Density of Numbers, Infinity & Limits

Ontario Grade 9 Mathematics

Number

Ontario MTH1W — Expectation B1.3

Have you ever tried to find the \"next\" number after 1 on a number line? With whole numbers, the answer is easy: it is 2. But what about fractions? After

\frac{1}{2}

, is the next fraction

\frac{3}{4}

? Or

\frac{51}{100}

? It turns out you can always squeeze another fraction in between any two fractions — forever. That surprising idea is called density. In this lesson you will explore three big ideas that describe how number sets behave: density (how tightly packed numbers are), infinity (the fact that some sets go on without end), and limits (the value a pattern of numbers keeps getting closer to). These ideas help mathematicians understand why different sets of numbers behave so differently from one another.

What you will learn

Explain what it means for a number set to be dense by finding numbers between any two given numbers.
Describe infinity as an endless process, not a number, and use it to explain why certain number sets never stop.
Identify the value that a pattern of numbers gets closer and closer to, and describe that value informally as the limit of the pattern.
Compare the natural numbers, integers, rational numbers, and irrational numbers using the ideas of density and infinity.

Prerequisite Review: Number Sets You Already Know

Before exploring new ideas, it helps to recall the number sets from earlier in this course. The natural numbers are the counting numbers:

1, 2, 3, 4, \ldots

The three dots (called an ellipsis) mean the list continues in the same pattern without stopping. The integers add zero and the negative whole numbers:

\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots

The rational numbers are all numbers that can be written as a fraction

\frac{a}{b}

, where

a

and

b

are integers and

b \neq 0

. This includes numbers like

\frac{3}{4}

,

-\frac{5}{2}

,

0.6

(which equals

\frac{3}{5}

), and even whole numbers like

4

(which equals

\frac{4}{1}

). The irrational numbers are numbers that cannot be written as a fraction of two integers. A familiar example is

\sqrt{2} \approx 1.414\ldots

, whose decimal digits continue forever with no repeating block.

Together, rational and irrational numbers fill the entire number line and form the real numbers. Keep these sets in mind — density, infinity, and limits each behave differently depending on which set you are working in.

Natural numbers: $1, 2, 3, \ldots$ — counting numbers only.
Integers: $\ldots, -2, -1, 0, 1, 2, \ldots$ — whole numbers, their negatives, and zero.
Rational numbers: any number of the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ .
Irrational numbers: real numbers whose decimal form never ends and never repeats.
Real numbers: all rational and irrational numbers together — every point on the number line.

Density: Always Room for One More

A number set is called dense when you can always find another number from that same set sitting between any two numbers you choose, no matter how close together they are. Think of it like this: between any two rational numbers, there is always at least one more rational number. That middle number is called the midpoint, and you find it by averaging the two numbers.

For example, take

0.1

and

0.2

. Their midpoint is

\frac{0.1 + 0.2}{2} = 0.15

. Now take

0.1

and

0.15

. Their midpoint is

\frac{0.1 + 0.15}{2} = 0.125

. You can repeat this process forever and always find a new rational number between the previous two. This is the defining property of a dense set — the gaps never become too small to fit another member of the set.

The natural numbers and integers do not have this property. Between the integers

3

and

4

there is no other integer. So the integers are not dense. The rational numbers, however, are dense. So are the real numbers. Understanding density explains why you can always zoom in further on a rational number line and find more rational numbers waiting there.

\text{Midpoint} = \frac{a + b}{2}

A set is dense if there is always another member of that set between any two chosen members.
Finding the midpoint — $\frac{a + b}{2}$ — always produces a rational number between $a$ and $b$ when $a$ and $b$ are rational.
The integers are not dense: there is no integer between $3$ and $4$ .
The rational numbers are dense: between any two rationals, infinitely many more exist.
Density is about how packed together numbers are — not just how many there are.

