Glossary

7.1 Rational and Irrational Numbers

2 min read•june 25, 2024

Numbers come in different flavors: rational and irrational. Rational numbers can be written as fractions, while irrational ones can't. This distinction is crucial for understanding the nature of numbers and their relationships.

Real numbers include both rational and irrational types. They form a continuous line, with rational numbers at specific points and irrational numbers filling the gaps. This concept helps us grasp the full spectrum of numbers we use in math.

Understanding Rational and Irrational Numbers

Rational vs irrational numbers

Top images from around the web for Rational vs irrational numbers

  • Decimals · Intermediate Algebra View original

    Is this image relevant?

  • Summary: Classes of Real Numbers | Developmental Math Emporium View original

    Is this image relevant?

  • Decimals · Intermediate Algebra View original

    Is this image relevant?

1 of 3

Top images from around the web for Rational vs irrational numbers

  • Decimals · Intermediate Algebra View original

    Is this image relevant?

  • Summary: Classes of Real Numbers | Developmental Math Emporium View original

    Is this image relevant?

  • Decimals · Intermediate Algebra View original

    Is this image relevant?

1 of 3
  • Rational numbers express as a ratio of two (ab\frac{a}{b}, b≠0b \neq 0) include integers (3), fractions (25\frac{2}{5}), and (0.75) or repeating decimals (0.3‾0.\overline{3})
  • Irrational numbers cannot express as a ratio of two integers have decimal expansions that neither terminate nor repeat (2\sqrt{2}, π\pi, [e](https://www.fiveableKeyTerm:e)[e](https://www.fiveableKeyTerm:e))

Decimal to fraction conversions

  • Terminating decimals to fractions write the decimal as a fraction over a power of 10, then simplify (0.25=25100=140.25 = \frac{25}{100} = \frac{1}{4})
  • Repeating decimals to fractions let xx equal the , multiply by a power of 10 to shift the decimal, and solve for xx (0.3‾=130.\overline{3} = \frac{1}{3})
    1. Let x=0.3‾x = 0.\overline{3}
    2. 10x=3.3‾10x = 3.\overline{3}
    3. 10x−x=3.3‾−0.3‾10x - x = 3.\overline{3} - 0.\overline{3}
    4. 9x=39x = 3
    5. x=13x = \frac{1}{3}
  • Fractions to decimals divide the numerator by the denominator (38=0.375\frac{3}{8} = 0.375)

Categorization of real numbers

  • Real numbers include all rational and irrational numbers
  • Subsets of real numbers
    • non-negative integers (0,1,2,3,…0, 1, 2, 3, \ldots)
    • Integers positive, negative, and zero (…,−3,−2,−1,0,1,2,3,…\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots)
    • Rational numbers express as ab\frac{a}{b}, aa and bb are integers and b≠0b \neq 0
    • Irrational numbers real numbers that are not rational (3\sqrt{3}, π\pi)

Relationships between Number Sets

  • Whole numbers ⊂\subset Integers ⊂\subset Rational numbers ⊂\subset Real numbers
  • Irrational numbers subset of real numbers but do not overlap with rational numbers
  • All real numbers can be represented on a , with rational numbers at specific points and irrational numbers filling the gaps

Advanced Number Classifications

  • : numbers that are roots of polynomial equations with integer coefficients (e.g., 2\sqrt{2}, 53\sqrt[3]{5})
  • : irrational numbers that are not algebraic (e.g., π\pi, ee)
  • The set of real numbers exhibits the property of , meaning between any two real numbers, there are infinitely many other real numbers

Key Terms to Review (21)

