Glossary

3.5 Irrational Numbers

3 min read•june 18, 2024

Numbers come in two flavors: rational and irrational. are fractions or decimals that end or repeat. , like or , go on forever without repeating. They're weird but important.

pop up in math and real life more than you'd think. They're crucial in geometry, physics, and even music theory. Learning to work with them opens up a whole new world of mathematical possibilities.

Understanding Irrational Numbers

Rational vs irrational numbers

Top images from around the web for Rational vs irrational numbers

  • Rational numbers in Mathematics ~ I Answer 4 U View original

    Is this image relevant?

  • Number Sets View original

    Is this image relevant?

  • Rational numbers in Mathematics ~ I Answer 4 U View original

    Is this image relevant?

1 of 3

Top images from around the web for Rational vs irrational numbers

  • Rational numbers in Mathematics ~ I Answer 4 U View original

    Is this image relevant?

  • Number Sets View original

    Is this image relevant?

  • Rational numbers in Mathematics ~ I Answer 4 U View original

    Is this image relevant?

1 of 3
  • Rational numbers expressed as a ratio of two integers ab\frac{a}{b}, where b≠0b \neq 0 (examples: 12\frac{1}{2}, 34\frac{3}{4}, −57\frac{-5}{7})
    • Decimal representations either terminate after a finite number of digits (0.50.5, 1.251.25) or repeat in a cyclical pattern (0.3333...0.3333..., 0.1212...0.1212...)
  • Irrational numbers cannot be expressed as a ratio of two integers
    • Decimal representations never terminate and never repeat in a cyclical pattern, continuing infinitely without any discernible pattern
    • Well-known examples include 2\sqrt{2}, 3\sqrt{3}, 5\sqrt{5}, π\pi (pi), and [e](https://www.fiveableKeyTerm:e)[e](https://www.fiveableKeyTerm:e) ()
  • Irrational numbers have unique decimal expansions that continue infinitely without repeating

Types of Irrational Numbers

  • : irrational numbers that are roots of polynomial equations with rational coefficients (e.g., 2\sqrt{2}, 53\sqrt[3]{5})
  • : irrational numbers that are not algebraic (e.g., π\pi, ee)
  • Both algebraic and transcendental numbers can be represented using , which are expressions involving infinite sequences of fractions

Simplification of square roots

  • are numbers that result from multiplying an integer by itself (11, 44, 99, 1616, 2525, etc.)
  • Simplifying a square root involves factoring out the largest perfect square from the (number under the square root symbol)
    1. Simplify the square root of the perfect square
    2. Leave the remaining factor under the square root symbol
  • Example: 48\sqrt{48}
    • 4848 factored as 16×316 \times 3, where 1616 is a perfect square (4×44 \times 4)
    • 48=16×3=16×3=43\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} (simplified form)

Arithmetic with irrational numbers

  • Addition and subtraction combine like terms, which are terms with the same irrational number (23+53=732\sqrt{3} + 5\sqrt{3} = 7\sqrt{3})
  • Multiplication involves multiplying the coefficients and irrational parts separately ((23)(35)=615(2\sqrt{3})(3\sqrt{5}) = 6\sqrt{15})
  • Division requires by multiplying numerator and denominator by the of the denominator
    • Conjugate of a+bca + b\sqrt{c} is a−bca - b\sqrt{c}
    • Example: 23=23×33=233\frac{2}{\sqrt{3}} = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} (rationalized form)

Applying Irrational Number Concepts

Solve problems involving irrational numbers in real-world contexts

  • states that in a right triangle with legs aa and bb and hypotenuse cc, a2+b2=c2a^2 + b^2 = c^2
    • If aa and bb are rational, cc may be irrational (32+42=523^2 + 4^2 = 5^2, where 55 is irrational)
  • Trigonometric ratios (, , ) of angles in a right triangle may be irrational (sin⁡(45∘)=22\sin(45^\circ) = \frac{\sqrt{2}}{2}, which is irrational)
  • and often involve the irrational number ee
    • ln⁡(1)=0\ln(1) = 0, but ln⁡(2)\ln(2) is irrational
    • e0=1e^0 = 1, but e1e^1 is irrational
  • Irrational numbers can be located on the , filling in the gaps between rational numbers

Key Terms to Review (25)

