Real Number System Foundations

Real Number System Hierarchy

In math, numbers can be grouped into different sets. A set is just a collection of things that belong together. The real number system is the collection of all numbers that can be placed on a number line.

Some number sets are small and fit inside bigger number sets. This is called a hierarchy. Learning this hierarchy helps you classify numbers correctly and understand how the number system is organized.

In this lesson, you will learn about these number sets:

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

1. Natural Numbers

Natural numbers are the counting numbers you use when you count objects.

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

These numbers do not include \(0\) or negative numbers.

2. Whole Numbers

Whole numbers are the natural numbers plus zero.

Examples: \(0, 1, 2, 3, 4, 5, \dots\)

So every natural number is a whole number, but \(0\) is a whole number that is not a natural number.

3. Integers

Integers include all whole numbers and their opposites.

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

Integers do not include fractions or decimals.

4. Rational Numbers

Rational numbers are numbers that can be written as a fraction of two integers, where the denominator is not zero.

In other words, a rational number can be written in the form

$$\frac{a}{b}$$

where \(a\) and \(b\) are integers and \(b \ne 0\).

Examples of rational numbers:

• \(\frac{1}{2}\)
• \(-\frac{3}{4}\)
• \(5\), because \(5 = \frac{5}{1}\)
• \(0\), because \(0 = \frac{0}{1}\)
• \(0.75\), because \(0.75 = \frac{3}{4}\)

A decimal is rational if it ends or repeats.

Examples:

• \(0.2\) ends, so it is rational.
• \(0.125\) ends, so it is rational.
• \(0.333\dots\) repeats, so it is rational.
• \(1.272727\dots\) repeats, so it is rational.

5. Irrational Numbers

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

Their decimals do not end and do not repeat.

Examples of irrational numbers:

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

For example, \(\pi = 3.14159265\dots\) goes on forever without a repeating pattern.

6. Real Numbers

Real numbers include all rational numbers and all irrational numbers. If a number can be placed on the number line, it is a real number.

That means natural numbers, whole numbers, integers, rational numbers, and irrational numbers are all part of the real number system.

The Hierarchy of the Real Number System

Here is the nesting, from smallest sets to larger sets:

• Natural numbers are inside whole numbers.
• Whole numbers are inside integers.
• Integers are inside rational numbers.
• Rational numbers and irrational numbers together make up real numbers.

You can think of it like this:

$$\text{Natural} \subset \text{Whole} \subset \text{Integers} \subset \text{Rational} \subset \text{Real}$$

Also, irrational numbers are part of the real numbers, but they are separate from rational numbers.

So the real numbers are split into two big parts:

$$\text{Real Numbers} = \text{Rational Numbers} \cup \text{Irrational Numbers}$$

Visualizing with a Venn Diagram

A Venn diagram helps show how sets fit inside each other.

Imagine one large rectangle labeled Real Numbers. Inside it are two regions:

• one region for Rational Numbers
• one separate region for Irrational Numbers

Inside the rational numbers region, you would place smaller nested circles:

• Integers
• inside integers: Whole Numbers
• inside whole numbers: Natural Numbers

This shows that every natural number is also a whole number, integer, rational number, and real number.

Important Ideas to Remember

• A number can belong to more than one set.
• When classifying a number, it is often best to name the smallest set it belongs to.
• All integers are rational because they can be written as fractions with denominator \(1\).
• Not all rational numbers are integers.
• Irrational numbers are real, but they are not rational.

Worked Example 1: Classify \(4\)

Step 1: Is \(4\) a natural number? Yes, because it is a counting number.

Step 2: Since it is natural, it is also in all the larger sets that contain natural numbers.

So \(4\) is:

• natural
• whole
• integer
• rational, because \(4 = \frac{4}{1}\)
• real

The smallest set it belongs to is natural numbers.

Worked Example 2: Classify \(0\)

Step 1: Is \(0\) a natural number? No, not in this lesson’s definition.

Step 2: Is \(0\) a whole number? Yes.

Step 3: Since whole numbers are also integers, rational numbers, and real numbers, \(0\) belongs to those sets too.

So \(0\) is:

• whole
• integer
• rational, because \(0 = \frac{0}{1}\)
• real

The smallest set it belongs to is whole numbers.

Worked Example 3: Classify \(-7\)

Step 1: Is \(-7\) a natural number or a whole number? No, because those sets do not include negative numbers.

Step 2: Is \(-7\) an integer? Yes.

Step 3: It is also rational because

$$-7 = \frac{-7}{1}$$

Step 4: All rational numbers are real numbers.

So \(-7\) is:

• integer
• rational
• real

The smallest set it belongs to is integers.

Worked Example 4: Classify \(\sqrt{2}\)

Step 1: \(\sqrt{2}\) cannot be written as a simple fraction of integers.

Step 2: Its decimal goes on forever and does not repeat.

So \(\sqrt{2}\) is irrational.

Since all irrational numbers are real numbers, \(\sqrt{2}\) is also real.

The smallest set it belongs to is irrational numbers.

How to Classify Any Number

1. Ask: Is it a counting number? If yes, it is natural.
2. If not, ask: Is it \(0\)? If yes, it is whole.
3. If not, ask: Is it a negative or positive number with no fraction or decimal? If yes, it is an integer.
4. If not, ask: Can it be written as a fraction of integers? If yes, it is rational.
5. If it cannot be written as a fraction and its decimal does not end or repeat, it is irrational.
6. Finally, remember that all of these are real numbers.

