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Irrational number

Irrational number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Irrational number rather than just read about it. In short: In mathematics, the irrational numbers are all the real numbers that are not rational numbers; that is, irrational numbers are those that cannot be expressed as the ratio of two integers. Geometrically, when the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common; that is, there is no length ("the…

Irrational number — main illustration
Irrational number — illustration

Key takeaways

  • Irrational number belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Irrational number to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Irrational number from memory before moving on to harder problems.

Reference excerpt

In mathematics, the irrational numbers are all the real numbers that are not rational numbers; that is, irrational numbers are those that cannot be expressed as the ratio of two integers. Geometrically, when the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common; that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like any real number, an irrational number can be expressed in positional notation; however, it does not terminate or end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics. Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic), and in many other ways. As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.

History

Ancient Greece The first proof of the existence of irrational numbers is attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram. The Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:

… excerpt ends here. Continue reading the full article.

Illustrations

Irrational number: The number 2 {\displaystyle {\sqrt {2}}} is irrational.
The number 2 {\displaystyle {\sqrt {2}}} is irrational.
Irrational number: An Euler diagram showing the set of real numbers ( R {\displaystyle \mathbb {R} } ), which include the rationals ( Q {\displaystyle \mathbb {Q} } ), which include the integers ( Z {\displaystyle \mathbb {Z} } ), which include the natural numbers ( N {\displaystyle \mathbb {N} } ). The real numbers also include the irrationals ( R {\displaystyle \mathbb {R} } \ Q {\displaystyle \mathbb {Q} } ).
An Euler diagram showing the set of real numbers ( R {\displaystyle \mathbb {R} } ), which include the rationals ( Q {\displaystyle \mathbb {Q} } ), which include the integers ( Z {\displaystyle \mathbb {Z} } ), which include the natural numbers ( N {\displaystyle \mathbb {N} } ). The real numbers also include the irrationals ( R {\displaystyle \mathbb {R} } \ Q {\displaystyle \mathbb {Q} } ).
Irrational number: Set inclusions between the natural numbers ( N {\displaystyle \mathbb {N} } ), the integers ( Z {\displaystyle \mathbb {Z} } ), the rational numbers ( Q {\displaystyle \mathbb {Q} } ), the real numbers ( R {\displaystyle \mathbb {R} } ), and the complex numbers ( C {\displaystyle \mathbb {C} } )
Set inclusions between the natural numbers ( N {\displaystyle \mathbb {N} } ), the integers ( Z {\displaystyle \mathbb {Z} } ), the rational numbers ( Q {\displaystyle \mathbb {Q} } ), the real numbers ( R {\displaystyle \mathbb {R} } ), and the complex numbers ( C {\displaystyle \mathbb {C} } )

Worked examples

Example 1 — a first encounter with Irrational number

Start with the simplest possible case. Write down what Irrational number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Irrational number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Irrational number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Irrational number

In research
Irrational number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Irrational number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Irrational number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Irrational numbers, Sets of real numbers, so understanding it makes those chapters shorter.
In everyday life
Look for Irrational number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Irrational number in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Irrational number means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Irrational number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Irrational number in simple terms?

In mathematics, the irrational numbers are all the real numbers that are not rational numbers; that is, irrational numbers are those that cannot be expressed as the ratio of two integers. Geometrically, when the ratio of lengths of two line segments is an irrational number, the line segments are al…

Why does Irrational number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Irrational number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Irrational number.

Tags

  • Irrational numbers
  • Sets of real numbers

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