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5.10 RADICALS, IRRATIONAL NUMBERS, AND THE CLOSURE AXIOMS RADICALS A radical is an expression that has a root There is no real-number answer for the square root of a negative number because no real number squared gives a negative answer RATIONAL AND IRRATIONAL NUMBERS A rational number is a number that can be written as a ratio of two integers. An irrational number is a real number that cannot be written as a ratio of two integers. SETS OF NUMBERS Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers TELL WHETHER EACH RADICAL REPRESENTS A RATIONAL NUMBER, AN IRRATIONAL NUMBER, OR NEITHER 1. 2. 3. 4. CLOSURE A given set of numbers is closed under an operation if there is just one answer, and that answer is in the given set whenever the operation is performed with numbers that are in that set. The braces { } mean “the set of” or “the set containing” (number is set)(operation)(number in set)=(unique number in set) CLOSURE AXIOMS Closure Under Multiplication The set of real numbers is closed under multiplication If x and y are any real numbers, then xy is a unique real number. CLOSURE AXIOMS Closure Under Addition The set of real numbers is closed under addition If x and y are any real numbers, then x + y is a unique real number. TELL WHETHER EACH SET OF NUMBERS IS CLOSED UNDER THE GIVEN OPERATION. GIVE AN EXAMPLE. {integers}, subtraction {rational numbers}, addition TELL WHETHER EACH SET OF NUMBERS IS CLOSED UNDER THE GIVEN OPERATION. GIVE AN EXAMPLE. {negative numbers}, multiplication {0, 1}, addition TELL WHETHER EACH SET OF NUMBERS IS CLOSED UNDER THE GIVEN OPERATION. GIVE AN EXAMPLE. {rational numbers}, square root {real numbers} addition TELL WHETHER EACH SET OF NUMBERS IS CLOSED UNDER THE GIVEN OPERATION. GIVE AN EXAMPLE. {negative numbers}, subtraction {perfect squares}, square root HOMEWORK P 209 #1 – 43 odd