More Quick Examples

• \(12\): natural, whole, integer, rational, real
• \(-2\): integer, rational, real
• \(\frac{5}{8}\): rational, real
• \(0.4\): rational, real
• \(0.121212\dots\): rational, real
• \(\pi\): irrational, real

Common Mistakes

• Mistake: Thinking all decimals are irrational.
  Some decimals are rational if they end or repeat.
• Mistake: Forgetting that integers are rational.
  Any integer can be written over \(1\).
• Mistake: Saying irrational numbers are not real.
  Irrational numbers are part of the real number system.
• Mistake: Putting \(0\) in the natural numbers in this lesson.
  Here, \(0\) is classified as a whole number, not a natural number.

Summary

The real number system is organized like a set of nested groups. Natural numbers are inside whole numbers, whole numbers are inside integers, and integers are inside rational numbers. Rational numbers and irrational numbers together make up all real numbers.

If you can decide whether a number is counting, whole, integer, rational, or irrational, then you can place it correctly in the real number system hierarchy.

Rational Numbers as Division

In math, a fraction does more than show a part of a whole. A fraction can also show a division problem.

For example, the fraction \(\frac{3}{4}\) means 3 divided by 4. We can write that as:

This idea is very important because it helps us understand rational numbers.

A rational number is any number that can be written as a fraction in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\).

Since a fraction is really division, every rational number can be thought of as a quotient, which is the answer to a division problem.

Here is the main idea:

• The numerator is the number being divided.
• The denominator is the number you divide by.
• The denominator can never be 0, because division by 0 is not possible.

So if we see \(\frac{a}{b}\), we can read it as:

This means fractions, decimals, and some whole numbers can all be different ways to show the same rational number.

For example:

• \(\frac{1}{2}\)
• \(1 \div 2\)
• \(0.5\)

These all represent the same value.

Why is this useful?

When you understand fractions as division, you can:

• change fractions into decimals,
• understand where rational numbers belong on a number line,
• see that whole numbers can also be written as fractions,
• compare numbers in different forms.

Fractions, division, and decimals

When you divide the numerator by the denominator, you get a decimal. Sometimes the decimal ends, and sometimes it repeats.

Examples:

• \(\frac{3}{5} = 3 \div 5 = 0.6\)
• \(\frac{1}{4} = 1 \div 4 = 0.25\)
• \(\frac{1}{3} = 1 \div 3 = 0.333\ldots\)

All of these are rational numbers because they can be written as fractions.

Whole numbers as division

Whole numbers are also rational numbers because they can be written as fractions.

For example:

This shows that rational numbers include more than just fractions like \(\frac{2}{3}\). They also include integers and whole numbers.

Negative rational numbers

Rational numbers can be negative too. A negative fraction is still division.

For example:

You can also write it as:

These forms all mean the same negative rational number.

Worked Example 1: Simple fraction as division

Write \(\frac{6}{3}\) as a division problem and find the value.

Step 1: Rewrite the fraction as division.

Step 2: Divide.

Answer: \(\frac{6}{3} = 2\)

This fraction is a rational number, and its value is a whole number.

Worked Example 2: Fraction to decimal

Write \(\frac{3}{8}\) as division and find its decimal form.

Step 1: Rewrite as division.

Step 2: Divide.

Answer: \(\frac{3}{8} = 0.375\)

This shows that a fraction and a decimal can name the same rational number.

Worked Example 3: A repeating decimal

Write \(\frac{2}{3}\) as division and describe the result.

Step 1: Rewrite as division.

Step 2: Divide.

The 6 repeats forever, so this is a repeating decimal.

Answer: \(\frac{2}{3} = 0.666\ldots\)

Even though the decimal does not end, it is still rational because it comes from a fraction.

Worked Example 4: Negative rational number

Write \(-\frac{5}{2}\) as division and find the value.

Step 1: Rewrite as division.

Step 2: Divide.

Answer: \(-\frac{5}{2} = -2.5\)

This is a rational number because it can be written as a fraction of two integers.

Connecting to the number line

Because fractions are division, they have exact locations on the number line.

For example:

• \(\frac{1}{2} = 0.5\), so it is halfway between 0 and 1.
• \(\frac{3}{2} = 1.5\), so it is halfway between 1 and 2.
• \(-\frac{1}{4} = -0.25\), so it is between 0 and \(-1\).

This helps us see that rational numbers fill many points on the number line, not just whole-number points.

Important rules to remember

• A fraction means division.
• \(\frac{a}{b} = a \div b\)
• The denominator cannot be 0.
• Rational numbers can be written as fractions, decimals, or integers.
• Decimals from rational numbers either end or repeat.

Common mistakes to avoid

• Mixing up numerator and denominator: \(\frac{2}{5}\) means \(2 \div 5\), not \(5 \div 2\).
• Using 0 in the denominator: \(\frac{4}{0}\) is not allowed.
• Thinking only ending decimals are rational: repeating decimals like \(0.333\ldots\) are also rational.

Quick check

1. What division problem does \(\frac{7}{4}\) represent?
2. What decimal do you get from \(\frac{1}{5}\)?
3. Is \(-3\) a rational number? Why?

Answers:

1. \(7 \div 4\)
2. \(0.2\)
3. Yes. It can be written as \(\frac{-3}{1}\).

Summary

A fraction is another way to write division. When you see \(\frac{a}{b}\), it means \(a \div b\), as long as \(b \ne 0\).

This idea helps us understand rational numbers in different forms, such as fractions, decimals, and integers. Rational numbers can be positive, negative, ending decimals, or repeating decimals, but they can all be written as a quotient of two integers.

Coordinate and Number Line Representations help us show where numbers belong and how far apart they are.

In 7th grade, you will work with positive numbers, negative numbers, fractions, and decimals. You will place them on a number line and on axes, which are the lines used in a coordinate plane.

This lesson will show you how to locate numbers, compare their positions, and find distance on both horizontal and vertical lines.

1. Understanding the number line

A number line is a straight line with numbers placed in order. The numbers increase as you move to the right and decrease as you move to the left.

