The Real Number System and Approximations

Rational Numbers and Decimal Expansions

In this lesson, you will learn how rational numbers are connected to decimals. You will see why every rational number has a decimal that either terminates (ends) or repeats in a pattern forever.

You will also learn how to turn repeating decimals back into fractions. This is an important skill because it helps show the connection between fractions, decimals, and the real number system.

1. What is a rational number?

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

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

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

Examples of rational numbers include:

• \(\frac{3}{4}\)
• \(-2 = \frac{-2}{1}\)
• \(0.6 = \frac{3}{5}\)
• \(0.272727\ldots\)

This means rational numbers include many fractions, whole numbers, integers, terminating decimals, and repeating decimals.

2. Types of decimal expansions

A decimal expansion is the decimal form of a number. Rational numbers can have two kinds of decimal expansions:

• Terminating decimals: the digits end.
• Repeating decimals: one digit or a group of digits repeats forever.

Examples of terminating decimals:

• \(\frac{1}{2} = 0.5\)
• \(\frac{3}{4} = 0.75\)
• \(\frac{7}{20} = 0.35\)

Examples of repeating decimals:

• \(\frac{1}{3} = 0.3333\ldots = 0.\overline{3}\)
• \(\frac{2}{11} = 0.181818\ldots = 0.\overline{18}\)
• \(\frac{5}{6} = 0.83333\ldots = 0.8\overline{3}\)

The bar over the digits shows the repeating part. For example, \(0.\overline{27}\) means \(0.27272727\ldots\)

3. Why do rational numbers terminate or repeat?

When you change a fraction into a decimal, you are really dividing the numerator by the denominator. In long division, there are only a limited number of possible remainders.

For example, if you divide by \(7\), the remainder must be one of:

$$0,1,2,3,4,5,6$$

There cannot be any other remainder.

Two things can happen during division:

• If the remainder becomes \(0\), the decimal stops. So the decimal terminates.
• If a remainder repeats, the same division steps will repeat again and again. So the decimal repeats.

Since there are only finitely many possible remainders, one of these two things must happen. That is why every rational number has a decimal expansion that either terminates or repeats.

4. When does a decimal terminate?

A fraction in simplest form has a terminating decimal only when the denominator has no prime factors other than \(2\) and/or \(5\).

This is because decimals are based on powers of 10, and

$$10 = 2 \times 5$$

So denominators that fit into a power of 10 will produce terminating decimals.

Examples:

• \(\frac{3}{8}\), since \(8 = 2^3\), terminates.
• \(\frac{7}{20}\), since \(20 = 2^2 \times 5\), terminates.
• \(\frac{2}{3}\), since \(3\) is not a factor of 10, repeats.
• \(\frac{5}{12}\), since \(12 = 2^2 \times 3\), repeats because of the factor \(3\).

Worked Example 1: Does the decimal terminate or repeat?

Decide whether \(\frac{9}{40}\) has a terminating or repeating decimal.

Step 1: Factor the denominator.

$$40 = 2^3 \times 5$$

Step 2: Check the prime factors.

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

Step 3: Find the decimal.

$$\frac{9}{40} = 0.225$$

So \(\frac{9}{40}\) is a rational number with a terminating decimal expansion.

5. Repeating decimals are also rational numbers

It may seem surprising, but a repeating decimal is always rational. That means it can always be written as a fraction.

We can prove this using algebra.

Worked Example 2: Convert a simple repeating decimal to a fraction

Write \(0.\overline{4}\) as a fraction.

Step 1: Let

$$x = 0.\overline{4}$$

Step 2: Multiply by 10 so one repeating digit moves left of the decimal point.

$$10x = 4.\overline{4}$$

Step 3: Subtract the original equation.

$$10x - x = 4.\overline{4} - 0.\overline{4}$$ $$9x = 4$$

Step 4: Solve for \(x\).

$$x = \frac{4}{9}$$

So,

$$0.\overline{4} = \frac{4}{9}$$

6. Converting repeating decimals with more than one repeating digit

If two digits repeat, multiply by \(100\). If three digits repeat, multiply by \(1000\), and so on. The idea is to line up the repeating parts so they cancel when you subtract.

Worked Example 3: Convert a repeating decimal with two repeating digits

Write \(0.\overline{27}\) as a fraction.

