REAL NUMBERS as opposed to fake numbers Two

REAL NUMBERS (as opposed to fake numbers? )

Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers

Rational Numbers • A rational number is a number that can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. • The number 8 is a rational number because it can be written as the fraction 8/1. • Likewise, 3/4 is a rational number because it can be written as a fraction. • Even a big, clunky fraction like 7, 324, 908/56, 003, 492 is rational, simply because it can be written as a fraction. • Every whole number is a rational number, because any whole number can be written as a fraction. For example, 4 can be written as 4/1, 65 can be written as 65/1, and 3, 867 can be written as 3, 867/1.

Rational Numbers • A rational number is a real number that can be written as a ratio of two integers. • A rational number written in decimal form is terminating or repeating.

Examples of Rational Numbers • 16 • 1/2 • 3. 56 • -8 • 1. 3333… • - 3/4

Irrational Numbers • An irrational number is a number that cannot be written as a ratio of two integers. • Irrational numbers written as decimals are non-terminating and non-repeating.

Irrational Numbers • All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. • An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers: • π = 3. 141592… = 1. 414213… • Although irrational numbers are not often used in daily life, they do exist on the number line. In fact, between 0 and 1 on the number line, there an infinite number of irrational numbers!

Examples of Irrational Numbers • Square roots of non-perfect “squares” 17 • Pi

Irrational Numbers • Every positive real number has two real roots – one positive (principal root) and one negative. • Ex: √ 16 = 4 and – 4 because 4 x 4 = 16 and -4 x -4 = 16 Negative real numbers have negative roots: Ex: -√ 16 = -4

Your Turn Which of the following numbers are rational? 1 √ 3 − 6 3½ 305. 83 √ 17 -2 3. 1415926535897932384626433

Compare and Order • Since irrational numbers never terminate, we can compare and order irrational numbers by locating them between two consecutive integers. • For example, the irrational number, √ 21 can be found between the perfect squares of √ 16 and √ 25. So, we know that the value of √ 21 is between 4 and 5. We can estimate the value at 4. 5. Check it out with your calculator!

Your Turn • Locate the following irrational numbers between two consecutive integers. • • √ 232 -√ 14 -√ 75 √ 600

Equivalent Forms • You can also simplify real numbers to find equivalent forms. • Ex: √ 12 is not a perfect square but it can be simplified by finding perfect squares within the number. For example √ 12 = √ 4 x √ 3. Since √ 4 is a perfect square, it can be simplified as 2. Therefore, √ 12 can be expressed as 2√ 3.

Your Turn • Simplify the following irrational numbers: • √ 18 = √ 9 x √ 2 = • √ 40 = √ 4 x √ 10 = • √ 72 = √ 9 x √ 4 x √ 2 = • √ 120 = √ 12 x √ 10 = √ 4 x √ 3 x √ 10