The center point is usually 0. Numbers to the right of 0 are positive, and numbers to the left of 0 are negative.

For example:

$$-4 \quad -3 \quad -2 \quad -1 \quad 0 \quad 1 \quad 2 \quad 3 \quad 4$$

Important ideas to remember:

• A number farther right is always greater.
• A number farther left is always less.
• Negative numbers are less than 0.
• Positive numbers are greater than 0.

2. Plotting integers on a number line

Integers are whole numbers and their opposites, including 0.

Examples of integers are \(-5\), \(0\), and \(8\).

To plot an integer:

1. Find 0 on the number line.
2. Count equal spaces.
3. Move right for positive numbers.
4. Move left for negative numbers.

If you want to plot \(-3\), start at 0 and move 3 units left. If you want to plot \(4\), start at 0 and move 4 units right.

3. Plotting fractions and decimals

Fractions and decimals fit between integers. A number line is not only for whole numbers. It represents all points between the numbers too.

To place a fraction or decimal correctly, first look at the interval it belongs in.

For example:

• \(\frac{1}{2}\) is between 0 and 1.
• \(2.7\) is between 2 and 3.
• \(-1.5\) is between \(-1\) and \(-2\).

Notice that \(-1.5\) is halfway between \(-1\) and \(-2\). On the negative side, numbers become smaller as you move left.

Here are some helpful fraction-decimal matches:

• \(\frac{1}{2}=0.5\)
• \(\frac{1}{4}=0.25\)
• \(\frac{3}{4}=0.75\)
• \(\frac{1}{10}=0.1\)

4. Comparing numbers by location

You can compare numbers by looking at their positions on the number line.

If one number is to the right of another, it is greater. If it is to the left, it is less.

Examples:

• \(-2 < 1\) because \(-2\) is left of 1.
• \(0.6 > 0.4\) because 0.6 is right of 0.4.
• \(-3.2 < -3\) because \(-3.2\) is a little left of \(-3\).

5. Distance on a number line

Distance tells how far apart two numbers are. Distance is always positive because it measures space, not direction.

To find distance on a number line, count the units between the numbers.

Examples:

• The distance between \(2\) and \(5\) is 3.
• The distance between \(-4\) and \(-1\) is 3.
• The distance between \(-2\) and \(3\) is 5.

You can think of it as the number of units from one point to the other.

6. Horizontal and vertical axes

In addition to a number line, numbers can be shown on axes.

A horizontal axis goes left and right, just like a regular number line. A vertical axis goes up and down.

On both types of axes:

• Positive numbers are on the positive direction of the axis.
• Negative numbers are on the opposite side of 0.
• The spaces must be equal.

For a horizontal axis:

• Right is positive.
• Left is negative.

For a vertical axis:

• Up is positive.
• Down is negative.

7. The coordinate plane

A coordinate plane is made by crossing a horizontal axis and a vertical axis.

The horizontal axis is called the x-axis. The vertical axis is called the y-axis.

The point where they cross is called the origin, and its coordinates are \((0,0)\).

A point on the coordinate plane is written as an ordered pair:

$$ (x,y) $$

The first number tells how far to move left or right on the x-axis. The second number tells how far to move up or down on the y-axis.

For example, the point \((3,-2)\) means:

• Move 3 units right.
• Then move 2 units down.

8. How to plot a point on a coordinate plane

1. Start at the origin, \((0,0)\).
2. Look at the first number, the x-value.
3. Move right if it is positive or left if it is negative.
4. Look at the second number, the y-value.
5. Move up if it is positive or down if it is negative.
6. Mark the point.

Examples:

• \((2,4)\): right 2, up 4
• \((-3,1)\): left 3, up 1
• \((1.5,-2)\): right 1.5, down 2

9. Reading a point from a graph

To name a plotted point, first read the horizontal position, then the vertical position.

Always write coordinates in the order \((x,y)\), not \((y,x)\).

If a point is 4 units left and 2 units up from the origin, its coordinates are \((-4,2)\).

Worked Example 1: Plotting integers on a number line

Plot \(-5\), \(0\), and \(3\) on a number line.

Step 1: Find 0.

Step 2: For \(-5\), move 5 units left of 0.

Step 3: For \(0\), mark the center point.

Step 4: For \(3\), move 3 units right of 0.

Answer: The points appear in this order from left to right: \(-5\), \(0\), \(3\).

Worked Example 2: Plotting fractions and decimals

Place \(\frac{3}{2}\), \(0.25\), and \(-1.5\) on a number line.

Step 1: Rewrite what you can in a familiar form.

\(\frac{3}{2}=1.5\)

Step 2: Place each number.

• \(0.25\) is one-fourth of the way from 0 to 1.
• \(1.5\) is halfway between 1 and 2.
• \(-1.5\) is halfway between \(-1\) and \(-2\).

Answer: From left to right, the numbers are \(-1.5\), \(0.25\), and \(1.5\).

Worked Example 3: Finding distance on a number line

What is the distance between \(-3\) and \(4\)?

Step 1: Locate both numbers on the number line.

Step 2: Count the units from \(-3\) to 4.

From \(-3\) to 0 is 3 units. From 0 to 4 is 4 units.

Step 3: Add the distances.

$$3+4=7$$

Answer: The distance is \(7\) units.

Worked Example 4: Plotting points on axes

Plot the point \((-2,3)\) on a coordinate plane.

Step 1: Start at the origin, \((0,0)\).

Step 2: The x-value is \(-2\), so move 2 units left.

Step 3: The y-value is \(3\), so move 3 units up.

Answer: The point is 2 units left and 3 units up from the origin.