Step 1: Let

$$x = 0.\overline{27}$$

Step 2: Multiply by 100 because 2 digits repeat.

$$100x = 27.\overline{27}$$

Step 3: Subtract the original equation.

$$100x - x = 27.\overline{27} - 0.\overline{27}$$ $$99x = 27$$

Step 4: Solve.

$$x = \frac{27}{99}$$

Step 5: Simplify.

$$\frac{27}{99} = \frac{3}{11}$$

So,

$$0.\overline{27} = \frac{3}{11}$$

7. Repeating decimals that start after some non-repeating digits

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

For example, in \(0.1\overline{6}\), the \(1\) does not repeat, but the \(6\) repeats forever:

$$0.16666\ldots$$

These can also be changed into fractions using algebra.

Worked Example 4: Convert a mixed repeating decimal to a fraction

Write \(0.1\overline{6}\) as a fraction.

Step 1: Let

$$x = 0.1\overline{6}$$

Step 2: Move the decimal so the repeating part starts right after the decimal point.

$$10x = 1.\overline{6}$$

Step 3: Now multiply again so one repeating digit moves left of the decimal point.

$$100x = 16.\overline{6}$$

Step 4: Subtract the smaller shifted equation from the larger one.

$$100x - 10x = 16.\overline{6} - 1.\overline{6}$$ $$90x = 15$$

Step 5: Solve.

$$x = \frac{15}{90} = \frac{1}{6}$$

So,

$$0.1\overline{6} = \frac{1}{6}$$

8. A useful pattern to notice

Here are some common repeating decimals and fractions:

• \(0.\overline{1} = \frac{1}{9}\)
• \(0.\overline{2} = \frac{2}{9}\)
• \(0.\overline{3} = \frac{1}{3}\)
• \(0.\overline{7} = \frac{7}{9}\)
• \(0.\overline{09} = \frac{1}{11}\)
• \(0.\overline{18} = \frac{2}{11}\)

These patterns can help you check your work, but it is still important to know the algebraic method.

9. Important ideas to remember

• A rational number can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\).
• Every rational number has a decimal expansion that either terminates or repeats.
• If a fraction in simplest form has a denominator with only factors of \(2\) and/or \(5\), the decimal terminates.
• If the denominator has any other prime factor, the decimal repeats.
• Every repeating decimal can be written as a fraction, so every repeating decimal is rational.

10. Quick check questions

1. Is \(\frac{7}{25}\) terminating or repeating?
2. Is \(\frac{4}{9}\) terminating or repeating?
3. Write \(0.\overline{5}\) as a fraction.
4. Write \(0.\overline{12}\) as a fraction.

Answers:

1. Terminating, because \(25 = 5^2\).
2. Repeating, because \(9 = 3^2\).
3. \(\frac{5}{9}\)
4. \(\frac{12}{99} = \frac{4}{33}\)

Summary

Rational numbers are numbers that can be written as fractions. When written as decimals, they always either end or repeat. Terminating decimals come from fractions whose simplified denominators have only factors of \(2\) and \(5\), and repeating decimals can always be converted back into fractions using algebra.

Lesson: Irrational Numbers

In math, all the numbers we use on the number line are part of the real number system. Two important groups inside the real numbers are rational numbers and irrational numbers.

This lesson focuses on irrational numbers. By the end, you should be able to tell what an irrational number is, recognize common examples, and understand how irrational numbers look as decimals.

What is an irrational number?

An irrational number is a number that cannot be written as a fraction of two integers. In other words, it cannot be written in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\).

Irrational numbers have decimal forms that go on forever without ending and without repeating in a pattern.

For example, the decimal for \(\pi\) begins like this:

\(\pi = 3.1415926535\ldots\)

The digits continue forever, and there is no repeating block of digits.

How are irrational numbers different from rational numbers?

A rational number can be written as a fraction. Its decimal form will either:

• terminate, which means it ends, such as \(0.75\) or \(2.4\), or
• repeat, which means a digit or group of digits repeats forever, such as \(0.333\ldots\) or \(1.272727\ldots\)

An irrational number does neither. Its decimal form does not end and does not repeat.

Here is a quick comparison:

• Rational: \(\frac{1}{2} = 0.5\)
• Rational: \(\frac{1}{3} = 0.333\ldots\)
• Irrational: \(\sqrt{2} = 1.4142135\ldots\)
• Irrational: \(\pi = 3.1415926\ldots\)

Common examples of irrational numbers

Some irrational numbers appear often in math. You should especially know these:

• \(\pi\)
• \(\sqrt{2}\)
• \(\sqrt{3}\)
• \(\sqrt{5}\)
• Any square root of a number that is not a perfect square

Remember that a perfect square is a number like \(1, 4, 9, 16, 25\), because each is the square of an integer:

$$ 1 = 1^2,\quad 4 = 2^2,\quad 9 = 3^2,\quad 16 = 4^2,\quad 25 = 5^2 $$

If you take the square root of a perfect square, you get an integer, which is rational.

