Learn the Subsets of Real Numbers Real Number System Learn the Subsets of Real Numbers

Learning Goal 1 (HS.N-RN.B3 and HS.A-SSE.A.1): The student will be able to use properties of rational and irrational numbers to write, simplify, and interpret expressions based on contextual situations. 4 3 2 1 In addition to level 3.0 and above and beyond what was taught in class,  the student may: ·         Make connection with other concepts in math ·         Make connection with other content areas. The student will be able to use properties of rational and irrational numbers to write, simplify, and interpret expressions on contextual situations. - justify the  sums and products of rational and irrational numbers -interpret expressions within the context of a problem The student will be able to use properties of rational and irrational numbers to write and simplify  expressions based on contextual situations. -identify parts of an expression  as related to the context and to each part With help from the teacher, the student has partial success with real number expressions. Even with help, the student has no success with real number expressions.

Notice that both Rational and Irrational numbers are a part of the Real Number system.

Rational Numbers A rational number is a number that can be expressed as a fraction (ratio) in the form a/b, where a and b are integers and b is not zero. Examples: ½, 8, 5/3, √4, 71/9, -12, 6.25, 0.311 When a rational number fraction is divided to form a decimal value, it becomes a terminating or repeating decimal. 2/5 can be represented as which is a terminating decimal. 1/3 can be represented as which is a repeating decimal.

Subsets of Rational Numbers Natural Numbers: Counting numbers. 1, 2, 3, 4, 5… Whole Numbers: Counting numbers plus zero. 0, 1, 2, 3, 4, 5… Integers: Whole numbers and their opposites. …-4, -3, -2, -1, 0, 1, 2, 3, 4…

Irrational Numbers An irrational number is a number that is NOT rational. It cannot be expressed as a fraction with integer values in the numerator and denominator. Examples: When an irrational number is expressed in decimal form, it goes on forever without repeating. Irrational numbers are the “ugly” numbers. They need symbols like π and √.

What about Pi (π)? Your previous math classes have instructed you to use 3.14 or 22/7 to represent π. These are only approximations or estimates. They do not change the fact that “pi” is an irrational number. Π = 3.14159265358979323846264338327950288419716939937510582097…

Properties of Rational and Irrational Numbers 1. The sum of two rational numbers is rational. Rational numbers can be written as fractions and integers. So if I add two fractions, an integer and a fraction, an integer and an integer, or basically, two rational numbers, the result is a rational number. Come up with two examples to support this property.

Properties of Rational and Irrational Numbers The product of two rational numbers is rational. This is similar to the previous reasoning. If you multiply a fraction by a fraction, an integer by a fraction, an integer by an integer or any rational number by another rational number, the result will be rational. Come up with two examples to support this property.

Properties of Rational and Irrational Numbers The sum of two irrational numbers is SOMETIMES irrational. The only time when you add two irrational numbers and do NOT get an irrational answer is when the irrational parts of the number have a sum of zero (zero is rational). Irrational Answer Rational Answer Come up with one example of each to support this property.

Properties of Rational and Irrational Numbers The product of two irrational numbers is SOMETIMES irrational. For example: Irrational Answer Rational Answer Come up with one example of each to support this property.

Properties of Rational and Irrational Numbers The sum of a rational number and an irrational number is irrational. Remember an irrational number in decimal form goes on forever without repeating and a rational number in decimal form either terminates or repeats. If you add the two situations together, you will still have a non-repeating, non- terminating decimal (an irrational number). For example: Come up with an example to support this property.

Properties of Rational and Irrational Numbers The product of a non-zero rational number and an irrational number is irrational. For example: Come up with two examples to support this property.

Recap of Rational and Irrational Numbers Watch a short video on rational and irrational numbers.