Common mistakes to avoid

• Mixing up left and right: Positive is right on a horizontal axis, negative is left.
• Mixing up up and down: Positive is up on a vertical axis, negative is down.
• Forgetting the order of coordinates: Always read and write \((x,y)\).
• Placing fractions unevenly: Make sure intervals are divided into equal parts.
• Thinking distance can be negative: Distance is always positive.

Quick check

1. Which number is farther right: \(-1\) or \(2\)?
2. Where is \(0.75\) located between 0 and 1?
3. What is the distance between \(-2\) and \(2\)?
4. How would you plot \((4,-1)\)?

Answers:

1. \(2\)
2. Three-fourths of the way from 0 to 1
3. \(4\) units
4. Move 4 units right and 1 unit down from the origin

Summary

A number line shows numbers in order from left to right. Negative numbers are left of 0, positive numbers are right of 0, and fractions and decimals fit between whole numbers.

Distance on a number line tells how far apart two numbers are and is always positive.

On a coordinate plane, points are written as \((x,y)\). The x-value tells left or right movement, and the y-value tells up or down movement.

When you understand number lines and axes, you can locate and compare all kinds of real numbers more easily.

Absolute Value and Distance

In math, numbers can show both how much and which direction. For example, on a number line, positive numbers are to the right of 0 and negative numbers are to the left of 0.

Sometimes, we only care about how far a number is from 0, not whether it is left or right. That idea is called absolute value.

The absolute value of a number is its distance from 0 on the number line. Because distance is never negative, absolute value is always 0 or positive.

We write absolute value using vertical bars. For example, the absolute value of 5 is written as \(|5|\), and the absolute value of negative 5 is written as \(|-5|\).

Here are some basic facts:

• \(|5| = 5\) because 5 is 5 units from 0.
• \(|-5| = 5\) because -5 is also 5 units from 0.
• \(|0| = 0\) because 0 is 0 units from 0.

You can think of absolute value as asking: “How far is this number from 0?”

On a number line, numbers that are the same distance from 0 but on opposite sides are called opposites. For example, 4 and -4 are opposites. They have the same absolute value:

$$|4| = 4 \qquad |-4| = 4$$

This shows that absolute value does not tell direction. It only tells distance.

Absolute Value and Distance Between Two Numbers

Absolute value can also help us find the distance between any two numbers on a number line.

To find the distance between two numbers, subtract them and take the absolute value:

$$\text{Distance between } a \text{ and } b = |a-b|$$

This works because distance must always be positive or 0.

For example, the distance between 2 and 7 is:

$$|2-7| = |-5| = 5$$

The distance between 7 and 2 is:

$$|7-2| = |5| = 5$$

Notice that the distance is the same either way, because distance does not depend on order.

Important Ideas to Remember

• Absolute value means distance from 0.
• Distance is never negative.
• A positive number and its opposite have the same absolute value.
• To find the distance between two numbers, use absolute value.

Worked Examples

Example 1: Find the absolute value of a positive number

Find \(|8|\).

8 is 8 units from 0, so:

$$|8| = 8$$

Example 2: Find the absolute value of a negative number

Find \(|-11|\).

-11 is 11 units from 0, so:

$$|-11| = 11$$

Even though the number is negative, its absolute value is positive because distance cannot be negative.

Example 3: Find the distance between two numbers

What is the distance between -3 and 5?

Use the distance formula:

$$|-3-5| = |-8| = 8$$

So the distance between -3 and 5 is 8 units.

You can also see this on the number line: from -3 to 0 is 3 units, and from 0 to 5 is 5 units. Altogether, that is 8 units.

Example 4: Find a missing number using absolute value

Solve \(|x| = 6\).

This means the distance from \(x\) to 0 is 6.

There are two numbers that are 6 units from 0:

• 6
• -6

So the solutions are:

$$x = 6 \text{ or } x = -6$$

A Common Mistake

Some students think a negative number stays negative inside absolute value bars. That is not correct.

For example:

$$|-9| = 9$$

The answer is not \(-9\), because absolute value is distance, and distance cannot be negative.

Try These Questions

1. Find \(|12|\).
2. Find \(|-7|\).
3. Find the distance between 4 and -2.
4. Solve \(|x| = 3\).

Answers

1. \(|12| = 12\)
2. \(|-7| = 7\)
3. $$|4-(-2)| = |6| = 6$$ so the distance is 6.
4. \(x = 3\) or \(x = -3\)

Summary

Absolute value tells how far a number is from 0 on the number line. It does not show direction, so the answer is never negative. You can also use absolute value to find the distance between any two numbers.

Rational Number Density is the idea that between any two rational numbers, there is always another rational number.

This means rational numbers are packed very closely together on the number line. No matter how near two rational numbers seem, you can always find one more rational number between them.

In this lesson, you will learn what rational number density means, how to find a rational number between two rational numbers, and why this is always possible.

First, remember: a rational number is any number that can be written as a fraction of two integers, where the denominator is not 0. Examples include \, \(\frac{1}{2}\), \(-3\), \(0.75\), and \(\frac{7}{4}\).

Integers and terminating or repeating decimals are all rational numbers because they can be written as fractions.

Main Idea: If you have two rational numbers, you can find a number between them by finding their midpoint.

The midpoint of two numbers is the number exactly halfway between them. To find it, add the two numbers and divide by 2:

If \(a\) and \(b\) are rational numbers, then \(a+b\) is rational, and dividing by 2 still gives a rational number. So the midpoint is also rational.

This shows that between any two rational numbers, there is at least one more rational number.

And it does not stop there. Once you find one rational number between them, you can repeat the process again and again. So there are actually infinitely many rational numbers between any two rational numbers.

Let us look at this on the number line.

Suppose you have \(1\) and \(2\). A number between them is \(1.5\), or \(\frac{3}{2}\).

