Elementary Algebra
Integers are the foundation of algebra, encompassing positive and negative whole numbers. They're crucial for representing real-world quantities like temperature, elevation, and finances. Understanding integers helps us grasp more complex mathematical concepts.
Operations with integers, like addition and subtraction, follow specific rules based on signs and absolute values. Mastering these operations is key to solving equations, working with algebraic expressions, and applying math to everyday situations.
Understanding Integers and Their Operations
Negative numbers in context
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- Represent quantities below zero or in opposite directions
- Temperature: -5°C signifies 5 degrees below freezing point
- Elevation: -50 m indicates 50 meters below sea level (Death Valley)
- Finance: -100 from an account
- Opposite of a number has same absolute value but different sign
- Opposite of 7 is -7 and vice versa
- Opposite of -2.5 is 2.5
Applications of absolute value
- Measures distance of a number from zero on number line regardless of sign
- Notation: where x is the number
- since 8 is 8 units from zero (positive direction)
- because -4 is 4 units from zero (negative direction)
- Useful in solving equations and inequalities
- means the distance between x and 3 is 6 units
- implies -5 ≤ y ≤ 5 since y can be at most 5 units from zero
Performing Operations with Integers
Addition with integers
- Same signs: add absolute values, keep common sign
- Different signs: subtract smaller absolute value from larger, result takes sign of larger absolute value
- Additive inverse property: for any integer a
Subtraction as adding opposites
- Definition: for any integers a and b
- Subtracting a positive = adding a negative
- Subtracting a negative = adding a positive
- Examples with different signs:
Integer Arithmetic and Algebraic Expressions
Order of operations
- Follows the PEMDAS rule: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
- Essential for correctly evaluating expressions with multiple operations
Signed numbers
- Integers and their operations form the basis for working with all signed numbers, including fractions and decimals
Algebraic expressions
- Combine variables and constants using integer arithmetic
- Example: 2x - 3y + 5 represents addition and subtraction of terms with variables
Key Terms to Review (22)
Absolute Value: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of whether the number is positive or negative. It is a fundamental operation that is used in various mathematical contexts, including integer operations, real number properties, and more.
Negative Integers: Negative integers are whole numbers that are less than zero. They represent quantities below the starting point or origin, and are used to denote losses, decreases, or positions to the left of zero on the number line.
Algebraic Expressions: Algebraic expressions are mathematical representations that combine variables, constants, and operations to express relationships and quantities. They are a fundamental component of algebra, used to model and solve a variety of problems involving unknown values.
Additive Inverse: The additive inverse of a number is the value that, when added to the original number, results in a sum of zero. It represents the opposite or negative value of a given number.
Addition: Addition is a fundamental mathematical operation that combines two or more numbers or quantities to find their sum. It is a crucial concept that underpins various algebraic and numerical operations, allowing for the representation and manipulation of quantities in a meaningful way.
Order of Operations: The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed to ensure consistent and unambiguous results. It is a fundamental concept in algebra that governs how expressions are evaluated.
Integers: Integers are a subset of the real number system, consisting of all positive and negative whole numbers, including zero. They are fundamental to various mathematical operations and concepts, particularly in the context of addition, subtraction, and the properties of real numbers.
Signed Numbers: Signed numbers are a way of representing positive and negative quantities using a sign (+ or -) to indicate the direction or value of a number. They are essential in the context of adding and subtracting integers, as the sign determines whether the number is to be added or subtracted.
Sum: The sum is the result of adding two or more numbers together. It represents the total or combined value of the addends, which are the individual numbers being added.
Opposite: The opposite of a number or quantity is the value that is equidistant from zero on the number line but in the opposite direction. In the context of addition and subtraction of integers, the opposite of a number represents the additive inverse, which when combined with the original number, results in a sum of zero.
+: The plus sign (+) is a mathematical symbol used to indicate addition, which is the process of combining two or more numbers or quantities to find their sum. This fundamental operation is essential in various areas of mathematics, including whole numbers, integers, equations, polynomials, and factoring.
Zero: Zero is a numerical value that represents the absence of quantity or magnitude. It is the starting point from which positive and negative numbers are defined, serving as a crucial concept in mathematics and various mathematical operations.
Subtraction: Subtraction is a fundamental mathematical operation that involves removing or taking away a quantity from another quantity. It is one of the four basic arithmetic operations, along with addition, multiplication, and division. Subtraction is essential in various mathematical contexts, including algebra, integers, and real numbers.
Positive Integers: Positive integers are the set of whole numbers greater than zero. They are the most fundamental numbers used in arithmetic and algebra, forming the basis for many mathematical operations and concepts.
Opposite Method: The opposite method is a strategy used in integer addition and subtraction operations. It involves applying the opposite operation to the given numbers in order to solve the problem, relying on the inverse relationship between addition and subtraction.
-: The minus sign, or negative sign, is a mathematical symbol used to indicate subtraction or a negative value. It is a fundamental operation in algebra and arithmetic, and it plays a crucial role in understanding and manipulating integers, equations, and polynomials.
Number Line: A number line is a visual representation of the number system, where numbers are arranged sequentially along a straight line. It serves as a fundamental tool in understanding and working with various mathematical concepts, including whole numbers, integers, fractions, and real numbers.
Counting Method: The counting method is a fundamental technique used in mathematics to determine the number of possible outcomes or arrangements in a given scenario. It involves systematically enumerating all possible cases to arrive at the total count.
Difference: The difference between two numbers is the amount by which one number exceeds or falls short of the other. It is the result of subtracting one number from another and represents the distance or separation between the two values.
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 the terms in an expression to be rearranged without changing the final value.
Associative Property: The associative property is a fundamental mathematical concept that describes how grouping of operands in an expression does not affect the final result. It is a key property that holds true for various mathematical operations, including addition, multiplication, and others.
Integer Arithmetic: Integer arithmetic refers to the basic operations of addition, subtraction, multiplication, and division performed on integers, which are whole numbers that can be positive, negative, or zero. It forms the foundation for more advanced mathematical concepts and is essential for understanding and working with numbers in various contexts.