Rational & Irrational Numbers

Rational Numbers The real number system consists of rational and irrational numbers. Rational numbers can be expressed in fractional form, , where a (the numerator) and b (the denominator) are both integers and b = 0. This means that the decimal form of the number either terminates or repeats. Counting numbers, whole numbers, integers, and non-integers are all rational numbers. a b

Counting numbers {1, 2, 3, 4, 5, 6, …} Whole numbers consist of the counting numbers and zero. {0, 1, 2, 3, 4, 5, …} Integers consist of the counting numbers, their opposites, and zero. {…, -3, -2, -1, 0, 1, 2, 3, …}

Non-integers consist of fractions that can be written as terminating or repeating decimals. A terminating decimal comes to a complete stop. A repeating decimal continues the same digit or block of digits forever. 1 3 2 5.25 0.6 -9.261 7 3

Irrational Numbers Irrational numbers are numbers that cannot be written as a ratio of two integers. Irrational numbers are non-repeating and non-terminating decimals because the decimal form of the number never ends and never repeats. The most common irrational number is pi (п). The value of п is 3.141592654…

Example 1 Tell whether each real number is rational or irrational. -23.75 rational decimal terminates 4.750918362… irrational decimal does not terminate 5 9 √15 irrational decimal form does not terminate rational number is in fraction form

Rational and Irrational Numbers Combining Rationals and Irrationals Addition and subtraction of any number to an irrational number gives another irrational number Examples of irrationals

Rational and Irrational Numbers Combining Rationals and Irrationals Multiplication and division of an irrational number by another irrational can often lead to a rational number. (but not always) Examples of Rationals 21 26 8 1 -13

Rational and Irrational Numbers Combining Rationals and Irrationals Determine whether the following are rational or irrational. (a) 0.73 (b) (c) 0.666…. (d) 3.142 (e) rational irrational rational rational irrational (f) (g) (h) (i) (j) irrational irrational rational rational irrational (j) (k) (l) irrational rational rational