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3.4 Rational Numbers

3 min read•june 18, 2024

Rational numbers are the building blocks of everyday math. They're fractions, decimals, and percentages we use to measure, compare, and calculate. Understanding how to work with them is crucial for solving real-world problems.

From adding fractions to converting percentages, rational numbers are versatile tools. We'll learn how to perform operations, switch between different forms, and apply these skills to practical situations like calculating discounts or converting units.

Rational Number Operations and Applications

Operations with rational numbers

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  • and of rational numbers involves finding a for fractions with unlike denominators (e.g., 12+13\frac{1}{2} + \frac{1}{3}), then adding or subtracting the numerators while keeping the common
  • of rational numbers is performed by multiplying the numerators and denominators separately (e.g., 23×34=612=12\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}), then simplifying the resulting if possible
  • of rational numbers is carried out by multiplying the dividend by the of the divisor (e.g., 34÷25=34×52=158\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}), then simplifying the resulting fraction if possible
  • () dictates the sequence in which operations should be performed: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)

Representations of rational numbers

  • Converting fractions to decimals is done by dividing the by the denominator (e.g., 34=0.75\frac{3}{4} = 0.75)
  • Converting decimals to fractions involves writing the as a fraction over a power of 10, then simplifying if possible (e.g., 0.6=610=350.6 = \frac{6}{10} = \frac{3}{5})
  • Converting fractions to percentages is achieved by multiplying the fraction by 100 and adding the % symbol (e.g., 25=40%\frac{2}{5} = 40\%)
  • Converting percentages to fractions requires writing the as a fraction over 100, then simplifying if possible (e.g., 75%=75100=3475\% = \frac{75}{100} = \frac{3}{4})
  • Converting mixed numbers to improper fractions involves multiplying the whole number by the denominator, adding the numerator, and keeping the denominator (e.g., 213=732\frac{1}{3} = \frac{7}{3})
  • Converting improper fractions to mixed numbers is done by dividing the numerator by the denominator to get the whole number and remainder, then writing the remainder as a fraction (e.g., 114=234\frac{11}{4} = 2\frac{3}{4})

Properties of rational numbers

  • : Two rational numbers are equivalent if they represent the same value (e.g., 12\frac{1}{2} and 24\frac{2}{4})
  • : Between any two rational numbers, there exists infinitely many other rational numbers
  • : Rational numbers can be arranged in order on a , with each number having a unique position
  • Rational numbers form a subset of the real , which also includes

Applications of rational numbers

  • Percentage problems involve:
    1. Finding a percentage of a number: number×percentage100\text{number} \times \frac{\text{percentage}}{100} (e.g., 25% of 80 is 80×25100=2080 \times \frac{25}{100} = 20)
    2. Finding what percentage one number is of another: partwhole×100%\frac{\text{part}}{\text{whole}} \times 100\% (e.g., 15 is 30% of 50 because 1550×100%=30%\frac{15}{50} \times 100\% = 30\%)
    3. Finding the whole given a part and a percentage: partpercentage100\frac{\text{part}}{\frac{\text{percentage}}{100}} (e.g., if 40 is 80% of a number, the number is 4080100=50\frac{40}{\frac{80}{100}} = 50)
  • Unit conversions require setting up a with the given units and the desired units, then solving the proportion for the unknown quantity (e.g., converting 5 feet to inches: 5 feet1=x inches12 inches/foot\frac{5 \text{ feet}}{1} = \frac{x \text{ inches}}{12 \text{ inches/foot}}, so x=60x = 60 inches)
  • Proportions are solved by writing the proportion as two equivalent fractions, cross-multiplying and solving for the unknown variable, and verifying the solution by plugging it back into the original proportion (e.g., if 23=x12\frac{2}{3} = \frac{x}{12}, then 2×12=3×x2 \times 12 = 3 \times x, so x=8x = 8)

Key Terms to Review (39)