But there are many others too, such as \(1.1\), \(1.25\), \(\frac{4}{3}\), and \(1.75\). All of these are rational and lie between 1 and 2.

This is what makes rational numbers dense on the number line.

Important Note: Dense does not mean every point on the number line is rational. It means that no matter which two rational numbers you choose, there is always another rational number between them.

Method 1: Find the midpoint

1. Add the two numbers.
2. Divide the sum by 2.
3. The result is a rational number between them.

Method 2: Rewrite the numbers with a common denominator

This method is helpful when the numbers are fractions.

If you can rewrite two fractions so there is space between their numerators, you may be able to choose a new fraction between them.

For example, between \(\frac{1}{3}\) and \(\frac{2}{3}\), you can rewrite them as:

Now \(\frac{3}{6}=\frac{1}{2}\) is between them.

Let us work through some examples.

Worked Example 1: Between two whole numbers

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

Use the midpoint formula:

So, \(7\) is a rational number between \(4\) and \(10\).

Check: \(4<7<10\), so it works.

Worked Example 2: Between two fractions

Find a rational number between \(\frac{1}{4}\) and \(\frac{3}{4}\).

Use the midpoint:

So, \(\frac{1}{2}\) is between \(\frac{1}{4}\) and \(\frac{3}{4}\).

Check: \(\frac{1}{4}<\frac{1}{2}<\frac{3}{4}\).

Worked Example 3: Between decimals

Find a rational number between \(2.3\) and \(2.5\).

Use the midpoint:

So, \(2.4\) is a rational number between \(2.3\) and \(2.5\).

Since terminating decimals are rational, \(2.4\) is rational.

Worked Example 4: Between negative rational numbers

Find a rational number between \(-\frac{5}{2}\) and \(-\frac{1}{2}\).

Use the midpoint:

So, \(-\frac{3}{2}\) is a rational number between \(-\frac{5}{2}\) and \(-\frac{1}{2}\).

Check in decimal form:

And indeed, \(-2.5<-1.5<-0.5\).

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

Suppose you start with two rational numbers, such as \(0\) and \(1\).

The midpoint is \(\frac{1}{2}\).

Now look between \(0\) and \(\frac{1}{2}\). The midpoint there is \(\frac{1}{4}\).

Now look between \(\frac{1}{2}\) and \(1\). The midpoint there is \(\frac{3}{4}\).

You can keep doing this forever:

This shows there is never just one number between two rational numbers. There are always more.

A useful shortcut: Sometimes you do not need the exact midpoint. You only need a rational number between the two given numbers.

For example, between \(\frac{2}{5}\) and \(\frac{4}{5}\), you might quickly notice that \(\frac{3}{5}\) is between them.

Between \(1.2\) and \(1.3\), you could choose \(1.25\).

As long as the number is rational and lies between the two numbers, it is correct.

Common Mistakes to Avoid

• Forgetting to divide by 2 when finding the midpoint.
• Choosing an endpoint instead of a number between them. The answer must be greater than the smaller number and less than the larger number.
• Mixing up negative numbers. On the number line, numbers farther left are smaller.
• Thinking there is only one answer. There are many possible rational numbers between two rational numbers.

Try these ideas on your own:

• Between \(3\) and \(4\), you could choose \(3.5\), \(\frac{13}{4}\), or \(3.1\).
• Between \(-1\) and \(0\), you could choose \(-\frac{1}{2}\) or \(-0.25\).
• Between \(\frac{2}{3}\) and \(\frac{5}{6}\), you could use the midpoint or rewrite with common denominators.

Summary

Rational number density means that between any two rational numbers, there is always another rational number.

A simple way to find one is to use the midpoint formula:

Because rational numbers stay rational when you add them and divide by 2, the midpoint will also be rational.

This can be repeated again and again, so there are infinitely many rational numbers between any two rational numbers on the number line.

Terminating and Repeating Decimals are two important kinds of decimal numbers. In this lesson, you will learn how to tell the difference between them, how they connect to fractions, and how to decide whether a fraction will have a terminating decimal or a repeating decimal.

This topic is part of the real number system. Both terminating and repeating decimals are rational numbers, which means they can be written as fractions.

Understanding this idea helps you move between fractions and decimals with confidence. It also helps you see patterns in the number system.

What is a terminating decimal?

A terminating decimal is a decimal that ends. It has a last digit.

• \(0.5\)

• \(1.25\)

• \(3.875\)

These decimals stop, so they are terminating decimals.

What is a repeating decimal?

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

• \(0.3333\ldots\) where the \(3\) repeats

• \(0.121212\ldots\) where the block \(12\) repeats

• \(2.1666\ldots\) where the \(6\) repeats

A repeating decimal does not end, but it follows a pattern.

Sometimes a bar is written over the repeating digits. For example:

• \(0.\overline{3} = 0.3333\ldots\)

• \(0.\overline{12} = 0.121212\ldots\)

• \(2.1\overline{6} = 2.1666\ldots\)

How are fractions connected to terminating and repeating decimals?

Every rational number can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\).

When you change a fraction into a decimal, one of two things happens:

• The decimal terminates, or

• The decimal repeats.

For example:

• \(\frac{1}{4} = 0.25\), which terminates

• \(\frac{1}{3} = 0.3333\ldots\), which repeats

How can you predict whether a fraction will terminate or repeat?

This is where prime factorization of the denominator helps.

First, write the fraction in simplest form. Then look at the denominator.

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

Why? Because our decimal system is based on powers of \(10\), and

So denominators made from only \(2\)'s and \(5\)'s can be turned into tenths, hundredths, thousandths, and so on.

