260 CHAPTER 5 Number Theory and the Real Number System Is the sum of any two natural numbers a natural number? The answer is yes. Thus, we say that the natural numbers are closed under the operation of addition. Are the natural numbers closed under the operation of subtraction? If we subtract one natural number from another natural number, must the difference always be a natural number? The answer is no. For example, − = − 3 5 2, and −2 is not a natural number. Therefore, the natural numbers are not closed under the operation of subtraction. Definition: Closure If an operation is performed on any two elements of a set and the result is always an element of the set, we say that the set has closure or is closed under that given operation. Example 1 Closure of Sets Determine whether the set of whole numbers is closed under the operations of (a) multiplication and (b) division. Solution a) If we multiply any two whole numbers, will the product always be a whole number? You may want to try multiplying several whole numbers. You will see that yes, the product of any two whole numbers is a whole number. Thus, the set of whole numbers is closed under the operation of multiplication. b) If we divide any two whole numbers, will the quotient always be a whole number? The answer is no. For example ÷ = 1 2 , 1 2 and 1 2 is not a whole number. Therefore, the set of whole numbers is not closed under the operation of division. 7 Now try Exercise 9 Next we will discuss three important properties: the commutative property, the associative property, and the distributive property of multiplication over addition. A knowledge of these properties is essential for the understanding of algebra. We begin with the commutative property. Commutative Property Addition Multiplication + = + a b b a ⋅ = ⋅ a b b a for any real numbers a and b. The commutative property states that the order in which two numbers are added or multiplied is not important. For example, + = + = 45 54 9and ⋅ = ⋅ = 3 6 6 3 18. Note that the commutative property does not hold for the operations of subtraction or division. For example, − ≠ − ÷ ≠ ÷ 47 74and9339 RECREATIONAL MATH KenKen Like Sudoku and Kakuro (see the Recreational Math boxes on page 100 and page 133), KenKen is a puzzle that requires logic. However, KenKen also requires you to use arithmetic facts. To play KenKen, fill in the blank squares with the digits 1 through the number of rows or columns contained in the puzzle. For example, the puzzle above has 4 rows and 4 columns; therefore, you fill in the blank squares with the digits 1 through 4. Like Sudoku, you are not allowed to repeat a digit in any row or column. The heavily outlined sets of squares are called cages. The numbers placed in each cage must combine, in any order , to produce the given number using the given mathematical operation. For example, the cage in the lower right corner of the puzzle above must contain two digits whose quotient is 2. Cages with just one box should be filled in with the given number in the top corner. A number can be repeated within a cage as long as it is not in the same row or column. For more information and more puzzles, see Kenken.com. Complete the above puzzle. The solution to the above puzzle can be found in the answer section in the back of this book. For an additional puzzle, see Exercise 66. Note: KenKen ® is a registered trademark of Nextoy, LLC. Puzzle content © 2014 KenKen Puzzle LLC. All rights reserved. 163 71 22 4 24 24 123 © KenKen Puzzle LLC. Kenken.com
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