1.1 Real Numbers: Algebra Essentials
3 min read•june 24, 2024
form the foundation of algebra, encompassing rational and . Understanding their subsets and representations on a is crucial for solving equations and graphing functions.
Algebraic operations and expressions build on , introducing variables and rules for manipulation. Mastering order of operations, properties of real numbers, and simplification techniques is essential for solving complex mathematical problems.
Real Number System
Subsets of real numbers
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- Real numbers encompass all rational and
- can be expressed as a ratio of two where
- are and their negatives ()
- include and zero ()
- Natural numbers (counting numbers) start at 1 and continue infinitely ()
- include and zero ()
- include () and ()
- are and their negatives ()
- Irrational numbers cannot be expressed as a ratio of two integers
- Include non-terminating, non-repeating decimals ()
- can be expressed as a ratio of two where
Representation of Real Numbers
- Number line: A visual representation of real numbers on a horizontal line
- : A way to represent a set of numbers using parentheses or brackets
- : A branch of mathematics that deals with the properties of collections of objects (including real numbers)
Algebraic Operations and Expressions
Order of operations application
- mnemonic device guides order of operations
- Parentheses: Perform operations within parentheses first
- : Evaluate exponents, powers, and roots
- Multiplication and Division: Multiply and divide from left to right
- Addition and Subtraction: Add and subtract from left to right
Properties of real numbers
- allows changing order of operands
- Addition:
- Multiplication:
- allows grouping operands differently
- Addition:
- Multiplication:
- distributes multiplication over addition
- leaves value unchanged when operating with identity element
- Addition:
- Multiplication:
- results in identity element when operating with inverse
- Addition:
- Multiplication: for
Evaluation of algebraic expressions
- Substitute given values for variables in expression
- Apply order of operations to simplify resulting expression
- Evaluate simplified expression to find final value
Simplification of complex expressions
- Combine by adding or subtracting coefficients of terms with same variables and exponents
- Factor out common factors from terms
- Simplify fractions by reducing numerator and denominator by (GCF)
- Apply properties of exponents
- Multiply powers with same base:
- Divide powers with same base:
- Power of a power:
- Power of a product:
- Power of a quotient: for
- Rationalize denominators containing by multiplying numerator and denominator by of denominator
- Use to represent the distance of a number from zero on the number line
Arithmetic Operations and Polynomials
- (addition, subtraction, multiplication, and division) are fundamental in algebraic manipulations
- Polynomials are expressions consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents
Key Terms to Review (55)
Absolute maximum: The absolute maximum of a function is the highest value that the function attains over its entire domain. It represents the peak point on the graph of the function.
Absolute value: Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is always non-negative and is denoted by vertical bars, e.g., $|x|$.
Absolute Value: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of the number's sign. It is a fundamental operation that is essential in understanding and working with real numbers, radicals, and sequences.
Algebraic expression: An algebraic expression is a mathematical phrase that includes numbers, variables, and operational symbols. It does not include an equality sign and represents a value.
Algebraic Expressions: Algebraic expressions are mathematical representations that combine variables, numbers, and operations to express relationships and quantities. They are a fundamental component of algebra, allowing for the manipulation and analysis of unknown values.
Arithmetic Operations: Arithmetic operations are the fundamental mathematical procedures used to manipulate numbers. They form the foundation for performing calculations and solving algebraic equations, which are essential in the study of real numbers and algebra essentials.
Associative Property: The associative property is a fundamental mathematical concept that describes the behavior of certain operations, such as addition and multiplication, where the grouping of the operands does not affect the final result. It allows for the rearrangement of the order of operations without changing the outcome.
Associative property of addition: The associative property of addition states that the way in which numbers are grouped when adding does not change their sum. Mathematically, this is expressed as $(a + b) + c = a + (b + c)$.
Commutative Property: The commutative property is a fundamental mathematical principle that states the order of the operands in an addition or multiplication operation does not affect the result. It allows for the rearrangement of terms without changing the overall value of the expression.
