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Study Guides > College Algebra

Summary: Real Numbers

Key Concepts

  • Rational numbers may be written as fractions or terminating or repeating decimals.
  • Determine whether a number is rational or irrational by writing it as a decimal.
  • The rational numbers and irrational numbers make up the set of real numbers. A number can be classified as natural, whole, integer, rational, or irrational.
  • The order of operations is used to evaluate expressions.
  • The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties.
  • Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. They take on a numerical value when evaluated by replacing variables with constants.
  • Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression.

Glossary

algebraic expression constants and variables combined using addition, subtraction, multiplication, and division associative property of addition the sum of three numbers may be grouped differently without affecting the result; in symbols, [latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex] associative property of multiplication the product of three numbers may be grouped differently without affecting the result; in symbols, [latex]a\cdot \left(b\cdot c\right)=\left(a\cdot b\right)\cdot c[/latex] base in exponential notation, the expression that is being multiplied commutative property of addition two numbers may be added in either order without affecting the result; in symbols, [latex]a+b=b+a[/latex] commutative property of multiplication two numbers may be multiplied in any order without affecting the result; in symbols, [latex]a\cdot b=b\cdot a[/latex] constant a quantity that does not change value distributive property the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, [latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex] equation a mathematical statement indicating that two expressions are equal exponent in exponential notation, the raised number or variable that indicates how many times the base is being multiplied exponential notation a shorthand method of writing products of the same factor formula an equation expressing a relationship between constant and variable quantities identity property of addition there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, [latex]a+0=a[/latex] identity property of multiplication there is a unique number, called the multiplicative identity, 1, which, when multiplied by a number, results in the original number; in symbols, [latex]a\cdot 1=a[/latex] integers the set consisting of the natural numbers, their opposites, and 0: [latex]\{\dots ,-3,-2,-1,0,1,2,3,\dots \}[/latex] inverse property of addition for every real number [latex]a[/latex], there is a unique number, called the additive inverse (or opposite), denoted [latex]-a[/latex], which, when added to the original number, results in the additive identity, 0; in symbols, [latex]a+\left(-a\right)=0[/latex] inverse property of multiplication for every non-zero real number [latex]a[/latex], there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\frac{1}{a}[/latex], which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, [latex]a\cdot \frac{1}{a}=1[/latex] irrational numbers the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers natural numbers the set of counting numbers: [latex]\{1,2,3,\dots \}[/latex] order of operations a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations rational numbers the set of all numbers of the form [latex]\frac{m}{n}[/latex], where [latex]m[/latex] and [latex]n[/latex] are integers and [latex]n\ne 0[/latex]. Any rational number may be written as a fraction or a terminating or repeating decimal. real number line a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left. real numbers the sets of rational numbers and irrational numbers taken together variable a quantity that may change value whole numbers the set consisting of 0 plus the natural numbers: [latex]\{0,1,2,3,\dots \}[/latex]

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