For example:

• \(\sqrt{9} = 3\), which is rational
• \(\sqrt{16} = 4\), which is rational

But if the number is not a perfect square, its square root is irrational.

For example:

• \(\sqrt{2}\) is irrational
• \(\sqrt{7}\) is irrational
• \(\sqrt{10}\) is irrational

How to recognize irrational numbers

Here are some good questions to ask yourself:

1. Can the number be written as a fraction of integers?
2. Does the decimal terminate?
3. Does the decimal repeat?
4. Is it the square root of a non-perfect square?

If the decimal goes on forever and does not repeat, the number is irrational.

If it is the square root of a number like \(2, 3, 6, 7, 8\), or any number that is not a perfect square, it is irrational.

Worked Example 1: Decide whether a decimal is irrational

Is \(0.125\) irrational?

Step 1: Check whether the decimal ends.

Yes, \(0.125\) terminates.

Step 2: Decide what that means.

A terminating decimal is rational, so \(0.125\) is not irrational.

Answer: \(0.125\) is rational.

Worked Example 2: Repeating versus irrational

Is \(0.27272727\ldots\) irrational?

Step 1: Look for a repeating pattern.

The digits \(27\) repeat over and over.

Step 2: Use the rule.

A repeating decimal is rational, not irrational.

Answer: \(0.27272727\ldots\) is rational.

Worked Example 3: Square roots

Is \(\sqrt{18}\) irrational?

Step 1: Ask whether \(18\) is a perfect square.

No. \(16\) and \(25\) are perfect squares, but \(18\) is not.

Step 2: Use the rule.

The square root of a non-perfect square is irrational.

Answer: \(\sqrt{18}\) is irrational.

Worked Example 4: Approximate an irrational number

Between which two integers does \(\sqrt{11}\) lie?

Step 1: Find perfect squares around \(11\).

$$ 9 < 11 < 16 $$

Step 2: Take square roots.

$$ \sqrt{9} < \sqrt{11} < \sqrt{16} $$ $$ 3 < \sqrt{11} < 4 $$

So \(\sqrt{11}\) is between \(3\) and \(4\).

Its decimal approximation is about \(3.32\).

Answer: \(\sqrt{11}\) is irrational and lies between \(3\) and \(4\).

Important idea: Approximations

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

For example:

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

These are not exact values. They are close values that help us work with irrational numbers.

Common mistakes to avoid

• Mistake 1: Thinking all decimals are irrational. Some decimals terminate or repeat, so they are rational.
• Mistake 2: Thinking every square root is irrational. Square roots of perfect squares are rational.
• Mistake 3: Thinking a long decimal is automatically irrational. A decimal can be long and still terminate or repeat.

Quick practice ideas

Decide whether each number is rational or irrational:

• \(0.4\)
• \(0.666\ldots\)
• \(\sqrt{49}\)
• \(\sqrt{6}\)
• \(\pi\)

The answers are:

• \(0.4\): rational
• \(0.666\ldots\): rational
• \(\sqrt{49} = 7\): rational
• \(\sqrt{6}\): irrational
• \(\pi\): irrational

Summary

Irrational numbers are real numbers that cannot be written as fractions of integers. Their decimal forms go on forever without terminating or repeating.

Common examples include \(\pi\) and square roots of non-perfect squares, such as \(\sqrt{2}\) and \(\sqrt{11}\). Learning to tell the difference between rational and irrational numbers helps you understand the real number system more clearly.

Lesson: The Real Number Continuum

Have you ever noticed that a number line looks completely filled in, with no gaps? That idea is called the real number continuum. It means that every point on the number line matches exactly one real number, and every real number has a place on the number line.

To understand this, we need to learn how real numbers are grouped and how they fit together. In this lesson, you will classify numbers into sets, compare rational and irrational numbers, and see why the number line is continuous.

1. What are real numbers?

Real numbers are all the numbers that can be placed on a number line. They include numbers less than zero, zero itself, and numbers greater than zero.

Some examples of real numbers are:

• , 7, 0, 4, 19
• \(\frac{1}{2}\), \(-\frac{3}{4}\), \(2.5\)
• \(\sqrt{9}=3\), \(\sqrt{2}\), \(\pi\)

All of these belong somewhere on the number line, so they are all real numbers.

2. Subsets of the real number system

The real number system is made of smaller groups, or subsets. You can think of these sets as nesting inside one another.

• Natural numbers: counting numbers such as \(1,2,3,4,\dots\)
• Whole numbers: natural numbers and zero, so \(0,1,2,3,\dots\)
• Integers: whole numbers and their opposites, so \(\dots,-3,-2,-1,0,1,2,3,\dots\)
• Rational numbers: numbers that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\neq 0\)
• Irrational numbers: numbers that cannot be written as a fraction of integers

Together, rational numbers and irrational numbers make up the set of real numbers.

