Real Numbers

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Real Numbers Terminating Decimals Repeating Decimals 27 Rational Numbers Irrational Numbers 4 = 6.75 0.3 4.5 −10 11 = .90 17 = 4.123105626... 4 = 4.0 5 Integers 9 = .5 € -3 € 2 =1.414213562... -2 € € − 11 = −3.31662479... Whole Numbers € -1 0 € Π = 3.141592654... e = 2.7182818284... Natural Numbers € € 2 1 3 € • Natural numbers are the counting numbers: 1,2,3,… • Whole Numbers are the natural numbers and zero: 0,1,2,3,… • Integers are whole numbers and their opposites: …,-3,-2,-1,0,1,2,3… a • Rational numbers can be expressed in the form b, where a and b are both 1 7 9 integers and b ≠ 0 : , , 2 1 10 • Terminating decimals are rational numbers in decimal form that have a finite number of digits: 1.5, 2.75, 4.0 € € • Repeating decimals are rational numbers in decimal form that have a block of one or more digits that repeat continuously: 1. 3,0.6,2.14,6.27 a • Irrational numbers cannot be expressed in the form b. They include square roots of whole numbers that are not perfect squares and non-terminating decimals that do not repeat: 2, €11 ,Π € € EXAMPLES • ALL NUMBERS ARE PART OF THE REAL NUMBER SYSTEM • ALL NUMBERS ARE EITHER RATIONAL OR IRRATIONAL • RATIONAL NUMBERS HAVE EITHER TEMINATING OR REPEATING DECIMALS • IRRATIONAL NUMBERS CONTINUE TO INFINITY WITHOUT A PATTERN 27 IS A REAL NUMBER AND A RATIONAL # SINCE THE DECIMAL TERMINATES 4 = 6.75 −10 IS A REAL NUMBER AND A RATIONAL # SINCE THE DECIMAL REPEATS 11 = .90 € 17 = 4.123105626... IS A REAL NUMBER AND AN IRRATIONAL # SINCE THE DECIMAL PLACES CONTINUE TO INFINITY WTHOUT A PATTERN € -1 = -1.0 IS A REAL NUMBER AND A RATIONAL # SINCE THE DECIMAL TERMINATES € IT IS ALSO AN INTEGER SINCE IT IS THE OPPOSITE OF 1 IS A REAL NUMBER AND A RATIONAL # SINCE THE DECIMAL TERMINATES 0 = 0.0 IT IS ALSO AN INTEGER SINCE IT CAN BE PLACED ON A NUMBER LINE IT IS ALSO A WHOLE # IS A REAL NUMBER AND A RATIONAL # SINCE THE DECIMAL TERMINATES 1 = 1.0 IT IS ALSO AN INTEGER SINCE IT CAN BE PLACED ON A NUMBER LINE IT IS ALSO A WHOLE # IT IS ALSO A NATURAL # .

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The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom
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