Addition: Addition is the mathematical operation of combining two or more numbers to find their total. This basic arithmetic function forms the foundation for more complex calculations, including those involving rational numbers, and is crucial for establishing order in mathematical expressions through operations.
Associative property: The associative property is a fundamental principle in mathematics that states the way numbers are grouped in addition or multiplication does not affect the final result. This means that when adding or multiplying three or more numbers, you can change the grouping of the numbers without changing the sum or product. This property emphasizes the flexibility in computations and helps simplify expressions, making it crucial in various mathematical contexts.
Closure Property: The closure property states that when you perform an operation on two elements from a set, the result is also an element of that same set. This concept is essential in understanding how different sets of numbers behave under operations like addition, subtraction, multiplication, and division. For rational numbers specifically, it helps to establish that performing these operations will yield another rational number, reinforcing the integrity of this number system.
Common denominator: A common denominator is a shared multiple of the denominators of two or more fractions, allowing those fractions to be compared or combined through addition or subtraction. By finding a common denominator, fractions can be converted into equivalent forms that make it possible to perform arithmetic operations. This concept is essential for understanding how to work with rational numbers and helps simplify complex fraction problems.
Commutative Property: The commutative property is a fundamental mathematical principle stating that the order in which two numbers are added or multiplied does not change the result. This property highlights that rearranging the elements in addition or multiplication still yields the same outcome, making calculations more flexible and efficient. It plays a crucial role in simplifying expressions and solving equations across various mathematical contexts.
Dealer preparation fee: A dealer preparation fee is an additional charge by a car dealership to cover the costs of making a new vehicle ready for sale. This typically includes cleaning, inspecting, and detailing the car before delivery to the buyer.
Decimal: A decimal is a number that represents a fraction whose denominator is a power of ten, typically expressed in the form of digits with a decimal point separating the whole number part from the fractional part. Decimals are essential for expressing rational numbers in a format that allows for easier calculations, comparisons, and conversions, especially in various base systems and when working with ratios and proportions.
Denominator: The denominator is the bottom part of a fraction that indicates how many equal parts the whole is divided into. In the context of rational numbers, it plays a crucial role in determining the value of the fraction and can affect operations like addition, subtraction, multiplication, and division. Understanding the denominator helps in recognizing how different fractions relate to one another, particularly when comparing sizes or performing calculations.
Density: Density is a measure of how much mass is contained within a given volume, typically expressed as mass per unit volume. It plays a crucial role in understanding how substances interact with one another, such as whether they will float or sink in a fluid, and is fundamental in various scientific calculations and applications.
Division: Division is a mathematical operation that represents the process of splitting a quantity into equal parts or determining how many times one number is contained within another. This operation plays a critical role in various mathematical concepts, providing a foundation for understanding relationships between numbers and forming the basis for operations involving fractions, ratios, and proportions. Division interacts with other operations like multiplication and is essential in contexts like calculating averages and understanding rational numbers.
Division in a base: Division in a base is the process of dividing numbers expressed in a specific numeral system, such as binary, octal, or hexadecimal. It follows similar principles to division in the decimal system but requires adjustments based on the base.
Equivalence: Equivalence refers to the concept of two objects or quantities being equal in value, meaning, or function, even if they may appear different. In the context of rational numbers, equivalence can be seen through different representations of the same value, such as fractions that simplify to the same number or decimal representations that convey the same quantity. This idea helps establish a deeper understanding of relationships between numbers and how they can be transformed or represented in various forms without changing their inherent value.
Fraction: A fraction represents a part of a whole and is expressed as the quotient of two integers, where the numerator indicates how many parts are being considered and the denominator indicates the total number of equal parts that make up the whole. Fractions are fundamental in understanding rational numbers, which encompass all numbers that can be expressed as fractions, including proper, improper, and mixed numbers. Additionally, fractions play a crucial role in comparing quantities and understanding ratios and proportions, which involve relationships between two or more numbers.
Fractional form: Fractional form is the representation of a number as a ratio of two integers, where the numerator is divided by the denominator. It is often used to express parts of a whole or to simplify complex calculations involving percentages and ratios.
Improper fraction: An improper fraction is a type of fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means that it represents a value that is equal to or greater than one whole. Improper fractions can also be converted into mixed numbers, which combine a whole number and a proper fraction.
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.
Mixed number: A mixed number is a number that combines a whole number and a proper fraction, representing values greater than one. This format allows for a more intuitive understanding of quantities that are not whole, making it easier to visualize and use in various mathematical contexts. Mixed numbers are often used in everyday situations like cooking measurements and construction.
Multiplication: Multiplication is a mathematical operation that combines groups of equal sizes to find the total quantity. It serves as a fundamental building block in mathematics, allowing us to simplify and solve problems involving repeated addition, scaling quantities, and working with rational numbers. This operation is also essential for understanding various mathematical concepts, including the order of operations, base systems, and financial calculations such as interest.
Multiplication and division in bases: Multiplication and division in bases involve performing these arithmetic operations within non-decimal numeral systems, such as binary (base-2) or hexadecimal (base-16). These operations follow similar principles to base-10 but require adjustments for different digits and place values.
