Rational and Irrational Numbers: Cheat Sheet

Essential Concepts

  • Rational numbers are numbers that can be expressed as a fraction where both the numerator (top number) and the denominator (bottom number) are whole numbers, and the denominator is not zero.
  • Rational numbers come in various forms, including whole numbers, fractions, and decimals that repeat or terminate.
  • Simplifying a fraction means rewriting it in its simplest form where the numerator and denominator share no common factors other than [latex]1[/latex]. This involves identifying the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number.
  • Equivalent fractions are different fractions that represent the same quantity or value. They may look different but have the same value when simplified. This property is based on the idea that multiplying or dividing both the numerator and denominator of a fraction by the same non-zero number results in a fraction equivalent to the original.
  • The least common multiple of two or more numbers is the smallest number that is a multiple of all the given numbers.
  • The prime factorization method involves breaking down each number into its prime factors (the prime numbers that multiply together to give the original number). The product obtained by this process is the least common multiple of the original set of numbers.
  • Arithmetic Operations with Fractions:
    • Addition and Subtraction: This involves finding a common denominator for the fractions involved and then adding or subtracting the numerators while keeping the denominator the same. It’s important to simplify the resulting fraction if possible.
    • Multiplication: Multiplying fractions requires multiplying the numerators together and the denominators together. The product should be simplified to its lowest terms.
    • Division: Dividing fractions involves multiplying the first fraction (dividend) by the reciprocal (inverse) of the second fraction (divisor). Simplification of the result is often necessary.
  • An improper fraction, where the numerator is larger than the denominator, can be converted to a mixed number. This is done by dividing the numerator by the denominator to find the whole number part and then expressing the remainder as a fraction.
  • To convert a mixed number to an improper fraction, multiply the whole number part by the denominator of the fractional part, add this to the numerator of the fractional part, and place the result over the original denominator.
  • Irrational numbers are numbers that cannot be expressed as a simple fraction, meaning their decimal form is non-repeating and non-terminating.
  • Irrational numbers can be distinguished since rational numbers can be written as fractions. Examples of irrational numbers include [latex]\sqrt{2}, \pi, e.[/latex] Recognizing patterns, such as non-repeating, non-ending decimals, is key.
  • A non-perfect square number is a number that does not have a square root that is a whole number. The square root of such a number is an irrational number (a non-terminating, non-repeating decimal).
  • Non-perfect squares can often be partially simplified using radicals. This involves expressing the square root in the simplest radical form by factoring out the square root of any perfect square factors.
  • Arithmetic Operations with Irrational Numbers:
    • Addition and Subtraction: These operations can be performed when irrational numbers are in a similar form. Often, it involves combining like terms or using approximation.
    • Multiplication and Division: Multiplying and dividing irrational numbers typically involves applying the properties of radicals and, in some cases, rationalizing the denominator in division.
    • Handling Mixed Operations: The operations with irrational numbers might require a combination of simplification, rationalization, and approximation, especially when mixed with rational numbers.
  • Real numbers are numbers that are either rational or irrational.
  • Categorizing Real Numbers:
    • Counting Numbers (Natural Numbers): These are the basic numbers used for counting ([latex]1, 2, 3, ...[/latex]). They do not include zero or any negative numbers, fractions, or decimals.
    • Whole Numbers: This category includes all counting numbers plus zero ([latex]0, 1, 2, 3, ...[/latex]). Like counting numbers, whole numbers do not include fractions, decimals, or negatives.
    • Integers: Integers extend whole numbers to include negative numbers. This set includes [latex]..., -3, -2, -1, 0, 1, 2, 3, ...[/latex]
    • Rational Numbers: These are numbers that can be expressed as a fraction of two integers (where the denominator is not zero), including integers, whole numbers, counting numbers, and fractions or decimals that terminate or repeat.
    • Irrational Numbers: Numbers that cannot be expressed as a simple fraction, with non-repeating, non-terminating decimals. Examples include [latex]\sqrt{2}, \pi, e.[/latex]
  • Properties of Real Numbers:
    • Commutative Property: This property states that the order in which two numbers are added or multiplied does not change the sum or product. For addition, [latex]a + b = b + a[/latex], and for multiplication, [latex]a \times b = b\times a[/latex].
    • Associative Property: This property suggests that the way numbers are grouped in addition or multiplication does not change their sum or product. For addition, [latex](a + b) + c = a + (b + c)[/latex], and for multiplication, [latex]a \times (b \times c) = (a \times b) \times c[/latex].
    • Distributive Property: This property demonstrates how multiplication is distributed over addition or subtraction. It is expressed as [latex]a \times (b + c) = a \times b + a \times c[/latex].
    • Identity Property: This property identifies an element that, when combined with another number, does not change the original number. For addition, the additive identity is [latex]0[/latex], shown as [latex]a + 0 = a[/latex]. For multiplication, the multiplicative identity is [latex]1[/latex], shown as [latex]a \times 1 = a[/latex].
    • Inverse Property: This property involves an element that, when combined with another number, results in the identity element. For addition, the additive inverse of a number is its negative, which sums to zero: [latex]a + (-a) = 0[/latex]. For multiplication, the multiplicative inverse is its reciprocal, which multiplies to 1: [latex]a \times \frac{1}{a} = 1[/latex] (for [latex]a \neq 0[/latex]).

Glossary

common denominator

when two rational numbers have the same bottom number

improper fraction

when the numerator in a fraction is larger than the denominator

irrational number

a number that cannot be written as the ratio of two integers

mixed number

an improper fraction written as an integer plus a fraction

rational number

a number that can be written in the form [latex]{\Large\frac{p}{q}}[/latex], where [latex]p[/latex] and [latex]q[/latex] are integers and [latex]q\ne o[/latex]

real numbers

numbers that are either rational or irrational

reciprocal

[latex]1[/latex] divided by the number

Key Equations

adding rational numbers with a common denominator

if [latex]b[/latex] is a non-zero integer, then [latex]\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}[/latex]

dividing rational numbers

if [latex]b[/latex], [latex]c[/latex] and [latex]d[/latex] are non-zero integers, then [latex]\frac{b}{a} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}[/latex]

dividing two square roots

[latex]\sqrt{a} \div \sqrt{b} = \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}[/latex]

equivalent fractions property

if [latex]a,b,c[/latex] are numbers where [latex]b\ne 0,c\ne 0[/latex], then [latex]{\Large\frac{a}{b}}={\Large\frac{a\cdot c}{b\cdot c}}\text{ and }{\Large\frac{a\cdot c}{b\cdot c}}={\Large\frac{a}{b}}[/latex]

multiplying more than two rational numbers

[latex]\frac{a}{b} \times \frac{c}{d} \times \frac{e}{f}=\frac{a \times c \times e}{b \times d \times f}[/latex]

multiplying rational numbers

if [latex]b[/latex] and [latex]d[/latex] are non-zero integers, then [latex]\frac{a}{b} \times \frac{c}{d}=\frac{a \times c}{b \times d}[/latex]

multiplying two square roots

[latex]\sqrt{a \times {b}}=\sqrt{a} \times \sqrt{b}[/latex]

subtracting rational numbers with a common denominator

if [latex]c[/latex] is a non-zero integer, then [latex]\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}[/latex]