Field Properties of Real Numbers

What are the field properties for addition of real numbers?

Mathematicians organize numbers into various sets, or structures, in which all the numbers satisfy a particular group of rules. There are several field properties for the addition of real numbers. Mathematicians call any set of numbers that satisfies the following properties a field: closure, commutativity, associativity, distributivity, identity elements, and inverses.

• The closure property for addition states that if a and b are real numbers,  is a real number. For example, in the equation , 3, , and  are all real numbers.
• The identity property for addition states that . For example in the equations  and , 8 and e are real numbers.
• The inverse property of addition states that for each real number a, there is a real number , such that . For example, in the equations  and , 8, , and  are all real numbers.
• The commutative property of addition states that if a and b are real numbers, . For example, in the equations  and , 8, , 2, and  are all real numbers.
• The associative property of addition states that if a, b, and c are real numbers, . Consider the equations  and . In these equations, 8, 3, 5, , , and  are all real numbers.

What are the field properties for multiplication of real numbers?

• • There are similar field properties for the multiplication of real numbers. The closure property for multiplication states that if a and b are real numbers,  is a real number. For example, in the equation , 3, , and  are all real numbers.
  • The identity property for multiplication states that . For example, in  and , 8 and e are real numbers.
  • The inverse property of multiplication states that for each real number a (except 0) , there is a real number  such that . For example, in the equations  and , 8, , and  are all real numbers.
  • The commutative property of multiplication states that if a and b are real numbers, . For example, in the equation  and , 8, , 2, and  are all real numbers.
  • The associative property of multiplication states that if a, b, and c are real numbers, . Consider the equations,  and . In these equations, 8, 3, 5, , , and  are all real numbers.

What is the distributive property?

The distributive property states that for any numbers a, b, and c, . For example, .

Review of New Vocabulary and Concepts

• An algorithm is a finite set of instructions for accomplishing some task when given a problem in its initial state.
• To compare numbers that are in different forms, change the numbers to the same form.
• To order numbers that are in different forms, write all numbers in the same form, then place them in the correct order.
• Add whole numbers from right to left, carrying as needed.
• Subtract whole numbers from right to left, borrowing as needed.
• Multiply whole numbers from right to left, carrying as needed. Insert a place holder for the second row of multiplication. Add the numbers to arrive at the answer.
• To divide whole numbers, determine the number of times the divisor goes into the dividend. Subtract the result. Carry down the next digit. Determine the number of times the divisor goes into the dividend. Subtract the result. Repeat until there is nothing left to bring down. If there is a remainder place it over the divisor.
• To add decimals, line up the decimals. Fill in holes, using zeros as place holders. Add as usual, carrying as needed. Bring the decimal straight down.
• To subtract decimals, line up the decimals. Fill in holes, using zeros as place holders. Subtract as usual, borrowing as needed. Bring the decimal straight down.
• To multiply decimals, do not line up the decimals. Multiply as usual, carrying as needed and ignoring the decimal. Count the number of digits after the decimal in the multiplicand. Move the decimal left that many places in the answer.
• To divide decimals, move the decimal to the right of the divisor to create a whole number. Move the decimal the same number of places in the dividend. Divide as usual. Move the decimal straight up, filling in with zeros as needed.
• The closure property for addition states that if a and b are real numbers, then  is a real number.
• The identity property for addition states that .
• The inverse property of addition states that for each real number a, there is a real number  such that .
• The commutative property of addition states that if a and b are real numbers, .
• The associative property of addition states that if a, b, and c are real numbers, .
• The closure property for multiplication states that if a and b are real numbers,  is a real number.
• The identity property for multiplication states that .
• The inverse property of multiplication states that for each real number a, there is a real number  such that .
• The commutative property of multiplication states that if a and b are real numbers, .
• The associative property of multiplication states that if a, b, and c are real numbers, .
• The distributive property states that for any numbers a, b, and c, .

Further Reading in Number Sense

1. 1. CRC Concise Encyclopedia of Mathematics. (Eric W. Weisstein): Chapman & Hall, 2002.
   2. Fundamental College Math. (Sandra P. Clarkson): Houghton Mifflin College Div., 1994.
   3. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes. (Daniel Solow) John Wiley and Sons, 2001.
   4. Don’t forget to test your knowledge with the Number Sense Chapter Quiz; click on the Quizzes link in the left-hand navigation bar to take the quiz.