We can show the relationship like this:

Natural numbers Whole numbers Integers Rational numbers Real numbers

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

3. Rational numbers

A number is rational if it can be written as a fraction.

Examples:

• \(5=\frac{5}{1}\)
• \(-2=\frac{-2}{1}\)
• \(0.75=\frac{3}{4}\)
• \(0.333\dots=\frac{1}{3}\)

Rational numbers have decimal forms that either:

• terminate (end), like \(0.4\) or \(2.75\)
• repeat, like \(0.121212\dots\) or \(0.333\dots\)

4. Irrational numbers

A number is irrational if it cannot be written as a fraction of integers.

Its decimal form goes on forever without repeating in a pattern.

Examples:

• \(\sqrt{2}=1.4142135\dots\)
• \(\sqrt{3}=1.7320508\dots\)
• \(\pi=3.1415926\dots\)

Some square roots are rational, but only when the number inside the root is a perfect square. For example:

• \(\sqrt{16}=4\), which is rational
• \(\sqrt{25}=5\), which is rational
• \(\sqrt{10}\), which is irrational

5. Why is the number line called a continuum?

The word continuum means something that continues without gaps. The real number line is a continuum because between any two real numbers, there are always more real numbers.

For example, between \(1\) and \(2\), we can find:

• \(1.5\)
• \(1.1\)
• \(1.01\)
• \(\sqrt{2}\approx 1.414\)

Even if you choose two numbers that are very close together, there is always another real number between them.

For instance, between \(0.4\) and \(0.5\), the number \(0.45\) is between them. Between \(0.45\) and \(0.46\), the number \(0.455\) is between them. This can continue forever.

That is why every point on the number line has a real number, and there are no empty spots between real numbers.

6. Classifying numbers in the real number system

When classifying a number, place it in the smallest set it belongs to.

For example:

• \(4\) is a natural number. It is also whole, integer, rational, and real.
• \(0\) is a whole number. It is also an integer, rational, and real.
• \(-7\) is an integer. It is also rational and real.
• \(\frac{2}{3}\) is a rational number. It is also real.
• \(\sqrt{5}\) is an irrational number. It is also real.

7. Worked Examples

Example 1: Classify \(8\)

Step 1: Ask whether \(8\) is a counting number. Yes.

So the smallest set it belongs to is natural numbers.

It is also a whole number, integer, rational number, and real number.

Answer: \(8\) is a natural number.

Example 2: Classify \(-3\)

Step 1: Is it a natural number? No, because it is negative.

Step 2: Is it a whole number? No, because whole numbers are \(0\) and positive numbers.

Step 3: Is it an integer? Yes.

Since \(-3\) is an integer, it is also rational because \(-3=\frac{-3}{1}\).

Answer: \(-3\) is an integer.

Example 3: Classify \(0.625\)

Step 1: Does the decimal end? Yes.

A terminating decimal is rational.

We can write it as a fraction:

$$0.625=\frac{625}{1000}=\frac{5}{8}$$

So it is a rational number.

Answer: \(0.625\) is rational.

Example 4: Classify \(\sqrt{7}\)

Step 1: Ask whether \(7\) is a perfect square. It is not.

So \(\sqrt{7}\) cannot be written as a whole number or integer.

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

Answer: \(\sqrt{7}\) is an irrational number.

8. Real numbers and points on a number line

Every real number has exactly one location on the number line.

• Positive numbers are to the right of \(0\)
• Negative numbers are to the left of \(0\)
• Fractions and decimals fit between integers
• Irrational numbers also fit on the line, even if their decimal form never ends

For example, \(\sqrt{2}\) is between \(1\) and \(2\) because:

$$1^2=1 \quad \text{and} \quad 2^2=4$$

Since \(2\) is between \(1\) and \(4\), \(\sqrt{2}\) must be between \(1\) and \(2\).

This shows that irrational numbers are just as real as integers or fractions. They also have exact places on the number line.

9. Important ideas to remember

• Real numbers are all numbers on the number line.
• Rational numbers can be written as fractions and have decimals that end or repeat.
• Irrational numbers cannot be written as fractions and have decimals that do not end or repeat.
• The number line is a continuum, meaning it has no gaps.
• Between any two real numbers, there is always another real number.

Brief Summary

The real number continuum means that the number line is completely filled with real numbers. Real numbers include rational numbers, like fractions and terminating or repeating decimals, and irrational numbers, like \(\sqrt{2}\) and \(\pi\). By learning the subsets of the real number system, you can classify numbers correctly and understand that every point on the number line represents a real number.