Infinity: A Process That Never Ends

Infinity is not a number you can reach or write on a number line. Instead, it describes an endless process. When mathematicians write

1, 2, 3, \ldots

, the ellipsis signals that the pattern continues without ever stopping. No matter how large a natural number you name, there is always a larger one. This is what it means for a set to be infinite.

Both the natural numbers and the integers are infinite — they have no largest or smallest member. The rational numbers are also infinite. Even in the tiny interval between

0

and

1

, there are infinitely many rational numbers:

\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots

Each new fraction is smaller than the one before, and the list never stops.

It is important not to treat infinity as though it were a very large number. You cannot compute

\infty + 1

or

\frac{\infty}{2}

in ordinary arithmetic. Infinity is a description of a process — the process of continuing a pattern with no end. When you see the ellipsis in a number list or on a number line, that ellipsis is pointing at infinity.

\frac{1}{2}

,\

\frac{1}{3}

,\

\frac{1}{4}

,\

\frac{1}{5}

,\ \ldots (infinitely many terms, each positive)

Infinity describes a process that continues without end — it is not a number on the number line.
A set is infinite when it has no last member; you can always find one more.
The natural numbers, integers, rational numbers, and real numbers are all infinite sets.
Even a small interval like $0$ to $1$ contains infinitely many rational numbers.
The symbol $\infty$ represents endlessness, not a reachable value.

Limits: What a Pattern Gets Closer and Closer To

Look again at the list

\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots

Each term is a positive number, but each one is smaller than the one before it. As you go further along the list, the terms get closer and closer to

0

. They never actually equal

0

— every term in the list is still a positive fraction — but

0

is the value they are approaching. That target value is called the limit of the pattern.

You can see the same idea with decimals. The pattern

0.9, 0.99, 0.999, 0.9999, \ldots

grows closer and closer to

1

with each new term. Each step adds another

9

in the decimal, cutting the remaining gap by a factor of ten. The terms never reach

1

within the list, but

1

is clearly the value the pattern is heading toward — the limit.

A limit is described in plain language as: the value that the terms of a pattern get closer and closer to, without necessarily reaching it. Recognizing limits helps you make sense of infinite patterns — instead of being confused by a list that never ends, you can ask 'what value is this list approaching?' and often find a clear, finite answer.

\frac{1}{1}

,\

\frac{1}{2}

,\

\frac{1}{3}

,\

\frac{1}{4}

,\ \ldots gets closer and closer to 0

A limit is the value that the terms of an infinite pattern get closer and closer to.
The terms may never actually equal the limit — they only approach it.
For the pattern $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots$ , the limit is $0$ .
For the pattern $0.9, 0.99, 0.999, \ldots$ , the limit is $1$ .
Identifying a limit means watching the trend — ask yourself what value the terms are clearly heading toward.

Connecting the Three Ideas

Density, infinity, and limits work together to describe the structure of number sets. Because the rational numbers are dense, you can build an infinite pattern of rational numbers that homes in on a single target. For example, the pattern

1.4,\ 1.41,\ 1.414,\ 1.4142,\ \ldots

is a sequence of rational numbers (each is a terminating decimal, so each is rational) whose limit is

\sqrt{2}

, an irrational number. This shows something remarkable: an infinite pattern of rational numbers can approach a limit that is not rational.

This connection between density, infinity, and limits helps explain why the real number line has no gaps. Every irrational number — like

\sqrt{2}

,

\sqrt{3}

, or

\pi

— can be approached as closely as you like by choosing rational numbers that are dense around it. The real numbers together (rational and irrational) leave no gaps at all on the number line.

In summary: density tells you there is always room between numbers, infinity tells you the room never runs out, and a limit tells you the value an infinite pattern is heading toward. Together, these three ideas give mathematicians precise language for describing how number sets fill the number line.

1.4,\ 1.41,\ 1.414,\ 1.4142,\ \ldots gets closer and closer to

\sqrt{2}

Density, infinity, and limits are connected: a dense set allows infinite patterns to be built that home in on a limit.
Rational numbers are dense, so you can get as close as you like to any real number using only rationals.
A pattern of rational numbers can have an irrational limit, such as $\sqrt{2}$ .
The real numbers (rational + irrational together) leave no gaps on the number line.
Limit language — 'gets closer and closer to' — is the informal bridge between infinite patterns and a target value.