Examples of denominators that give terminating decimals:

• \(2\)

• \(4 = 2 \times 2\)

• \(5\)

• \(8 = 2 \times 2 \times 2\)

• \(10 = 2 \times 5\)

• \(20 = 2 \times 2 \times 5\)

• \(25 = 5 \times 5\)

If the denominator in simplest form has any prime factor other than \(2\) or \(5\), then the decimal will repeat.

Examples of denominators that give repeating decimals:

• \(3\)

• \(6 = 2 \times 3\)

• \(7\)

• \(9 = 3 \times 3\)

• \(12 = 2 \times 2 \times 3\)

Even if a denominator has a \(2\) or a \(5\), the decimal will still repeat if there is also a different prime factor, like \(3\) or \(7\).

For example, \(\frac{1}{6}\) repeats because

The factor \(3\) causes the decimal to repeat.

Worked Example 1: Decide whether \(\frac{3}{8}\) terminates or repeats

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

\(\frac{3}{8}\) is already in simplest form.

Step 2: Prime factorize the denominator.

Step 3: Look at the prime factors.

The denominator has only \(2\)'s, so the decimal terminates.

Step 4: Convert to a decimal if needed.

So, \(\frac{3}{8}\) is a terminating decimal.

Worked Example 2: Decide whether \(\frac{5}{12}\) terminates or repeats

Step 1: Check simplest form.

\(\frac{5}{12}\) is in simplest form.

Step 2: Prime factorize the denominator.

Step 3: Look at the prime factors.

The denominator has a \(3\). Since it has a prime factor other than \(2\) or \(5\), the decimal repeats.

If you divide, you get

So, \(\frac{5}{12}\) is a repeating decimal.

Important reminder: Always simplify first

You must look at the denominator after the fraction is simplified.

For example, consider \(\frac{6}{15}\).

If you only look at \(15\), you might notice

That suggests the decimal repeats. But let us simplify first:

Now the denominator is \(5\), which has only the prime factor \(5\).

So the decimal actually terminates:

Worked Example 3: Simplify first, then decide for \(\frac{6}{15}\)

Step 1: Simplify.

Step 2: Factor the denominator.

Step 3: Decide.

Since the denominator has only \(5\), the decimal terminates.

Step 4: Write the decimal.

How to convert a repeating decimal into a fraction

Repeating decimals are rational numbers, so they can be written as fractions.

A helpful method is to use a variable and subtraction.

Worked Example 4: Convert \(0.\overline{3}\) to a fraction

Let

That means

Since one digit repeats, multiply both sides by \(10\).

Now subtract the original equation from the new equation.

Now divide both sides by \(9\).

So,

Another repeating decimal example

Convert \(0.\overline{12}\) to a fraction.

Let

Now two digits repeat, so multiply by \(100\).

Subtract the original equation.

Divide by \(99\).

Simplify the fraction.

So,

Quick pattern to notice

• If 1 digit repeats, the denominator is often a form of \(9\): \(9\), \(18\), \(27\), and so on before simplifying.

• If 2 digits repeat, the denominator is often a form of \(99\) before simplifying.

• If 3 digits repeat, the denominator is often a form of \(999\) before simplifying.

For example:

• \(0.\overline{7} = \frac{7}{9}\)

• \(0.\overline{45} = \frac{45}{99} = \frac{5}{11}\)

Terminating decimals can also be written as fractions

This is usually easier. Use place value.

For example,

Another example:

How this fits into the real number system

Terminating decimals and repeating decimals are both rational numbers.

They can all be placed on the number line, and they can all be written as fractions.

This is different from irrational numbers, which have decimals that do not end and do not repeat.

For example, numbers like \(\pi\) and \(\sqrt{2}\) are irrational.

So remember:

• Terminating decimal: ends

• Repeating decimal: has a repeating pattern forever

• Both are rational numbers

Common mistakes to avoid

• Do not forget to simplify first. The denominator in simplest form is what matters.

• Do not think every long decimal repeats. A decimal is repeating only if there is a clear repeating pattern.

• Do not confuse repeating with terminating. A terminating decimal stops. A repeating decimal continues forever.

• When converting repeating decimals to fractions, multiply by the correct power of 10. Use \(10\) for 1 repeating digit, \(100\) for 2 repeating digits, and so on.

Summary

A decimal can either terminate or repeat if it comes from a rational number. To predict which one a fraction will have, simplify the fraction and prime factorize the denominator.

If the denominator has only \(2\) and/or \(5\) as prime factors, the decimal terminates. If it has any other prime factor, the decimal repeats.

You also learned that repeating decimals can be changed into fractions by using a variable, multiplying by a power of \(10\), and subtracting.

When you understand these patterns, fractions and decimals become much easier to connect.

Irrational Number Intuition

In math, all the numbers you usually work with belong to a big family called the real numbers. Real numbers include whole numbers, negative numbers, fractions, decimals, and some special numbers that cannot be written as simple fractions.

In this lesson, you will learn what irrational numbers are, how they are different from rational numbers, and how to estimate where irrational numbers belong on a number line.

1. Rational vs. Irrational Numbers

A rational number is any number that can be written as a fraction of two integers, like \(\frac{1}{2}\), \(\frac{3}{4}\), \(-5\), or \(2\).

Rational numbers include:

• Integers, such as \(-3\), \(0\), and \(7\)
• Fractions, such as \(\frac{2}{3}\) and \(\frac{9}{5}\)
• Decimals that stop, such as \(0.6\) and \(4.25\)
• Decimals that repeat, such as \(0.333\ldots\) and \(1.272727\ldots\)

An irrational number is a number that cannot be written as a fraction of two integers. Its decimal form goes on forever without repeating in a pattern.