Square Roots and Cube Roots

In this lesson, you will learn what square roots and cube roots mean, how to find them for numbers that are perfect squares or perfect cubes, and how to solve simple equations like \(x^2=p\) and \(x^3=p\).

Roots are the opposite of powers. For example, squaring means multiplying a number by itself, and cubing means multiplying a number by itself twice. A root asks, “What number was multiplied to make this result?”

Understanding roots is an important part of the real number system. Some roots are whole numbers, and some are not. In 8th grade, you should be able to recognize and evaluate roots of perfect squares and perfect cubes.

1. What is a square root?

A square root of a number is a value that, when multiplied by itself, gives that number.

For example, since

$$4^2=16$$

we say that \(4\) is a square root of \(16\).

Because

$$(-4)^2=16$$

\(-4\) is also a square root of \(16\).

So the number \(16\) has two square roots: \(4\) and \(-4\).

When you see the radical sign \(\sqrt{\phantom{x}}\), it means the principal square root, which is the nonnegative square root.

So, even though both \(4\) and \(-4\) are square roots of \(16\), we write

$$\sqrt{16}=4$$

Common perfect squares are numbers made by squaring whole numbers:

• \(1^2=1\)
• \(2^2=4\)
• \(3^2=9\)
• \(4^2=16\)
• \(5^2=25\)
• \(6^2=36\)
• \(7^2=49\)
• \(8^2=64\)
• \(9^2=81\)
• \(10^2=100\)

If a number is a perfect square, its square root is a whole number.

2. What is a cube root?

A cube root of a number is a value that, when multiplied by itself three times, gives that number.

For example, since

$$3^3=27$$

we say that \(3\) is the cube root of \(27\).

We write this as

$$\sqrt[3]{27}=3$$

Cube roots are different from square roots in an important way. A positive number has two square roots, but a number has only one real cube root.

For example:

• \(2^3=8\), so \(\sqrt[3]{8}=2\)
• \((-2)^3=-8\), so \(\sqrt[3]{-8}=-2\)

Common perfect cubes are:

• \(1^3=1\)
• \(2^3=8\)
• \(3^3=27\)
• \(4^3=64\)
• \(5^3=125\)
• \((-1)^3=-1\)
• \((-2)^3=-8\)
• \((-3)^3=-27\)
• \((-4)^3=-64\)

3. Square roots and cube roots are opposites of exponents

Square roots undo squaring, and cube roots undo cubing.

For square roots:

$$\sqrt{a^2}=|a|$$

This means the result is the distance from 0, so it is never negative.

For cube roots:

$$\sqrt[3]{a^3}=a$$

Cube roots can be negative, positive, or zero.

4. Solving equations with square roots

To solve an equation like \(x^2=p\), ask: What numbers squared equal \(p\)?

If \(p\) is positive, there are usually two solutions: one positive and one negative.

For example, if

$$x^2=49$$

then

$$x=7 \text{ or } x=-7$$

because both \(7^2\) and \((-7)^2\) equal \(49\).

If \(x^2=0\), then the only solution is \(x=0\).

If \(x^2\) equals a negative number, there is no real number solution. For example, \(x^2=-9\) has no real solution because no real number squared gives a negative result.

5. Solving equations with cube roots

To solve an equation like \(x^3=p\), ask: What number cubed equals \(p\)?

There is only one real solution.

For example, if

$$x^3=-27$$

then

$$x=-3$$

because

$$(-3)^3=-27$$

6. Worked Examples

Example 1: Evaluate a square root

Find \(\sqrt{81}\).

Ask: What positive number times itself equals \(81\)?

$$9^2=81$$

So,

$$\sqrt{81}=9$$

Answer: \(9\)

Example 2: Evaluate a cube root

Find \(\sqrt[3]{-64}\).

Ask: What number multiplied by itself three times equals \(-64\)?

$$(-4)^3=-64$$

So,

$$\sqrt[3]{-64}=-4$$

Answer: \(-4\)

Example 3: Solve a square equation

Solve \(x^2=36\).

Ask: What numbers squared equal \(36\)?

$$6^2=36 \quad \text{and} \quad (-6)^2=36$$

So the equation has two solutions:

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

Answer: \(x=\pm 6\)

Example 4: Solve a cube equation

Solve \(x^3=125\).

Ask: What number cubed equals \(125\)?

$$5^3=125$$

So,

$$x=5$$

Answer: \(5\)

7. Important things to remember

• \(\sqrt{25}=5\), not \(\pm 5\). The radical sign means the principal, or nonnegative, square root.
• But if you solve \(x^2=25\), the solutions are \(x=5\) and \(x=-5\).
• Square roots of positive perfect squares are whole numbers.
• Cube roots of perfect cubes are whole numbers.
• A negative number cannot have a real square root.
• A negative number can have a real cube root.