Comparing Number Sets: Density, Infinity, and Limits

Number Set	Infinite?	Dense?	Example of a Limit Approached
Natural numbers $\{1, 2, 3, \ldots\}$	Yes — no largest member	No — no natural number between 3 and 4	None within the set
Integers $\{\ldots, -1, 0, 1, \ldots\}$	Yes — no largest or smallest	No — no integer between 0 and 1	None within the set
Rational numbers	Yes — infinitely many in any interval	Yes — midpoint of any two rationals is rational	$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$ approaches $0$
Irrational numbers	Yes — infinitely many	No — not closed under averages	$1.4, 1.41, 1.414, \ldots$ approaches $\sqrt{2}$
Real numbers	Yes — every point on number line	Yes — densest of all: no gaps	Any infinite pattern of reals can approach a real limit

Worked example

Finding Numbers Between Two Rationals and Identifying a Limit

Part A — Density: Find two rational numbers between

\frac{1}{4}

and

\frac{1}{3}

. Part B — Limit: The pattern below continues without end. Identify the value the terms are getting closer and closer to, and explain how you know.

\frac{1}{4},\ \frac{1}{8},\ \frac{1}{16},\ \frac{1}{32},\ \ldots

Step 1 — Convert to a common denominator
To find a number between two fractions, it helps to rewrite them with a common denominator so you can compare them easily. The least common denominator of 4 and 3 is 12.
$\frac{1}{4} = \frac{3}{12} \qquad \frac{1}{3} = \frac{4}{12}$
Step 2 — Find the midpoint of the two fractions
Use the midpoint formula $\frac{a+b}{2}$ . Add the two fractions and divide by 2 (which is the same as multiplying by $\frac{1}{2}$ ).
$\frac{\frac{3}{12} + \frac{4}{12}}{2} = \frac{\frac{7}{12}}{2} = \frac{7}{24}$
Step 3 — Verify the midpoint sits between the original two
Check that $\frac{1}{4} < \frac{7}{24} < \frac{1}{3}$ by rewriting all three fractions with denominator 24.
$\frac{1}{4} = \frac{6}{24}, \quad \frac{7}{24} = \frac{7}{24}, \quad \frac{1}{3} = \frac{8}{24} \quad \Rightarrow \quad \frac{6}{24} < \frac{7}{24} < \frac{8}{24} \checkmark$
Step 4 — Find a second rational number between the originals
Apply the midpoint formula again, this time between $\frac{7}{24}$ and $\frac{1}{3}$ . This shows the process can be repeated — illustrating density.
$\frac{\frac{7}{24} + \frac{8}{24}}{2} = \frac{\frac{15}{24}}{2} = \frac{15}{48} = \frac{5}{16}$
Step 5 — Part B: List the pattern and watch the trend
Write out several terms as decimals so the trend is easy to see. Each term is half of the one before it.
$\frac{1}{4} = 0.25, \quad \frac{1}{8} = 0.125, \quad \frac{1}{16} = 0.0625, \quad \frac{1}{32} = 0.03125, \quad \ldots$
Step 6 — Identify the limit
Each term is positive, but the terms keep shrinking toward zero. No term in the list equals zero, but the pattern gets closer and closer to zero with each step. The limit of this pattern is $0$ .
The terms approach 0. The limit is 0.

Answer: Part A: Two rational numbers between

\frac{1}{4}

and

\frac{1}{3}

are

\frac{7}{24}

and

\frac{5}{16}

. Part B: The terms get closer and closer to

0

; the limit of the pattern is

0

.