Algebraic Numbers: Algebraic numbers are the roots of polynomial equations with integer coefficients. They include both rational numbers and irrational numbers that can be expressed as solutions to algebraic equations.
Cube Root: The cube root is a mathematical operation that finds the value that, when multiplied by itself three times, equals a given number. It is a way of reversing the process of raising a number to the power of three, and is represented by the symbol $\sqrt[3]{}$.
Decimal Expansion: A decimal expansion is the representation of a number in the decimal number system, where the number is expressed as a sequence of digits after the decimal point. It is a fundamental concept that connects the topics of decimals and rational and irrational numbers.
Density: Density is a physical property that describes the mass of a substance per unit volume. It is a fundamental concept in mathematics and science, particularly in the study of rational and irrational numbers, as it provides a way to quantify and compare the compactness or sparsity of different materials and numerical representations.
E: e, also known as Euler's number, is a fundamental mathematical constant that represents the base of the natural logarithm. It is an irrational number, meaning its decimal representation never repeats or terminates, and it is approximately equal to 2.71828. The constant e is widely used in various fields of mathematics, science, and engineering, and it has many important properties and applications.
Integers: Integers are a set of positive and negative whole numbers, including zero. They are the foundation for many mathematical operations and concepts, and are essential in understanding and working with various topics in pre-algebra.
Irrational Number: An irrational number is a type of real number that cannot be expressed as a simple fraction, meaning it cannot be written in the form of \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). These numbers have non-repeating, non-terminating decimal expansions, making them distinct from rational numbers. Common examples of irrational numbers include the square root of any prime number and the mathematical constant \(\pi\).
Natural Numbers: Natural numbers, also known as counting numbers, are the set of positive integers starting from 1 and continuing without end. They are the most basic and fundamental numbers used in mathematics, forming the foundation for numerical operations and quantification.
Non-terminating: A non-terminating number is a numerical value that does not have a finite number of digits after the decimal point. It continues on indefinitely without repeating in a pattern, making it impossible to express the number as a simple fraction. This concept is particularly relevant in the context of rational and irrational numbers.
Number Line: A number line is a visual representation of the number system, where numbers are arranged in a linear fashion along a horizontal or vertical axis. It serves as a fundamental tool in understanding and working with various numerical concepts, including whole numbers, integers, fractions, and rational and irrational numbers.
Pi: Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation never ends or repeats, and it is widely used in various mathematical and scientific applications, particularly in the study of geometry and trigonometry.
Pythagoras: Pythagoras was an ancient Greek mathematician and philosopher who is best known for his contributions to the study of rational and irrational numbers. He is particularly famous for the Pythagorean theorem, which describes the relationship between the sides of a right triangle.
Pythagorean Theorem: The Pythagorean Theorem is a fundamental relationship in geometry that describes the connection between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Rational Number: A rational number is a number that can be expressed as a ratio or fraction of two integers, where the denominator is not zero. Rational numbers are an important concept in both the topics of Rational and Irrational Numbers, as well as Integer Exponents and Scientific Notation.
Real Number: A real number is a number that can be represented on the number line. It includes all rational numbers (fractions and integers) as well as irrational numbers (such as pi and the square root of 2), and encompasses the complete set of numbers that we use in everyday life and mathematics.
Repeating Decimal: A repeating decimal is a decimal number in which one or more digits in the decimal part repeat infinitely. This pattern of repeating digits is a characteristic of certain fractions when expressed as a decimal.
Square Root: The square root, denoted by the symbol √, is a mathematical operation that finds the value that, when multiplied by itself, equals a given number. It represents the inverse operation of squaring a number.
Terminating: A terminating decimal is a decimal number that has a finite number of digits after the decimal point. It can be expressed as a fraction and eventually stops repeating. Terminating decimals are a type of rational number, as opposed to irrational numbers which have an infinite, non-repeating decimal representation.
Transcendental Numbers: Transcendental numbers are real numbers that are not the roots of any polynomial equation with integer coefficients. They cannot be expressed as a ratio of two integers, unlike rational numbers, and their digits do not repeat in a periodic pattern like algebraic irrational numbers. Transcendental numbers are an important concept in the study of rational and irrational numbers.
Whole Numbers: Whole numbers, also known as natural numbers, are the set of positive integers that begin with 1 and continue indefinitely. They are the most fundamental and commonly used numbers in mathematics, with wide applications in various fields.
π: π, also known as pi, is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation never repeats or terminates, and it is widely used in various mathematical and scientific applications.
© 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