Commutative property of addition: The commutative property of addition states that changing the order of addends does not change the sum. Mathematically, if $a$ and $b$ are real numbers, then $a + b = b + a$.
Commutative property of multiplication: The commutative property of multiplication states that the order in which two numbers are multiplied does not affect the product. Mathematically, for any real numbers $a$ and $b$, $a \times b = b \times a$.
Complex conjugate: A complex conjugate of a complex number is obtained by changing the sign of its imaginary part. If the complex number is $a + bi$, its complex conjugate is $a - bi$.
Conjugate: A conjugate is a pair of complex numbers or algebraic expressions that have the same magnitude but opposite signs in their imaginary parts. This concept is crucial in the understanding of real numbers and complex numbers, as conjugates are used to simplify calculations and perform operations such as division.
Distributive property: The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. It is expressed as $a(b + c) = ab + ac$.
Distributive Property: The distributive property is a fundamental algebraic rule that allows for the simplification of expressions involving multiplication and addition or subtraction. It states that the product of a number and a sum is equal to the sum of the products of the number with each addend.
E: e, also known as Euler's number, is a fundamental mathematical constant that is the base of the natural logarithm. It is an irrational number that is approximately equal to 2.71828 and is widely used in mathematics, science, and engineering. The term 'e' is central to the understanding of exponential functions, logarithmic functions, and their properties, which are crucial concepts in college algebra.
Equation: An equation is a mathematical statement that asserts the equality of two expressions. It consists of two expressions separated by an equals sign ($=$).
Exponential notation: Exponential notation is a mathematical way to represent numbers as a base raised to an exponent. It simplifies the expression of very large or very small numbers.
Exponents: Exponents are mathematical notations that represent repeated multiplication of a number by itself. They are used to express very large or very small numbers concisely and to simplify algebraic expressions involving powers of the same base.
Factor by grouping: Factor by grouping is a method used to factor polynomials that involves rearranging and combining terms into groups that have a common factor. This technique is particularly useful for polynomials with four or more terms.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
Formula: A formula is a mathematical expression that describes the relationship between different quantities. It often includes variables, constants, and arithmetic operations.
Greatest common factor: The greatest common factor (GCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. It is useful in simplifying fractions, factoring polynomials, and solving equations.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides two or more integers without a remainder. It is a fundamental concept in algebra that is essential for understanding real numbers, factoring polynomials, working with rational expressions, solving quadratic equations, and analyzing power and polynomial functions.
Identity Property: The identity property is a fundamental concept in algebra that states that adding or multiplying a number by the additive or multiplicative identity, respectively, will result in the original number. This property is crucial in understanding and manipulating algebraic expressions and equations.
Identity property of addition: The identity property of addition states that adding zero to any number will yield the same number. In other words, for any real number $a$, $a + 0 = a$.
Identity property of multiplication: The identity property of multiplication states that any number multiplied by 1 remains unchanged. Mathematically, for any real number $a$, $a \times 1 = a$.
Integers: Integers are whole numbers that can be positive, negative, or zero. They do not have fractional or decimal parts.
Integers: Integers are a subset of real numbers that represent the whole numbers, both positive and negative, including zero. They are the basic building blocks of mathematics and are essential in understanding and working with various algebraic concepts.
Interval notation: Interval notation is a mathematical shorthand used to represent subsets of the real number line. It uses parentheses and brackets to describe the set's endpoints and whether those endpoints are included or excluded.
Interval Notation: Interval notation is a way of representing sets of real numbers using ordered pairs or inequalities. It is a concise and efficient method for describing the range of values that a variable can take on, which is essential in understanding concepts related to real numbers, linear inequalities, and domain and range.
Inverse Property: The inverse property refers to the relationship between an operation and its inverse operation, where the result of applying an operation and then its inverse operation is the original value. This concept is fundamental in the study of real numbers and algebra essentials.
Inverse property of addition: The inverse property of addition states that for every real number $a$, there exists a number $-a$ such that $a + (-a) = 0$. This property is essential in solving equations and simplifying expressions.
