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  • https://math.libretexts.org/Bookshelves/Applied_Mathematics/Understanding_Elementary_Mathematics_(Harland)/05%3A_______Binary_Operations/5.01%3A_Operations_and_Properties
    This exercise set is designed to give you an understanding of what "binary operations" are, and to give you a deeper understanding for the commutative, associative and distributive properties. To do t...This exercise set is designed to give you an understanding of what "binary operations" are, and to give you a deeper understanding for the commutative, associative and distributive properties. To do this, we're going to define and work with some nonsense operations.
  • https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_1e_(OpenStax)/01%3A_Foundations/1.06%3A_Properties_of_Real_Numbers
    \[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \texti...\[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \textit{opposite } \text{add to zero.} \\ \\ \\ \textbf{of multiplication } \text{For any real number }a,a\neq 0 & a·\dfrac{1}{a}=1 \\ \;\;\;\;\;\dfrac{1}{a} \text{ is the } \textbf{multiplicative inverse} \text{ of }a \\ \;\;\;\; \text{A number and its } \textit{reciprocal} \text{ multiply to one.} \end…
  • https://math.libretexts.org/Courses/Las_Positas_College/Foundational_Mathematics/13%3A_Additional_Foundational_Content/13.07%3A_Foundations/13.7.06%3A_Properties_of_Real_Numbers
    \[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \texti...\[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \textit{opposite } \text{add to zero.} \\ \\ \\ \textbf{of multiplication } \text{For any real number }a,a\neq 0 & a·\dfrac{1}{a}=1 \\ \;\;\;\;\;\dfrac{1}{a} \text{ is the } \textbf{multiplicative inverse} \text{ of }a \\ \;\;\;\; \text{A number and its } \textit{reciprocal} \text{ multiply to one.} \end…
  • https://math.libretexts.org/Workbench/Intermediate_Algebra_2e_(OpenStax)/01%3A_Foundations/1.06%3A_Properties_of_Real_Numbers
    The Identity Property of Addition that states that for any real number a , a + 0 = a a , a + 0 = a and 0 + a = a . 0 + a = a . The Identity Property of Multiplication that states that for any real num...The Identity Property of Addition that states that for any real number a , a + 0 = a a , a + 0 = a and 0 + a = a . 0 + a = a . The Identity Property of Multiplication that states that for any real number a , a · 1 = a a , a · 1 = a and 1 · a = a . 1 · a = a . This leads to the Inverse Property of Multiplication that states that for any real number a , a ≠ 0 , a · 1 a = 1 . a , a ≠ 0 , a · 1 a = 1 .
  • https://math.libretexts.org/Bookshelves/PreAlgebra/Fundamentals_of_Mathematics_(Burzynski_and_Ellis)/02%3A_Multiplication_and_Division_of_Whole_Numbers/2.06%3A_Summary_of_Key_Concepts
    In a multiplication of whole numbers, the repeated addend is called the multiplicand, and the number that records the number of times the multiplicand is used is the multiplier. If three whole numbers...In a multiplication of whole numbers, the repeated addend is called the multiplicand, and the number that records the number of times the multiplicand is used is the multiplier. If three whole numbers are to be multiplied, the product will be the same if the first two are multiplied first and then that product is multiplied by the third, or if the second two are multiplied first and then that product is multiplied by the first.
  • https://math.libretexts.org/Courses/Monroe_Community_College/MTH_104_Intermediate_Algebra/1%3A_Foundations/1.6%3A_Properties_of_Real_Numbers
    \[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \texti...\[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \textit{opposite } \text{add to zero.} \\ \\ \\ \textbf{of multiplication } \text{For any real number }a,a\neq 0 & a·\dfrac{1}{a}=1 \\ \;\;\;\;\;\dfrac{1}{a} \text{ is the } \textbf{multiplicative inverse} \text{ of }a \\ \;\;\;\; \text{A number and its } \textit{reciprocal} \text{ multiply to one.} \end…
  • https://math.libretexts.org/Courses/Cosumnes_River_College/Corequisite_Codex/01%3A_Sets_and_Numbers/1.04%3A_Properties_of_Real_Numbers
    \[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \texti...\[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \textit{opposite } \text{add to zero.} \\ \\ \\ \textbf{of multiplication } \text{For any real number }a,a\neq 0 & a·\dfrac{1}{a}=1 \\ \;\;\;\;\;\dfrac{1}{a} \text{ is the } \textbf{multiplicative inverse} \text{ of }a \\ \;\;\;\; \text{A number and its } \textit{reciprocal} \text{ multiply to one.} \end…
  • https://math.libretexts.org/Bookshelves/PreAlgebra/Fundamentals_of_Mathematics_(Burzynski_and_Ellis)/01%3A_Addition_and_Subtraction_of_Whole_Numbers/1.06%3A_Properties_of_Addition
    If three whole numbers are to be added, the sum will be the same if the first two are added first, then that sum is added to the third, or, the second two are added first, and that sum is added to the...If three whole numbers are to be added, the sum will be the same if the first two are added first, then that sum is added to the third, or, the second two are added first, and that sum is added to the first. The fact that (a first number + a second number) + third number = a first number + (a second num­ber + a third number) is an example of the property of addi­tion.
  • https://math.libretexts.org/Courses/Coastline_College/Math_C104%3A_Mathematics_for_Elementary_Teachers_(Tran)/07%3A_______Binary_Operations/7.01%3A_Operations_and_Properties
    This exercise set is designed to give you an understanding of what "binary operations" are, and to give you a deeper understanding for the commutative, associative and distributive properties. To do t...This exercise set is designed to give you an understanding of what "binary operations" are, and to give you a deeper understanding for the commutative, associative and distributive properties. To do this, we're going to define and work with some nonsense operations.
  • https://math.libretexts.org/Workbench/Hawaii_CC_Intermediate_Algebra/01%3A_Algebra_Fundamentals/1.02%3A_Operations_with_Real_Numbers
    The result of adding real numbers is called the sum and the result of subtracting is called the difference. Given any real numbers a, b, and c, we have the following properties of addition: Additive I...The result of adding real numbers is called the sum and the result of subtracting is called the difference. Given any real numbers a, b, and c, we have the following properties of addition: Additive Identity Property,  Additive Inverse Property, Associative Property, Commutative Property
  • https://math.libretexts.org/Courses/Fresno_City_College/MATH_201%3A_Elementary_Algebra/01%3A_Foundations/1.05%3A_Properties_of_Real_Numbers
    \[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \texti...\[\begin{array}{lc} \textbf{of addition} \text{For any real number }a, & a+(−a)=0 \\ \;\;\;\; −a \text{ is the } \textbf{additive inverse }\text{ of }a & {} \\ \;\;\;\; \text{A number and its } \textit{opposite } \text{add to zero.} \\ \\ \\ \textbf{of multiplication } \text{For any real number }a,a\neq 0 & a·\dfrac{1}{a}=1 \\ \;\;\;\;\;\dfrac{1}{a} \text{ is the } \textbf{multiplicative inverse} \text{ of }a \\ \;\;\;\; \text{A number and its } \textit{reciprocal} \text{ multiply to one.} \end…

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