Check: For Part A, confirm by writing all values over 48:

\frac{1}{4} = \frac{12}{48}

,

\frac{7}{24} = \frac{14}{48}

,

\frac{5}{16} = \frac{15}{48}

,

\frac{1}{3} = \frac{16}{48}

. The order

12 < 14 < 15 < 16

confirms both numbers lie between the originals. For Part B, each term equals the previous term multiplied by

\frac{1}{2}

, so the values shrink without bound toward

0

but are always positive — consistent with a limit of

0

.

Common mistakes and how to avoid them

Treating infinity as a very large number, for example writing

\infty - 1

or saying 'the last number is

\infty

'.

Correction: Infinity is not a number — it describes a process that continues without end. There is no 'last' member of an infinite set, and ordinary arithmetic operations do not apply to

\infty

.

Saying that a pattern 'reaches' its limit. For example, claiming that

\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots

eventually equals

0

.

Correction: The terms get closer and closer to the limit, but no single term in the pattern equals the limit. The limit describes the value being approached, not a value that is reached within the list.

Assuming the integers are dense because there are infinitely many of them.

Correction: A set being infinite does not make it dense. Density requires that another member of the same set can always be found between any two chosen members. There is no integer between

5

and

6

, so the integers are not dense.

Using subtraction to find a number between two fractions — for example, writing

\frac{1}{3} - \frac{1}{4} = \frac{1}{12}

and claiming

\frac{1}{12}

is between the two.

Correction: The difference

\frac{1}{12}

is less than

\frac{1}{4}

, so it is not between the two original fractions. Use the midpoint formula

\frac{a+b}{2}

to guarantee the result is between

a

and

b

.

Thinking that a decimal that goes on forever must be irrational.

Correction: A repeating decimal such as

0.\overline{3} = \frac{1}{3}

goes on forever but is rational. A decimal is irrational only when it continues forever with no repeating block, like

\sqrt{2} = 1.41421\ldots

Lesson summary

A number set is dense if you can always find another member of that same set between any two chosen members. Rational and real numbers are dense; natural numbers and integers are not.
The midpoint formula $\frac{a+b}{2}$ always produces a rational number between two rational numbers $a$ and $b$ , and can be applied repeatedly — showing density in action.
Infinity describes a process that continues without end. It is not a number. A set is infinite when it has no last member, no matter how far you count or list.
A limit is the value that the terms of an infinite pattern get closer and closer to. The terms approach the limit but do not have to equal it.
A pattern of rational numbers can approach an irrational limit (for example, $1.4, 1.41, 1.414, \ldots$ approaches $\sqrt{2}$ ), which shows how density and limits connect.
Density, infinity, and limits together explain the structure of number sets: density guarantees room between numbers, infinity guarantees the room never runs out, and limits identify the target of an unending approach.

Check your understanding

Question 1

Which of the following number sets is dense?

The natural numbers $\{1, 2, 3, \ldots\}$
The integers $\{\ldots, -1, 0, 1, 2, \ldots\}$
The rational numbers
The set of perfect squares $\{1, 4, 9, 16, \ldots\}$

Show answer and explanation

The rational numbers

The rational numbers are dense: between any two rational numbers you can always find another rational number using the midpoint formula

\frac{a+b}{2}

. The natural numbers, integers, and perfect squares all have gaps — for example, there is no integer between 2 and 3, and no perfect square between 4 and 9.

Question 2

The pattern

0.9,\ 0.99,\ 0.999,\ 0.9999,\ \ldots

continues without end. What value are the terms getting closer and closer to?

$0.9$
$0.\overline{9}$ , which is a value just below $1$ but never equal to $1$
$1$
$\infty$

Show answer and explanation

1

With each new term, the gap between the term and

1

shrinks by a factor of ten. The terms are getting closer and closer to

1

, so the limit of the pattern is

1

. Infinity is not a number and cannot be a limit on the number line. The notation

0.\overline{9}

actually equals

1

, but the key point here is that the pattern approaches

1

.

Question 3

A student says: 'Infinity is just a very large number, bigger than any number I can write.' What is wrong with this statement?