8. Quick check

1. \(\sqrt{49}=\ ?\)
2. \(\sqrt[3]{8}=\ ?\)
3. Solve: \(x^2=64\)
4. Solve: \(x^3=-8\)

Answers:

1. \(7\)
2. \(2\)
3. \(x=8\) or \(x=-8\)
4. \(x=-2\)

Summary

A square root tells what number was multiplied by itself to make a given number, and a cube root tells what number was multiplied by itself three times. The symbol \(\sqrt{}\) gives the nonnegative square root, while solving \(x^2=p\) usually gives two solutions. Cube root equations like \(x^3=p\) have one real solution, and cube roots can be negative.

Approximating Irrational Numbers

In math, not every number can be written as a simple fraction or as a decimal that ends or repeats. Some numbers go on forever without repeating. These are called irrational numbers.

Common examples of irrational numbers are square roots of numbers that are not perfect squares, such as \(\sqrt{2}\), \(\sqrt{5}\), and \(\sqrt{10}\).

Since irrational numbers cannot be written exactly as simple decimals, we often use approximations. An approximation is a value that is close to the exact number.

Approximating irrational numbers helps us do three important things:

• compare their sizes,
• place them on a number line,
• estimate the value of expressions.

1. Use perfect squares to estimate square roots

A very useful strategy is to find the two perfect squares that your number is between.

For example, to estimate \(\sqrt{7}\), think about the perfect squares near 7:

$$ 4 = 2^2 \quad \text{and} \quad 9 = 3^2 $$

Since \(7\) is between \(4\) and \(9\), \(\sqrt{7}\) must be between \(2\) and \(3\).

This gives us a first estimate:

$$ 2 < \sqrt{7} < 3 $$

To get a better approximation, decide whether \(7\) is closer to \(4\) or \(9\). Since 7 is closer to 9 than to 4, \(\sqrt{7}\) is closer to 3 than to 2.

A decimal approximation is:

$$ \sqrt{7} \approx 2.65 $$

2. Approximating by testing decimals

You can test decimal numbers by squaring them.

Suppose we want a better estimate for \(\sqrt{7}\). Try \(2.6\):

$$ 2.6^2 = 6.76 $$

Try \(2.7\):

$$ 2.7^2 = 7.29 $$

Since \(7\) is between \(6.76\) and \(7.29\), we know:

$$ 2.6 < \sqrt{7} < 2.7 $$

If we want even more accuracy, test numbers like \(2.64\) and \(2.65\).

3. Comparing irrational numbers

Approximations help us compare irrational numbers with each other and with rational numbers.

For example, compare \(\sqrt{8}\) and \(2.9\).

We know:

$$ 4 < 8 < 9 $$

So:

$$ 2 < \sqrt{8} < 3 $$

A better approximation is:

$$ \sqrt{8} \approx 2.83 $$

Since \(2.83 < 2.9\), we can say:

$$ \sqrt{8} < 2.9 $$

4. Locating irrational numbers on a number line

To place an irrational number on a number line, first find the two integers it is between. Then use a decimal approximation to decide where it belongs.

For example, \(\sqrt{13}\):

$$ 9 < 13 < 16 $$

So:

$$ 3 < \sqrt{13} < 4 $$

A decimal approximation is:

$$ \sqrt{13} \approx 3.61 $$

On a number line, \(\sqrt{13}\) would be placed between 3 and 4, a little past 3.6.

5. Estimating expressions with irrational numbers

Sometimes you need to estimate expressions that include irrational numbers.

For example, estimate:

$$ \sqrt{5} + 2 $$

Since \(\sqrt{5} \approx 2.24\), then:

$$ \sqrt{5} + 2 \approx 2.24 + 2 = 4.24 $$

This is not exact, but it is a useful estimate.

Worked Example 1: Find two integers that \(\sqrt{11}\) is between

Step 1: Find perfect squares around 11.

$$ 9 = 3^2 \quad \text{and} \quad 16 = 4^2 $$

Step 2: Since \(11\) is between \(9\) and \(16\), \(\sqrt{11}\) is between \(3\) and \(4\).

$$ 3 < \sqrt{11} < 4 $$

Answer: \(\sqrt{11}\) is between 3 and 4.

