Mixed Expressions

6 Key excerpts on "Mixed Expressions"

• 

  eBook - ePub

  Introductory Algebra

  • Chris Nord(Author)
  • 2021(Publication Date)
  • Chemeketa Press

    (Publisher)

  CHAPTER 4

  Expressions

  What is a Number?

  The English language is full of figures of speech — such as “beating around the bush” — that we use to explain ideas. These are phrases that have meaning beyond their literal, word-for-word interpretation. When we say we’re “beating around the bush,” we mean avoiding a major, important topic by addressing minor, unimportant topics instead. If someone asks us what we’re beating, we say something like “Oh, that’s just an expression,” a combination of words that work together to create meaning.

  In math, an expression is a combination of numbers, variables, and operation symbols that together have meaning. In math, the meaning of an expression is a number. In this chapter, we will distinguish between

  arithmetic expressions

  , which are expressions that don’t contain variables, and

  algebraic expressions

  , which do contain variables.

  4.1 Introduction to Expressions

  In this section, you will learn how to evaluate arithmetic and algebraic expressions. You will also learn how to represent and interpret measurements in scientific notation. Finally, we will officially introduce formulas and learn two important ones — the area formula for a rectangle and the area formula for a triangle.

  A. Arithmetic Expressions

  An arithmetic expression is a combination of numbers and operation symbols. This can be as simple as a single number, such as 23, or it can be a much more complicated combination of numbers and operation symbols, such as .

  An arithmetic expression can always be evaluated by performing each of the operations correctly and in the correct order. Once an arithmetic expression has been evaluated, it can then be replaced with its number value. Both the original expression and its evaluated form have the same meaning.

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  eBook - PDF

  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  We will learn many new words in this chapter but since algebra is generalized arithmetic, some of what was said in Chapter 1 (such as rules of signs) will be re- peated here. I 2 R V 2 > R x n 3 2 x n  (x)(x)(x)(x)...(x) ¯˚˚˚˘˚˚˚˙ x 2  (x)(x) x 2 3 2  (3)(3) 3 2 2 FIGURE 2–1 V I R Section 1 ◆ Algebraic Expressions 63 We will redo the basic operations of addition, subtraction, and so on, but now with symbols rather than numbers. Learn this material well, for it is the foundation on which later chapters rest. 2–1 Algebraic Expressions Every field has its own special terms, and algebra is no exception. So let’s start by learning some new terms and new words that we’ll be using throughout our study of mathematics. Mathematical Expressions A mathematical expression is a grouping of mathematical symbols, such as signs of operation, numbers, and letters. ◆◆◆ Example 1: The following are mathematical expressions: (a) (b) 4 sin 3x (c) ◆◆◆ Algebraic Expressions An algebraic expression is one containing only algebraic symbols and operations (addition, subtraction, multiplication, division, roots, and powers), such as in Ex- ample 1(a). All other expressions are called transcendental, such as Examples 1(b) and (c). We will study those later. Terms The plus and the minus signs divide an expression into terms. ◆◆◆ Example 2: The expression has three terms. first term second term third term ◆◆◆ Equations None of the expressions in Example 1 contains an equal sign . When two ex- pressions are set equal to each other, we get an equation. ◆◆◆ Example 3: The following are equations: (a) (b) (c) ◆◆◆ Constants and Variables A constant is a quantity that does not change in value in a particular problem. It is usually a number, such as 8, 4.67, or . A variable is a quantity that may change during a particular problem. A variable is usually represented by a letter from the end of the alphabet (x, y, z, etc.).

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  eBook - PDF

  Technical Mathematics with Calculus

  • Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  What is Algebra? Understanding the fundamental rules of algebra is important in order to calculate results for a wide variety of situations in science and technology. Algebra is a language of mathematics that is based on generalized arithmetic using letters to represent numbers in expressions, equations, or formulas. 2 ⋅ 2 ⋅ 2 ⋅ 2 = 2 4 translates to a ⋅ a ⋅ a ⋅ a = a 4 and more generally a a a a a a n n factors        ( )( )( )( ) ( ) = The material in this chapter is the foundation on which other later chapters in this book are built, so your understanding of the concepts in this chapter is important. 2 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Identify and understand all the components (literal terms, constants, and variables) that make up algebra. • Add and subtract algebraic expressions and identify groups using symbols like parentheses and brackets. • Complete operations using integral exponents and remove groups of symbols in algebraic expressions. • Complete operations with algebraic expressions using multiplication. • Complete operations with algebraic expressions using division. Introduction to Algebra 2–1 Algebraic Expressions Mathematical Expressions A mathematical expression is a grouping of mathematical symbols, such as signs of operation, numbers, and letters. ◆◆◆ Example 1: The following are mathematical expressions: (a) x 2 - 2x + 3 (b) x 5 sin 3 ⋅ (c) + x e 5 log x 2 ◆◆◆ 41 Section 2–1 ◆ Algebraic Expressions Algebraic and Transcendental Expressions An algebraic expression is one containing algebraic symbols and operations (addition, subtrac- tion, multiplication, division, roots, and powers), such as in Example 1(a). A transcendental expression, such as in Examples 1(b) and (c), contains trigonometric, exponential, or logarith- mic functions. Equations None of the expressions in Example 1 contains an equal sign ( = ).