Nothing is wrong — infinity is the largest possible number.
Infinity is a number, but it can only be used in multiplication, not addition.
Infinity is not a number at all; it describes a process that continues without end, so ordinary arithmetic does not apply to it.
Infinity applies only to negative numbers, not positive ones.

Show answer and explanation

Infinity is not a number at all; it describes a process that continues without end, so ordinary arithmetic does not apply to it.

Infinity describes an endless process, not a position on the number line. You cannot add, subtract, or multiply with

\infty

the way you do with ordinary numbers. Saying it is a 'very large number' leads to errors, such as believing

\infty + 1 > \infty

, which is not meaningful in ordinary arithmetic.

Question 4

You want to find a rational number between

\frac{2}{5}

and

\frac{1}{2}

. You apply the midpoint formula. Which of the following is correct?

$\frac{2}{5} - \frac{1}{2} = -\frac{1}{10}$ , so the number between them is $-\frac{1}{10}$ .
$\frac{\frac{2}{5} + \frac{1}{2}}{2} = \frac{\frac{9}{10}}{2} = \frac{9}{20}$ , which is between $\frac{2}{5}$ and $\frac{1}{2}$ .
$\frac{\frac{2}{5} + \frac{1}{2}}{2} = \frac{\frac{9}{10}}{2} = \frac{9}{20}$ , but this is not rational so it cannot be between two rationals.
$\frac{2}{5} + \frac{1}{2} = \frac{9}{10}$ , and $\frac{9}{10}$ is between $\frac{2}{5}$ and $\frac{1}{2}$ .

Show answer and explanation

\frac{\frac{2}{5} + \frac{1}{2}}{2} = \frac{\frac{9}{10}}{2} = \frac{9}{20}

, which is between

\frac{2}{5}

and

\frac{1}{2}

.

The midpoint formula gives

\frac{\frac{2}{5}+\frac{1}{2}}{2}

. Converting:

\frac{2}{5} = \frac{4}{10}

and

\frac{1}{2} = \frac{5}{10}

, so the sum is

\frac{9}{10}

. Dividing by

2

gives

\frac{9}{20}

. Checking:

\frac{2}{5} = \frac{8}{20}

and

\frac{1}{2} = \frac{10}{20}

, and

8 < 9 < 10

, confirming

\frac{9}{20}

lies between the two. The fraction

\frac{9}{20}

is rational because both 9 and 20 are integers.

Key terms

Dense (density): A number set is dense if, between any two members of that set, there is always at least one more member of the same set. The rational numbers are dense; the integers are not.
Infinity: A description of a process that continues without end. Infinity is not a number; it means that a pattern or set has no final member.
Limit: The value that the terms of an infinite pattern get closer and closer to. The terms approach the limit but do not have to equal it within the list.
Rational number: Any number that can be written in the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ . Includes all fractions, terminating decimals, and repeating decimals.
Irrational number: A real number whose decimal form continues forever with no repeating block, and which cannot be written as a fraction of two integers. Example: $\sqrt{2}$ .
Ellipsis ( $\ldots$ ): Three dots used in a number list to show that the pattern continues in the same way without stopping.
Midpoint: The value exactly halfway between two numbers $a$ and $b$ , calculated using the formula $\frac{a+b}{2}$ . The midpoint of two rational numbers is always rational.
Real numbers: The complete set of all rational and irrational numbers together. Every point on the number line corresponds to a real number.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation B1.3. It is a study resource, not an official curriculum publication.

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B1.3 · Density of Numbers, Infinity & Limits

Ontario Grade 9 Mathematics

Ontario MTH1W — Expectation B1.3

What you will learn

Prerequisite Review: Number Sets You Already Know

Density: Always Room for One More

Infinity: A Process That Never Ends

Limits: What a Pattern Gets Closer and Closer To

Connecting the Three Ideas

Comparing Number Sets: Density, Infinity, and Limits

Finding Numbers Between Two Rationals and Identifying a Limit

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review