoint \(\rightarrow\) bracket \([\ ]\) \(\rightarrow\) closed circle \(\rightarrow\) \(\le\) or \(\ge\)
• Not included endpoint \(\rightarrow\) parenthesis \((\ )\) \(\rightarrow\) open circle \(\rightarrow\) \(<\) or \(>\)

9. How to convert step by step

From inequality to interval notation:

1. Find the endpoints.
2. Decide whether each endpoint is included.
3. Use a bracket if included and a parenthesis if not included.
4. Use \(-\infty\) or \(\infty\) if the set continues forever.

From interval notation to inequality:

1. Look at the left endpoint and symbol.
2. Look at the right endpoint and symbol.
3. Write an inequality using \(<\), \(>\), \(\le\), or \(\ge\).
4. If possible, write it as one combined inequality, such as \(a \le x < b\).

Summary

Interval notation and inequality notation both describe sets of real numbers. Parentheses mean an endpoint is not included, while brackets mean it is included. Inequalities use symbols such as \(<\), \(>\), \(\le\), and \(\ge\). When an interval extends forever, always use parentheses with \(\infty\) or \(-\infty\).

Complex Numbers Introduction

In earlier maths, you learned about real numbers. Real numbers include counting numbers, whole numbers, integers, fractions, decimals, and irrational numbers like \(\sqrt{2}\) and \(\pi\).

Real numbers are enough for many problems, but not for all of them. Some equations have no real number solution. This is where complex numbers begin.

In this lesson, you will learn why the real number system has a limit, what the imaginary unit is, and how mathematicians write numbers that include it.

1. The limit of the real number system

Think about squaring real numbers:

• \(3^2 = 9\)
• \((-3)^2 = 9\)
• \(0^2 = 0\)

Whenever you square a real number, the result is never negative.

That means an equation like

has no real solution, because there is no real number whose square is \(-1\).

This does not mean the equation is meaningless. It means the real number system is not large enough to include its solution.

2. Defining the imaginary unit

To solve equations like \(x^2 = -1\), mathematicians defined a new number called the imaginary unit.

The imaginary unit is written as \(i\), and it is defined by

This definition is the starting point for complex numbers.

Since \(i^2 = -1\), we can say that

So the equation \(x^2 = -1\) has solutions

3. What is a complex number?

A complex number is any number that can be written in the form

where:

• \(a\) is the real part
• \(b\) is the imaginary part
• \(i\) is the imaginary unit, with \(i^2 = -1\)

Examples of complex numbers include:

• \(3 + 2i\)
• \(-5 + i\)
• \(7 - 4i\)
• \(0 + 6i = 6i\)
• \(9 + 0i = 9\)

This last example is important: every real number can also be thought of as a complex number with imaginary part \(0\).

4. Real numbers and complex numbers

The real numbers are part of the complex numbers. You can think of it like this:

• All real numbers are complex numbers.
• Not all complex numbers are real numbers.

For example:

• \(4 = 4 + 0i\), so it is real and complex.
• \(-2.5 = -2.5 + 0i\), so it is real and complex.
• \(3i\) is complex, but not real.
• \(2 + i\) is complex, but not real.

5. Why square roots of negative numbers are different

In the real number system:

• \(\sqrt{9} = 3\)
• \(\sqrt{16} = 4\)

But \(\sqrt{-9}\) is not a real number. Using \(i\), we can rewrite it.

Since \(-9 = 9(-1)\), we have

Similarly,

and

This gives us a way to write solutions that were impossible in the real number system.

6. Powers of \(i\)

Because \(i^2 = -1\), powers of \(i\) follow a repeating pattern.

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = i^2 \cdot i = (-1)(i) = -i\)
• \(i^4 = i^2 \cdot i^2 = (-1)(-1) = 1\)

Then the pattern repeats:

• \(i^5 = i\)
• \(i^6 = -1\)
• \(i^7 = -i\)
• \(i^8 = 1\)

This pattern is useful when simplifying expressions with powers of \(i\).

7. Solving very simple equations with \(i\)

When an equation has a negative number under a square, complex numbers allow us to write the answer.

For example:

Take the square root of both sides:

So the solutions are \(2i\) and \(-2i\).

Another example:

Then

Worked Example 1: Deciding whether a number is real, complex, or both

Classify each number: \(5\), \(3i\), \(-2 + 4i\).

Step 1: Look for the form \(a + bi\).

• \(5 = 5 + 0i\)
• \(3i = 0 + 3i\)
• \(-2 + 4i\) is already in the form \(a + bi\)

Step 2: Classify them.

• \(5\): real and complex
• \(3i\): complex, not real
• \(-2 + 4i\): complex, not real

Worked Example 2: Simplifying a square root of a negative number

Simplify \(\sqrt{-36}\).

Step 1: Separate the negative part.

Step 2: Replace \(\sqrt{-1}\) with \(i\).

Answer: \(\sqrt{-36} = 6i\)

Worked Example 3: Solving a simple quadratic equation

Solve \(x^2 + 9 = 0\).

Step 1: Move 9 to the other side.

Step 2: Take square roots.

Step 3: Write the square root using \(i\).

Answer: \(x = 3i\) and \(x = -3i\)

Worked Example 4: Simplifying powers of \(i\)

Simplify \(i^7\).

Method 1: Use the repeating pattern.

The powers repeat every 4:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

Since \(7\) matches the third place in the pattern after dividing by 4,

Method 2: Multiply step by step.

Answer: \(i^7 = -i\)

8. Common mistakes to avoid

• Mistake: Saying \(\sqrt{-9} = -3\).
  Wrong, because \((-3)^2 = 9\), not \(-9\). The correct value is \(3i\).
• Mistake: Thinking complex numbers are not numbers.
  They are numbers in a larger number system than the real numbers.
• Mistake: Forgetting the \(\pm\) when solving \(x^2 = -a\).
  For example, if \(x^2 = -16\), then \(x = \pm 4i\).
• Mistake: Forgetting that a real number can be written as \(a + 0i\).

9. Key ideas to remember

• Some equations have no solution in the real numbers.
• The imaginary unit is defined by \(i^2 = -1\).
• A complex number has the form \(a + bi\).
• Real numbers are included inside the complex numbers.
• Square roots of negative numbers can be written using \(i\).

Brief Summary

The real number system has a boundary: no real number squares to a negative number. To go beyond that boundary, mathematicians defined the imaginary unit \(i\), where \(i^2 = -1\).

Using \(i\), we can write complex numbers in the form \(a + bi\). This lets us describe solutions to equations like \(x^2 = -1\), \(x^2 = -4\), and \(x^2 = -9\), which have no real solutions.