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  eBook - ePub

  Common Core Standards for Middle School Mathematics

  A Quick-Start Guide

  • Amitra Schwols, Kathleen Dempsey, John Kendall(Authors)
  • 2013(Publication Date)
  • ASCD

    (Publisher)

  Chapter 4

  Expressions and Equations

  . . . . . . . . . . . . . . . . . . . .

  The introduction to the Common Core standards' high school Algebra domain provides some useful definitions to help differentiate the two mathematical terms expression and equation:

  An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function…. An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. (CCSSI, 2010c, p. 62)

  The middle school Expressions and Equations (EE) domain provides a critical bridge between content in the Operations and Algebraic Thinking domain in earlier grades and algebraic content students will encounter in high school. Figure 4.1 shows an overview of the Expressions and Equations domain's clusters and standards by grade level.

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  eBook - PDF

  PreStatistics

  • Donald Davis, William Armstrong, Mike McCraith, , Donald Davis, William Armstrong, Mike McCraith(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  If we place a real number in front or behind a variable, we call that number a coefficient . For our purposes, many of the coefficients will be represented by rational numbers. Together, variables and coefficients make up algebraic terms . Terms that are separated with addition and subtraction signs are called algebraic expressions . Objectives 1 Differentiate between an Expression and an Equation 2 Translate English Sentences into Mathematical Equations 3 Identify Strict Inequalities 4 Classify Inclusive Inequalities 5 Determine Possible Variable Values Based on Inequalities and the Phrases “At Most” and “At Least” 6 Determine Possible Variable Values for Compound Inequalities ■ ■ A variable is a letter used to represent an unknown quantity. ■ ■ A coefficient is a real number that is multiplied by a variable. ■ ■ An algebraic term is the product or quotient of a variable and a coefficient. ■ ■ An algebraic expression is the sum or difference of algebraic terms. Vocabulary An equation is a mathematical statement in which one algebraic expression is set equal to a constant or another algebraic expression. Equation As an example, in the algebraic expression 4 2 3.5 x x is the variable and the number 2 3.5 is the coefficient. The number 4 is not multiplied by a variable, so we call it a constant . If we set one algebraic expression equal to a constant or another algebraic expression, the result is an equation . As a result we determine that 4 2 3.5 x 5 2 31 is an equation, as is 4 2 3.5 x 5 15.5 1 6.1 x . In Example 1, we practice differentiating between an algebraic expression and an equation. EXAMPLE 1 Differentiating between an Algebraic Expression and an Equation Classify each of the following as an algebraic expression or an equation. Then identify the variables, coefficients, and constants. a. 1.7 5 35.8 1 z (11.2) b. s 21.5 SECTION 2.1 • Translating English to Algebra: Expressions, Equations, and Inequalities 57 Perform the Mathematics a.

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  Available until 8 Feb |Learn more

  Introductory Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  2. –20° 40° Melbourne New Delhi London Toronto 30° 20° 10° –10° 0° (zero degrees) • • • • 48 Module 2 • The four basic algebraic operations Adding and subtracting in algebra So far, we have added and subtracted numbers. We know that in algebra we use letters as placeholders. We have also learnt in Module 1 that a term has a coefficient, a coefficient has a sign, a term has a base with or without an exponent and the base can be raised to a power. –3 a 2 term sign coefficient to the power exponent or index base Pre-knowledge Definitions Algebraic expression An algebraic expression is one or more algebraic terms in a phrase. It can include variables , constants and operating symbols, such as plus and minus signs. It is only a phrase, not a whole sentence, so it does not include an equal sign . Example: 3 x 2 – 2 y + 7 xy + y 2 – 5 Term Terms are elements that are separated by a plus or a minus sign only . The example above has five terms: 3 x 2 , –2 y , 7 xy , y 2 and –5. The sign in front of a term belongs to that term. Terms could consist of variables and coefficients, or constants. Variable A variable is a letter or symbol that represents an unknown value. In algebraic expressions letters represent variables. These letters are actually numbers in disguise. In the expression 3 x 2 – 2 y + 7 xy + y 2 – 5 the variables are x and y . We call these letters var iables because the numbers they represent can vary : we can substitute one or more numbers for the letters in the expression. Coefficient A coefficient is the number, together with its sign, which is multiplied by the variable in an algebraic expression. In 3 x 2 – 2 y + 7 xy + y 2 – 5 the coefficient of the first term is 3 , the coefficient of the second term is – 2 , the coefficient of the third term is 7 and the coefficient of the fourth term is 1 . If a term consists of only variables, as in y 2 , its coefficient is 1. Constant A constant is a number that cannot change its value.

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