Examples of irrational numbers include:

• \(\pi \approx 3.14\)
• \(\sqrt{2} \approx 1.41\)
• \(\sqrt{3} \approx 1.73\)
• \(\sqrt{5} \approx 2.24\)

2. Why Some Square Roots Are Irrational

Some square roots are easy because they come from perfect squares.

For example:

• \(\sqrt{1} = 1\)
• \(\sqrt{4} = 2\)
• \(\sqrt{9} = 3\)
• \(\sqrt{16} = 4\)

These are rational numbers because they are whole numbers.

But what about \(\sqrt{2}\)? There is no whole number that squares to make 2. Also, \(\sqrt{2}\) cannot be written exactly as a fraction. Its decimal keeps going forever without repeating:

$$\sqrt{2} \approx 1.41421356\ldots$$

That means \(\sqrt{2}\) is irrational.

In general, the square root of a number that is not a perfect square is irrational. For 7th grade, this is a very useful rule.

3. Understanding Decimal Clues

You can often tell whether a number is rational or irrational by looking at its decimal form.

• If the decimal ends, it is rational.
• If the decimal repeats, it is rational.
• If the decimal goes on forever and does not repeat, it is irrational.

Examples:

• \(0.75\) is rational because it ends.
• \(0.4444\ldots\) is rational because it repeats.
• \(3.14159265\ldots\) is irrational because it does not end and does not repeat.

4. Estimating Irrational Numbers

Even though irrational numbers cannot be written exactly as simple fractions or neat decimals, we can still estimate them.

To estimate a square root, find the two perfect squares it is between.

For example, to estimate \(\sqrt{10}\):

• \(9 < 10 < 16\)
• So, \(\sqrt{9} < \sqrt{10} < \sqrt{16}\)
• That means \(3 < \sqrt{10} < 4\)

So \(\sqrt{10}\) is between 3 and 4.

Since 10 is closer to 9 than to 16, \(\sqrt{10}\) is closer to 3 than to 4. A good estimate is:

$$\sqrt{10} \approx 3.16$$

5. Placing Irrational Numbers on a Number Line

To place an irrational number on a number line, first figure out which two whole numbers it lies between. Then estimate its decimal value.

For example, \(\sqrt{7}\):

• \(4 < 7 < 9\)
• So, \(2 < \sqrt{7} < 3\)

This tells us \(\sqrt{7}\) belongs between 2 and 3 on the number line.

Since \(\sqrt{7} \approx 2.65\), it should be placed a little past 2.6.

For \(\pi\), we use its common estimate:

$$\pi \approx 3.14$$

So on a number line, \(\pi\) is between 3 and 4, just a little to the right of 3.1.

Worked Example 1: Is the number rational or irrational?

Classify \(0.125\).

Step 1: Check the decimal.

The decimal ends.

Step 2: Decide the type of number.

A decimal that ends is rational.

Answer: \(0.125\) is rational.

Worked Example 2: Is the square root rational or irrational?

Classify \(\sqrt{36}\) and \(\sqrt{11}\).

For \(\sqrt{36}\):

Since 36 is a perfect square,

$$\sqrt{36} = 6$$

So \(\sqrt{36}\) is rational.

For \(\sqrt{11}\):

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

Answer:

• \(\sqrt{36}\) is rational
• \(\sqrt{11}\) is irrational

Worked Example 3: Estimate an irrational number

Estimate \(\sqrt{15}\) to decide where it belongs on the number line.

Step 1: Find nearby perfect squares.

$$9 < 15 < 16$$

Step 2: Take square roots.

$$3 < \sqrt{15} < 4$$

Step 3: Decide where it is between 3 and 4.

Since 15 is very close to 16, \(\sqrt{15}\) is very close to 4.

A good estimate is:

$$\sqrt{15} \approx 3.87$$

Answer: \(\sqrt{15}\) is irrational and should be placed between 3 and 4, close to 4.

Worked Example 4: Compare and place numbers

Order these numbers from least to greatest: \(3\), \(\sqrt{8}\), \(\pi\), \(3.2\).

Step 1: Estimate the irrational numbers.

• \(\sqrt{8} \approx 2.83\)
• \(\pi \approx 3.14\)

Step 2: Compare all values.

• \(\sqrt{8} \approx 2.83\)
• \(3 = 3.00\)
• \(\pi \approx 3.14\)
• \(3.2 = 3.20\)

Answer:

$$\sqrt{8} < 3 < \pi < 3.2$$

6. Quick Tips to Remember

• Rational numbers can be written as fractions.
• Irrational numbers cannot be written as fractions.
• Decimals that end or repeat are rational.
• Decimals that never end and never repeat are irrational.
• The square root of a perfect square is rational.
• The square root of a number that is not a perfect square is irrational.

7. Common Mistakes to Avoid

• Do not assume every decimal is irrational. Many decimals are rational.
• Do not think an irrational number cannot be placed on a number line. It can be placed by estimating.
• Do not confuse \(\sqrt{9}\) with \(\sqrt{10}\). \(\sqrt{9}=3\) is rational, but \(\sqrt{10}\) is irrational.
• Do not use \(3.14\) as the exact value of \(\pi\). It is only an estimate.

Summary

Irrational numbers are real numbers that cannot be written as fractions, and their decimals go on forever without repeating. Common examples are \(\pi\) and square roots of numbers that are not perfect squares, like \(\sqrt{2}\) and \(\sqrt{7}\). You can still work with irrational numbers by estimating their values and placing them on a number line between nearby whole numbers.

Magnitude and Scientific Notation Foundations

Sometimes numbers in math and science are very large, like the number of stars in a galaxy. Other times they are very small, like the size of a tiny particle. Writing all the zeros in these numbers can be hard to read and easy to mess up.

That is why we use powers of ten and scientific notation. These tools help us write very large and very small numbers in a shorter, clearer way.