√2: The symbol √2 represents the square root of 2, which is an important number in mathematics because it is the first known irrational number. Its value is approximately 1.41421356, and it cannot be expressed as a simple fraction. This characteristic makes √2 significant in various mathematical contexts, especially when discussing irrational numbers and their properties, as it highlights the distinction between rational and irrational values.
√3: The term √3 refers to the positive square root of 3, which is an irrational number approximately equal to 1.732. This number cannot be expressed as a simple fraction, and its decimal representation is non-terminating and non-repeating. Understanding √3 helps illustrate the properties of irrational numbers, especially in relation to geometry and algebra, where it often appears in calculations involving right triangles and the Pythagorean theorem.
√5: √5, or the square root of 5, is an irrational number that cannot be expressed as a fraction of two integers. It is approximately equal to 2.236 and falls under the category of non-repeating, non-terminating decimals. As an irrational number, √5 plays a significant role in various mathematical concepts, particularly in geometry, where it can appear in calculations involving right triangles or the Pythagorean theorem.
Algebraic numbers: Algebraic numbers are numbers that are solutions to polynomial equations with integer coefficients. This means that any number that can be expressed as a root of such equations, including both rational and certain irrational numbers, is considered algebraic. These numbers play a crucial role in number theory and help to bridge the gap between rational and irrational numbers.
Conjugate: In mathematics, a conjugate typically refers to a pair of expressions that are closely related through a change in sign or order. This term is especially relevant in the context of irrational numbers, where conjugates can simplify expressions, particularly when dealing with roots and complex numbers. Recognizing conjugates allows for easier manipulation of these expressions and aids in rationalizing denominators or simplifying computations.
Continued fractions: Continued fractions are expressions that represent numbers through a sequence of integer divisions, where each division can be expressed as a whole number plus a fraction. This format is particularly useful for approximating irrational numbers, allowing us to see their fractional parts in a structured way. By breaking down a number into continued fractions, we can gain insight into its properties and how closely it can be approximated by rational numbers.
Cosine: Cosine is a trigonometric function defined for an angle in a right triangle as the ratio of the length of the adjacent side to the length of the hypotenuse. This function is essential for understanding relationships between angles and sides in right triangles, and it also extends to the unit circle, where it represents the x-coordinate of a point on the circle corresponding to a given angle.
Decimal expansion: Decimal expansion refers to the representation of a number in the base-10 numeral system, using digits to express values after a decimal point. This method allows both rational and irrational numbers to be expressed in a format that can indicate their magnitude and precision. Decimal expansions for rational numbers are either terminating or repeating, while irrational numbers have non-terminating and non-repeating decimal expansions.
E: The number e is a fundamental mathematical constant approximately equal to 2.71828, which serves as the base of the natural logarithm. This unique number arises naturally in various mathematical contexts, especially in calculus, where it plays a crucial role in growth processes, compound interest, and differential equations. Understanding e is essential for comprehending concepts related to exponential growth and decay, as well as irrational numbers.
Euler's number: Euler's number, denoted as 'e', is an irrational number approximately equal to 2.71828, that serves as the base for natural logarithms. It is a crucial constant in mathematics, particularly in calculus, where it arises in processes of growth and decay, as well as in the context of complex numbers through Euler's formula.
Exponential Functions: Exponential functions are mathematical functions of the form $$f(x) = a imes b^x$$, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions are characterized by their rapid growth or decay and are widely used to model real-world phenomena, including populations, finance, and certain physical processes. The behavior of exponential functions is closely tied to irrational numbers when considering bases like $e$ (Euler's number), which is an important irrational number in mathematics.
Irrational numbers: Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. They have non-repeating, non-terminating decimal expansions.
Irrational Numbers: Irrational numbers are real numbers that cannot be expressed as a fraction of two integers, meaning they have non-repeating, non-terminating decimal expansions. These numbers include famous constants like $$\pi$$ and $$e$$, as well as square roots of non-perfect squares. The existence of irrational numbers is essential for understanding the completeness of the number system.
Natural Logarithms: Natural logarithms are logarithms with the base of the mathematical constant e, which is approximately equal to 2.71828. They are denoted as ln(x) and play a crucial role in various fields of mathematics, especially in calculus, due to their unique properties in relation to exponential functions. These logarithms are particularly important when dealing with growth processes and can be applied to solve equations involving exponentials.
Perfect Squares: Perfect squares are numbers that can be expressed as the square of an integer. This means that if you take a whole number and multiply it by itself, the result is a perfect square. Understanding perfect squares is crucial because they help in identifying rational and irrational numbers, particularly when determining whether the square root of a number is rational or irrational.
Pythagorean Theorem: The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.
Pythagorean theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This relationship connects geometry and algebra, forming the basis for understanding distances and relationships within triangular structures and applications in real-world scenarios.
Radicand: A radicand is the number or expression under a radical sign, which indicates the operation of taking a root, such as a square root or cube root. The value of the radicand is crucial because it determines the result of the root operation. For instance, in the expression $$\sqrt{9}$$, the number 9 is the radicand, and its properties affect whether the result is rational or irrational.
Rational Numbers: Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. This definition connects them to whole numbers, integers, and other types of numbers, making them a fundamental part of the number system. They can be positive, negative, or zero and can also be represented as terminating or repeating decimals.
Rationalizing the Denominator: Rationalizing the denominator is the process of eliminating any irrational numbers from the denominator of a fraction by multiplying both the numerator and the denominator by a suitable expression. This technique is important because it helps simplify expressions, making them easier to work with, particularly when dealing with irrational numbers such as square roots. It not only clarifies calculations but also facilitates comparisons and further mathematical operations involving fractions.
Real number line: The real number line is a visual representation of all real numbers arranged in a straight line, where each point on the line corresponds to a unique real number. This concept is essential for understanding various types of numbers, including integers, rational numbers, and especially irrational numbers, as it allows us to see how these numbers relate to one another in a continuous manner.
Sine: Sine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. It is used extensively in various mathematical contexts, especially when analyzing periodic phenomena, waveforms, and in navigation. Sine values can be expressed in both rational and irrational forms, particularly when dealing with angles that do not correspond to simple fractions.
Tangent: A tangent is a line that touches a curve at a single point without crossing it. In mathematics, this concept is crucial in understanding angles and slopes, especially in the context of circles and right triangles. Tangents also play a role in various fields such as calculus and trigonometry, particularly when dealing with the relationships between angles and side lengths in triangles.
Transcendental numbers: Transcendental numbers are a type of real number that cannot be the root of any non-zero polynomial equation with rational coefficients. This means they go beyond algebraic numbers, which can be expressed as solutions to such equations. Transcendental numbers are crucial in understanding the complexity of real numbers and illustrate the richness of the number system, encompassing well-known constants like $$e$$ and $$\pi$$.
π: π, or pi, is an irrational number that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14159 and is crucial in various mathematical contexts, particularly when dealing with circular shapes and their properties. Its unique characteristics as an irrational number mean that it cannot be expressed as a simple fraction, making it an interesting and essential component in geometry and calculus.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide