8th Grade Rational and Irrational Numbers Lesson (Free PDF)

Q: Is zero a rational number?

Yes. Zero equals \frac{0}{1}, a fraction of two integers, so it is rational. It is also a whole number and an integer.

Q: Is a repeating decimal always rational?

Yes. Any decimal with a block of digits that repeats forever can be turned into a fraction by multiplying by a power of 10 and subtracting. For example, 0.\overline{3} = \frac{1}{3}.

Q: Why is the square root of 2 irrational?

No fraction squared equals exactly 2, and its decimal form 1.41421356... never repeats. Mathematicians prove this with a contradiction argument that you will see in high school.

Q: Is 22/7 equal to pi?

No. \frac{22}{7} = 3.142857... repeats, while \pi = 3.14159... never repeats. The fraction is a handy rational approximation, slightly larger than \pi.

Q: How do I compare a square root and a decimal?

Square both numbers, or estimate the root to two decimal places. For instance \sqrt{30} \approx 5.48, which is less than 5.5, because 5.5^{2} = 30.25 > 30.

8th Grade math lesson: Rational and Irrational Numbers – Maths PDF

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This rational irrational lesson for 8th Grade covers rational and irrational numbers (Common Core 8.NS.A.1, 8.NS.A.2), with a free PDF to print.

A rational number is any number that can be written as a fraction of two integers; its decimal form stops or repeats. An irrational number, such as the square root of 2 or pi, has a decimal that never stops and never repeats. Together they make up the real numbers.

Grade 8 · Common Core State Standards: 8.NS.A.1, 8.NS.A.2

Key facts in this rational irrational lesson

Key facts

Terminating and repeating decimals are rational; every rational number can be written as a fraction.
Square roots of non-perfect squares, and \(\pi\), are irrational.
To write \(0.\overline{7}\) as a fraction, multiply by 10, subtract, and divide: \(\frac{7}{9}\).
Estimate \(\sqrt{n}\) by finding the two perfect squares around \(n\), then refine to tenths and hundredths.
Compare real numbers by converting them to decimals with the same number of places.

What is a rational number?

A rational number is any number you can write as a fraction of two integers. Fractions, integers, terminating decimals and repeating decimals are all rational.

Rational and irrational

A rational number can be written as \(\frac{a}{b}\) with integers \(a\) and \(b\), \(b \neq 0\). Its decimal form stops or repeats. An irrational number cannot be written that way; its decimal never stops and never repeats.

Every real number is either rational or irrational, never both.

How do you turn a repeating decimal into a fraction?

Let x equal the decimal, multiply by a power of 10 so the repeating blocks line up, and subtract. The repeating tail cancels.

Multiply by 10 so the repeating tails line up, then subtract.

Set \(x = 0.\overline{7}\).
Multiply by 10: \(10x = 7.\overline{7}\).
Subtract: \(10x – x = 7.\overline{7} – 0.\overline{7}\), so \(9x = 7\).
Divide: \(x = \frac{7}{9}\).

If two digits repeat, multiply by 100 instead. For example, \(0.\overline{45} = \frac{45}{99} = \frac{5}{11}\).

How do you estimate an irrational number?

Square roots of numbers that are not perfect squares are irrational. To place \(\sqrt{10}\), find the perfect squares on each side: \(3^{2} = 9\) and \(4^{2} = 16\), so \(3 < \sqrt{10} < 4\). Since 10 is only 1 more than 9, the root is close to 3. Testing \(3.1^{2} = 9.61\) and \(3.2^{2} = 10.24\) shows \(3.1 < \sqrt{10} < 3.2\).

√10 is between 3 and 4, close to 3.2

Keep zooming in and the same idea gives more digits: \(\sqrt{2} = 1.41421356…\) never ends and never repeats.

√2 ≈ 1.414 lies between 1.41 and 1.42

Which numbers are irrational?

Number	Type	Reason
\(\sqrt{49}\)	Rational	\(\sqrt{49} = 7\)
\(\sqrt{7}\)	Irrational	7 is not a perfect square
\(\pi\)	Irrational	Decimal never repeats
\(0.\overline{36}\)	Rational	\(= \frac{4}{11}\)

How do you compare and order real numbers?

Write each number as a decimal to the same number of places, then order the decimals. To order \(\sqrt{7}\), \(2.6\) and \(\frac{8}{3}\), find \(\sqrt{7} \approx 2.646\) and \(\frac{8}{3} = 2.666…\). Now the order is clear: \(2.6 < \sqrt{7} < \frac{8}{3}\).

Remember two safe rules. First, a sum or product of two rational numbers is always rational. Second, a nonzero rational number times an irrational number, such as \(3\pi\), is irrational.

Rational irrational lesson: questions and answers

Is zero a rational number?

Yes. Zero equals \(\frac{0}{1}\), a fraction of two integers, so it is rational. It is also a whole number and an integer.

Is a repeating decimal always rational?

Yes. Any decimal with a block of digits that repeats forever can be turned into a fraction by multiplying by a power of 10 and subtracting. For example, \(0.\overline{3} = \frac{1}{3}\).

Why is the square root of 2 irrational?

No fraction squared equals exactly 2, and its decimal form \(1.41421356…\) never repeats. Mathematicians prove this with a contradiction argument that you will see in high school.

Is 22/7 equal to pi?

No. \(\frac{22}{7} = 3.142857…\) repeats, while \(\pi = 3.14159…\) never repeats. The fraction is a handy rational approximation, slightly larger than \(\pi\).

How do I compare a square root and a decimal?

Square both numbers, or estimate the root to two decimal places. For instance \(\sqrt{30} \approx 5.48\), which is less than 5.5, because \(5.5^{2} = 30.25 > 30\).

How to use this rational irrational lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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