Inverse property of multiplication: The inverse property of multiplication states that any nonzero number multiplied by its reciprocal equals 1. Mathematically, for a number $a$, where $a \neq 0$, $a \cdot \frac{1}{a} = 1$.
Irrational numbers: Irrational numbers are real numbers that cannot be expressed as the ratio of two integers. Their decimal expansions are non-repeating and non-terminating.
Irrational Numbers: Irrational numbers are real numbers that cannot be expressed as a simple ratio of two integers. They have decimal expansions that never repeat or terminate, such as pi (π) and the square root of 2. Irrational numbers are an important concept in the study of real numbers and their properties, as well as in the context of radicals and rational exponents.
Like Terms: Like terms are algebraic expressions that have the same variable(s) raised to the same power. They can be combined by adding or subtracting their coefficients, as they represent the same type of quantity.
Natural Numbers: Natural numbers, also known as counting numbers, are the set of positive integers starting from 1 and continuing indefinitely. They are the most fundamental and intuitive numbers used for counting, ordering, and quantifying objects in the physical world.
Number Line: A number line is a visual representation of the real number system, where numbers are depicted as points along a horizontal line. It serves as a fundamental tool in understanding and working with various mathematical concepts, including real numbers, operations, and the relationships between them.
PEMDAS: PEMDAS is a mnemonic that represents the order of operations in mathematics, which is the sequence in which mathematical operations should be performed to solve an expression. The acronym stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
Pi (π): Pi (π) is a fundamental 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, including the study of real numbers, angles, trigonometry, and the graphing of trigonometric functions.
Polynomial: A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents. It can be written in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ where $a_n, a_{n-1}, ..., a_1, a_0$ are constants and $n$ is a non-negative integer.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers. Polynomials are fundamental in various areas of mathematics, including algebra, calculus, and the study of functions.
Radicals: Radicals, in the context of algebra, refer to the square root symbol (√) or other root symbols that indicate the extraction of a root from a number or expression. These symbols represent irrational numbers and are used to represent and manipulate quantities that cannot be expressed as simple fractions.
Rational numbers: Rational numbers are any numbers that can be expressed as the quotient or fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. They include fractions, integers, and finite or repeating decimals.
Rational Numbers: Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. This means any number that can be written in the form $$\frac{a}{b}$$, where 'a' and 'b' are integers and 'b' is not equal to zero, qualifies as a rational number. Rational numbers include integers, fractions, and terminating or repeating decimals, showcasing their diversity within the real number system.
Real numbers: Real numbers include all rational and irrational numbers, encompassing the entire set of numbers on the number line. They can be positive, negative, or zero.
Real Numbers: Real numbers are the set of all numbers that include both rational numbers (fractions and integers) and irrational numbers (numbers that cannot be expressed as a simple fraction). Real numbers are the foundation for algebra and are essential for understanding and working with complex numbers.
Repeating Decimals: Repeating decimals, also known as infinite decimals or recurring decimals, are decimal numbers where one or more digits in the decimal part repeat infinitely. They occur when a fraction cannot be expressed as a terminating decimal.
Set Theory: Set theory is a branch of mathematics that deals with the study of sets, which are collections of distinct objects. It provides a fundamental framework for understanding and analyzing relationships between different groups or categories of elements.
Terminating Decimals: Terminating decimals are numbers that have a finite number of digits after the decimal point. They are a type of rational number, meaning they can be expressed as the quotient of two integers. Terminating decimals are essential in understanding real numbers because they provide a clear representation of fractions, especially when comparing or performing operations with them.
Variable: A variable is a symbol used to represent an unknown or arbitrary number in mathematical expressions and equations. It allows for generalization and manipulation of algebraic formulas.
Whole numbers: Whole numbers are the set of non-negative integers, including zero. They are commonly used in counting and basic arithmetic operations.
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 and are essential for understanding the concepts of 1.1 Real Numbers: Algebra Essentials.