In this lesson, you will learn how to:

• understand magnitude, or how big or small a number is,
• use powers of ten to describe place value,
• write numbers in scientific notation,
• turn scientific notation back into standard form,
• compare numbers by their size.

1. Understanding Magnitude

Magnitude means the size of a number. A number with greater magnitude is farther from zero on the number line.

For this lesson, we will focus on positive numbers, so magnitude is the same as asking, “Which number is bigger?”

For example:

• \(1000\) has greater magnitude than \(100\).
• \(0.001\) has smaller magnitude than \(0.1\).

When numbers get extremely large or extremely small, it helps to notice how many places the decimal point would move. This is connected to powers of ten.

2. Powers of Ten and Place Value

Our number system is based on ten. Each place value is 10 times greater than the place to its right.

Here are some powers of ten:

• \(10^0 = 1\)
• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^3 = 1000\)
• \(10^4 = 10000\)

The exponent tells how many times to multiply by 10.

For example:

Powers of ten also help describe small decimals:

• \(10^{-1} = 0.1\)
• \(10^{-2} = 0.01\)
• \(10^{-3} = 0.001\)
• \(10^{-4} = 0.0001\)

A negative exponent means the number is less than 1. Each step to the right of the decimal point is another power of ten smaller.

For example:

3. What Is Scientific Notation?

Scientific notation is a short way to write numbers using a number between 1 and 10 multiplied by a power of ten.

The form looks like this:

where:

• \(a\) is at least 1 but less than 10,
• \(n\) is an integer exponent.

Examples of scientific notation:

• \(3.2 \times 10^4\)
• \(7.5 \times 10^{-3}\)
• \(1.09 \times 10^6\)

Not scientific notation:

• \(12.4 \times 10^3\) because \(12.4\) is not between 1 and 10
• \(0.6 \times 10^5\) because \(0.6\) is less than 1

4. Writing Large Numbers in Scientific Notation

To write a large number in scientific notation:

1. Move the decimal point so the number becomes a value between 1 and 10.
2. Count how many places you moved the decimal.
3. Use that count as a positive exponent on 10.

Why is the exponent positive? Because large numbers are made by multiplying by powers of ten.

Worked Example 1

Write \(45{,}000\) in scientific notation.

Step 1: Move the decimal so the first number is between 1 and 10.

\(45{,}000 \rightarrow 4.5\)

Step 2: Count the moves. The decimal moved 4 places to the left.

Step 3: Write the answer.

5. Writing Small Numbers in Scientific Notation

To write a very small decimal in scientific notation:

1. Move the decimal point so the number becomes a value between 1 and 10.
2. Count how many places you moved the decimal.
3. Use that count as a negative exponent on 10.

Why is the exponent negative? Because the original number is less than 1.

Worked Example 2

Write \(0.00072\) in scientific notation.

Step 1: Move the decimal to make a number between 1 and 10.

\(0.00072 \rightarrow 7.2\)

Step 2: Count the moves. The decimal moved 4 places to the right.

Step 3: Write the answer using a negative exponent.

6. Changing Scientific Notation Back to Standard Form

To turn scientific notation back into a regular number, use the exponent to decide which way to move the decimal point:

• If the exponent is positive, move the decimal to the right.
• If the exponent is negative, move the decimal to the left.

Worked Example 3

Write \(6.3 \times 10^5\) in standard form.

The exponent is 5, so move the decimal 5 places to the right.

\(6.3 \rightarrow 630000\)

So,

Worked Example 4

Write \(8.91 \times 10^{-3}\) in standard form.

The exponent is \(-3\), so move the decimal 3 places to the left.

\(8.91 \rightarrow 0.00891\)

So,

7. Comparing Magnitudes with Powers of Ten

Scientific notation also makes it easier to compare very large or very small numbers.

First compare the exponents:

• For large numbers, the greater positive exponent usually means the greater number.
• For small numbers with negative exponents, the exponent closer to zero usually means the greater number.

Examples:

• \(3 \times 10^6\) is greater than \(3 \times 10^5\) because \(10^6\) is 10 times larger than \(10^5\).
• \(4 \times 10^{-2}\) is greater than \(4 \times 10^{-5}\) because \(0.04\) is greater than \(0.00004\).

If the exponents are the same, compare the first numbers.

Example:

• \(7.1 \times 10^4\) is greater than \(2.9 \times 10^4\) because \(7.1 > 2.9\).

8. Helpful Patterns to Remember

• Moving the decimal left makes the number smaller in appearance, but it represents a large original number when paired with a positive exponent.
• Moving the decimal right gives a number between 1 and 10 for tiny decimals, and this uses a negative exponent.
• Positive exponents mean a number greater than or equal to 10.
• Negative exponents mean a number between 0 and 1.
• In scientific notation, the first factor must always be at least 1 and less than 10.

9. Common Mistakes to Avoid

• Forgetting the sign of the exponent: Large numbers use positive exponents, and small decimals use negative exponents.
• Not making the first number between 1 and 10: \(23 \times 10^4\) is not correct scientific notation.
• Moving the decimal the wrong direction: When changing from scientific notation to standard form, positive means right and negative means left.
• Miscounting decimal places: Count carefully one place at a time.

10. Quick Check Ideas

You can ask yourself these questions:

• Is my first number between 1 and 10?
• Does the exponent match whether the number is large or small?
• If I change it back to standard form, do I get the original number?

Summary

Magnitude tells how large or small a number is. Powers of ten help show how numbers grow larger or smaller in our base-ten system.

Scientific notation writes numbers as $$a \times 10^n$$ where \(a\) is at least 1 and less than 10. Large numbers use positive exponents, and small numbers use negative exponents.

By learning to move the decimal point correctly and count places carefully, you can read, write, and compare very large and very small numbers with confidence.