Worked Example 2: Approximate \(\sqrt{11}\) to the nearest tenth

We already know:

$$ 3 < \sqrt{11} < 4 $$

Now test tenths:

$$ 3.3^2 = 10.89 $$ $$ 3.4^2 = 11.56 $$

Since 11 is between 10.89 and 11.56, we know:

$$ 3.3 < \sqrt{11} < 3.4 $$

Because 11 is closer to 10.89 than to 11.56, \(\sqrt{11}\) is closer to 3.3.

Answer:

$$ \sqrt{11} \approx 3.3 $$

Worked Example 3: Compare \(\sqrt{18}\) and \(4.3\)

First find an approximation for \(\sqrt{18}\).

$$ 16 < 18 < 25 $$

So:

$$ 4 < \sqrt{18} < 5 $$

Now test a decimal or use a known approximation:

$$ \sqrt{18} \approx 4.24 $$

Now compare:

$$ 4.24 < 4.3 $$

Answer:

$$ \sqrt{18} < 4.3 $$

Worked Example 4: Estimate \(2\sqrt{3} + 1\)

First approximate \(\sqrt{3}\).

$$ 1^2 = 1 \quad \text{and} \quad 2^2 = 4 $$

So \(\sqrt{3}\) is between 1 and 2, and a common approximation is:

$$ \sqrt{3} \approx 1.73 $$

Now substitute into the expression:

$$ 2\sqrt{3} + 1 \approx 2(1.73) + 1 $$ $$ = 3.46 + 1 = 4.46 $$

Answer:

$$ 2\sqrt{3} + 1 \approx 4.46 $$

Helpful steps to remember

1. Find the perfect squares on either side of the number.
2. Use those perfect squares to find the two integers the square root is between.
3. Test decimal values if you need a better approximation.
4. Use the approximation to compare numbers, place them on a number line, or estimate expressions.

Common mistakes to avoid

• Do not assume a square root is rational just because it has a decimal approximation.
• Do not forget that \(\sqrt{n}\) is between the roots of the perfect squares around \(n\).
• When comparing numbers, make sure both numbers are written in a similar form, such as decimals.
• Remember that an approximation is close to the value, not the exact value.

Summary

Irrational numbers like \(\sqrt{2}\) and \(\sqrt{7}\) cannot be written as exact fractions or terminating decimals, so we approximate them.

To approximate a square root, find the perfect squares around the number and then test decimals if needed. These approximations help you compare irrational numbers, place them on a number line, and estimate the value of expressions.

Bounds and Error in Approximation

Sometimes in maths, we cannot write the exact value of a number as a simple decimal. This often happens with irrational numbers, such as \(\sqrt{2}\), \(\sqrt{5}\), or \(\pi\). Their decimal digits go on forever and do not repeat.

Because of this, we often use an approximation. An approximation is a value that is close to the real number, but not exactly equal to it.

When we approximate, it is important to know how close our answer is. This is where bounds and error come in.

1. What are bounds?

A lower bound is a value that is less than or equal to the actual number.

An upper bound is a value that is greater than or equal to the actual number.

If a number \(x\) lies between two values, we can write:

$$a < x < b$$

This means \(a\) is a lower bound and \(b\) is an upper bound.

For example, we know:

$$1^2 = 1 \quad \text{and} \quad 2^2 = 4$$

Since \(2\) lies between \(1\) and \(4\), the square root of 2 must lie between \(1\) and \(2\):

$$1 < \sqrt{2} < 2$$

So \(1\) is a lower bound for \(\sqrt{2}\), and \(2\) is an upper bound.

2. Improving bounds

We can make bounds more accurate by checking decimal numbers.

For \(\sqrt{2}\), try tenths:

$$1.4^2 = 1.96$$ $$1.5^2 = 2.25$$

Since \(1.96 < 2 < 2.25\), we know:

$$1.4 < \sqrt{2} < 1.5$$

These bounds are tighter than \(1 < \sqrt{2} < 2\), so they give a better approximation.

We can improve again by checking hundredths.

$$1.41^2 = 1.9881$$ $$1.42^2 = 2.0164$$

Since \(1.9881 < 2 < 2.0164\), we know:

$$1.41 < \sqrt{2} < 1.42$$

This process is called iterative decimal approximation. It means improving a decimal approximation step by step.

3. What is error?

The error tells us how far an approximation may be from the true value.

If we say \(\sqrt{2} \approx 1.4\), then the actual value is not exactly \(1.4\). The difference between the true value and the approximation is the error.

In school maths, we often talk about the maximum error, which is the greatest possible difference.

For example, if a number is rounded to 1 decimal place, it is accurate to the nearest tenth. That means the true value is within \(0.05\) of the rounded value.