Number line: A number line is a visual representation of numbers arranged in a straight horizontal line, where each point corresponds to a real number. It serves as a tool to understand numerical relationships and operations, providing a clear way to visualize integers, rational numbers, and real numbers. The concept of the number line is crucial for solving equations and inequalities, helping to illustrate how different types of numbers are related and how they can be compared or combined.
Number system: A number system is a way of representing and classifying numbers using a consistent set of symbols and rules. It provides the framework for understanding different types of numbers, such as whole numbers, integers, rational numbers, and real numbers. Number systems play a crucial role in mathematical operations and can be expressed in various bases, impacting how multiplication and division are performed.
Numerator: The numerator is the top part of a fraction that indicates how many parts of a whole are being considered. It plays a crucial role in determining the value of the fraction, as it tells us what portion of the whole is being taken or represented. The numerator works together with the denominator, which is the bottom part of the fraction, to provide a complete understanding of the ratio between two quantities.
Order of operations: Order of operations is a set of rules that specifies the correct sequence to evaluate a mathematical expression. It ensures consistency and avoids ambiguity in solving equations.
Order of Operations: The order of operations is a mathematical rule that dictates the sequence in which calculations should be performed to ensure consistent and correct results. This rule is essential in simplifying expressions, especially when they involve multiple operations like addition, subtraction, multiplication, division, and the use of parentheses. Understanding this concept is crucial for accurately solving problems involving rational numbers and algebraic expressions.
Ordering: Ordering refers to the arrangement or sequence of elements based on a certain criterion, which can be numerical, alphabetical, or based on other properties. This concept is fundamental in understanding how different entities relate to one another and is crucial for organizing information clearly and effectively. It allows for the comparison of values and the establishment of hierarchy within a set.
PEMDAS: PEMDAS is an acronym that represents the order of operations used in mathematics to solve expressions correctly. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Understanding this order is essential when dealing with calculations involving rational numbers and is foundational in systems like the Hindu-Arabic positional system, ensuring that calculations are performed systematically and accurately.
Percent: A percent is a way to express a number as a fraction of 100, often used to compare relative sizes or amounts. It represents a portion of a whole and is commonly utilized in various contexts such as finance, statistics, and everyday life to convey comparisons, rates, and changes. Understanding how to calculate and interpret percent is essential for making informed decisions based on quantitative data.
Percentage: A percentage is a way of expressing a number as a fraction of 100, denoting a proportion or ratio in relation to a whole. It is widely used in various calculations to determine parts of a total, making it crucial in fields such as finance, statistics, and everyday decision-making. Understanding percentages allows for clearer interpretations of data, comparisons, and assessments in different contexts.
Proper Fraction: A proper fraction is a type of fraction where the numerator (the top number) is less than the denominator (the bottom number). This means that proper fractions represent values less than one, indicating that the part is smaller than the whole. Understanding proper fractions is crucial as they are foundational in arithmetic and help in grasping more complex concepts in rational numbers.
Proportion: A proportion is an equation that states that two ratios are equal. It reflects the relationship between quantities and can be used to express how one quantity compares to another, whether through scaling, sharing, or finding parts of a whole. This concept connects to various mathematical applications, including rational numbers, where it helps understand comparisons, and visualizations, where proportions can illustrate data relationships.
Ratio: A ratio is a mathematical expression that compares two quantities by division, indicating how many times one value contains or is contained within the other. Ratios can represent relationships between numbers, allowing for the understanding of proportions, comparisons, and equivalences in various contexts, such as fractions and percentages. They serve as foundational concepts in various areas of mathematics, illustrating the relationship between parts and wholes.
Rational number: Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. They can be either positive, negative, or zero.
Reciprocal: A reciprocal is a mathematical term that refers to the inverse of a number, which is found by dividing 1 by that number. This concept is crucial in understanding how fractions, division, and multiplication interact within the realms of rational and real numbers. When dealing with reciprocals, it’s important to remember that every non-zero number has a reciprocal, and the product of a number and its reciprocal is always 1.
Repeating decimal: A repeating decimal is a decimal representation of a number that has a digit or group of digits that repeat infinitely. This type of decimal occurs when a fraction is expressed in decimal form and can be traced back to rational numbers, where the numerator and denominator have no common factors other than one. Repeating decimals highlight the relationship between fractions and their decimal equivalents, showcasing the nature of rational numbers as they can be expressed in both forms.
Simplification: Simplification is the process of reducing expressions to their simplest form by eliminating unnecessary elements or combining like terms. This technique allows for easier manipulation and understanding of mathematical expressions, making it crucial in solving problems accurately and efficiently. Simplification is commonly applied to rational numbers and algebraic expressions, ensuring that calculations can be performed more easily.
Subtraction: Subtraction is the mathematical operation of finding the difference between two numbers, where one number is taken away from another. It is often represented by the minus sign (-) and plays a crucial role in various mathematical processes, including simplifying expressions, solving equations, and managing real-world situations like budgeting and finance.
Subtraction in bases: Subtraction in bases involves performing arithmetic subtraction in number systems other than base 10, such as binary (base 2), octal (base 8), or hexadecimal (base 16). It requires understanding how to borrow and subtract digits within those specific bases.
Terminating decimal: A terminating decimal is a decimal representation of a number that has a finite number of digits after the decimal point. This means that the decimal expansion comes to an end, rather than continuing infinitely. Terminating decimals are a specific subset of rational numbers, which can be expressed as the quotient of two integers.
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3.5 Irrational Numbers