So if a length is given as \(3.2\) cm correct to 1 decimal place, the actual length lies between:

$$3.15 \leq \text{length} < 3.25$$

The maximum error is:

$$0.05$$

4. Bounds from rounding

When a number is rounded, we can work out its lower and upper bounds.

• Rounded to the nearest whole number: error is at most \(0.5\)
• Rounded to 1 decimal place: error is at most \(0.05\)
• Rounded to 2 decimal places: error is at most \(0.005\)

For example:

• If \(x = 7\) correct to the nearest whole number, then $$6.5 \leq x < 7.5$$
• If \(y = 4.8\) correct to 1 decimal place, then $$4.75 \leq y < 4.85$$
• If \(z = 2.34\) correct to 2 decimal places, then $$2.335 \leq z < 2.345$$

Notice that we usually include the lower bound with \(\leq\), but not the upper bound. That is because a number exactly at the upper bound would round up to the next value.

5. Using bounds for irrational numbers

Bounds are especially useful for irrational numbers because we cannot write their exact decimal values completely.

Suppose we want bounds for \(\sqrt{5}\).

First, use whole numbers:

$$2^2 = 4 \quad \text{and} \quad 3^2 = 9$$

So:

$$2 < \sqrt{5} < 3$$

Now use tenths:

$$2.2^2 = 4.84$$ $$2.3^2 = 5.29$$

Since \(4.84 < 5 < 5.29\), we get:

$$2.2 < \sqrt{5} < 2.3$$

Now use hundredths:

$$2.23^2 = 4.9729$$ $$2.24^2 = 5.0176$$

So:

$$2.23 < \sqrt{5} < 2.24$$

This tells us that \(\sqrt{5}\) is approximately \(2.24\), or more accurately about \(2.236\dots\).

Worked Example 1: Find whole-number bounds for \(\sqrt{7}\)

Step 1: Find perfect squares around 7.

$$2^2 = 4 \quad \text{and} \quad 3^2 = 9$$

Step 2: Since \(4 < 7 < 9\), take square roots.

$$2 < \sqrt{7} < 3$$

Answer: The lower bound is \(2\) and the upper bound is \(3\).

Worked Example 2: Improve the bounds for \(\sqrt{7}\) to 1 decimal place

We already know:

$$2 < \sqrt{7} < 3$$

Try tenths:

$$2.6^2 = 6.76$$ $$2.7^2 = 7.29$$

Since \(6.76 < 7 < 7.29\), we know:

$$2.6 < \sqrt{7} < 2.7$$

Answer: The bounds to 1 decimal place are \(2.6\) and \(2.7\).

Worked Example 3: Improve the bounds for \(\sqrt{7}\) to 2 decimal places

We know:

$$2.6 < \sqrt{7} < 2.7$$

Now test hundredths near the answer.

$$2.64^2 = 6.9696$$ $$2.65^2 = 7.0225$$

Since \(6.9696 < 7 < 7.0225\), we have:

$$2.64 < \sqrt{7} < 2.65$$

Answer: The bounds to 2 decimal places are \(2.64\) and \(2.65\).

Worked Example 4: Find the bounds and maximum error for a rounded number

A mass is given as \(12.7\) kg correct to 1 decimal place.

Step 1: Numbers correct to 1 decimal place are rounded to the nearest tenth.

Half of one tenth is:

$$0.05$$

Step 2: Subtract and add \(0.05\).

$$12.7 - 0.05 = 12.65$$ $$12.7 + 0.05 = 12.75$$

Step 3: Write the bounds.

$$12.65 \leq \text{mass} < 12.75$$

Step 4: State the maximum error.

$$\text{maximum error} = 0.05\text{ kg}$$

Answer: The mass is at least \(12.65\) kg and less than \(12.75\) kg. The maximum error is \(0.05\) kg.

6. Tips for solving bounds questions

• For square roots, find the perfect squares on either side first.
• To improve bounds, test decimals and square them.
• For rounded values, use half of the place value to find the error.
• Use \(\leq\) for the lower bound and \(<\) for the upper bound.
• Check that your approximation really lies between your bounds.

7. Common mistakes to avoid

• Mixing up upper and lower bounds: the lower bound is the smaller value.
• Using the wrong error amount: for 1 decimal place, use \(0.05\), not \(0.1\).
• Forgetting to square test values: when estimating \(\sqrt{n}\), compare the squares with \(n\).
• Including the upper bound incorrectly: the upper bound is usually not included.

Brief Summary

Bounds tell us the interval where the true value lies. A lower bound is below the value, and an upper bound is above it. For irrational numbers like square roots, we can improve bounds step by step by testing decimal numbers. Error tells us how far an approximation could be from the true value, and rounded numbers have bounds based on half of the place value used.