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806.2.1 Order and Compare Rational and Irrational numbers and Locate on the number line Rational Number ~ any number that can be made by dividing one integer by another. The word comes from the word "ratio". Examples: 1/2 is a rational number (1 divided by 2, or the ratio of 1 to 2) 0.75 is a rational number (3/4) 1 is a rational number (1/1) 2 is a rational number (2/1) 2.12 is a rational number (212/100) -6.6 is a rational number (-66/10) But Pi is not a rational number, it is an "Irrational Number". Famous Irrational Numbers Pi is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795 (and more ...) The number e (Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527 (and more ...) The Golden Ratio is an irrational number. The first few digits look like this: 1.61803398874989484820... (and more ...) Comparison Property ~ If you are given any two numbers a and b, then there are three possible relationships between them. Either: a = b a > b a < b • We are comparing the two numbers, or putting them in order. • To read < and > remember, we read from left to right. • In < the left side is smaller than the right side of the symbol, therefore the symbol is representing less than • In > the left side is bigger than the right side of the symbol, therefore the symbol is representing greater than • An inequality is a mathematical sentence using < or > to compare two expressions. • We can use any of the following inequality symbols to compare numbers: Symbol < > ≤ ≥ ≠ • Read as less than greater than less than or equal to greater than or equal to not equal to Along with =, we can make any sentence true by using the appropriate symbol. Step 1: To Compare Rational numbers, you must first make them all into decimals. The simplest method is to use a calculator. Example: What is 5/8 as a decimal? Type in 5 / 8 and hit the F The answer should be 0.625 D button If you can’t use a calculator, then you can do one of a couple things: 1. Make an equivalent fraction over 10 or 100 if possible. 75 = 100 ¾ 2. Divide the numerator by the denominator to make a decimal. Example: 1/3 Step 2: Line up your numbers, making sure you line up the decimals. Example: 5/8, .125, .4 5/8 = .625 .125 3.4 Step 3: Bring all your decimal numbers out to the same place value by adding zeros. Once you do this, it’s easy to see the order. 0.625 0.125 0.400 Step 4: List them in order using the correct inequality symbols. 0.125 < .4 < 5/8 For irrational numbers, you must estimate the number in relationship to the other numbers using the largest place value necessary. Practice: 1. Write the following rational numbers in order from greatest to least: 19 /25, 0.33, 0.68, 1 , 0.5 2. Write the following rational numbers in descending order: 9 , 64/16, 3.63, 25/8, 2.125 3. Write the following rational numbers in order from greatest to least: 31 /6, 4.121, 38/9, 47/12, 16 Now, let’s do some practice ☺ 806.2.1 ~ Order rational and irrational numbers Directions: Write the following sets of rational and irrational numbers in order from least to greatest. 1) 16 , 4 2 , 3.4, 3 5 2) 6.5, 6 3 , 16 , 8 3) 1.25, 2 , 10 20 , 3 6) π 6.3333… 2 , 1 3 , 1.875, 1 3 8 5 20 , -2 3 , 5 10 , 3 7) -4.5, -4 1 , 7 , 5 , -3.875, 9) 1.75, 5) 4 5 , 4.375, 4.3, 4 3 , 4.161616… 10) 5.838383…, 5 3 , 8 9 4 4.182182… 29 , 8 4) 8 1 , 8.22, 8 1 , 8.3, 8.35235246… 5 14 , 5 8 23 , 5 5 8) 3 2 , 3.9, 3 7 3.6 5 , 1.9, 1 2 3 8 25 , Locate rational/irrational numbers on a number line: Let's think about where 4.5, 1.838383... and π should be placed on a number line. 1.838383... is placed closer to the 2 because as a rounded number it would be rounded to 2. π is placed closer to the 3 because π is approximately 3.1416. 4.5 is halfway between 4 and 5. Let's place rational and irrational numbers on a number line. Draw a number line for each practice problem and place the number given on the number line. 1. Place -1.4, placement. 2 , and on a number line and justify their -1.4 is almost halfway between -1 and -2 (closer to -1) 2 is approximately 1.4 so is almost halfway between 1 and 2 is closer to the 2 than the 3 but not near the halfway point 2. Place 3.9, 9/3, and -0.3 on a number line and justify their placement. 3.9 is approximately 4 9/3 is equal to 3 -0.3 is close to the halfway point between 0 and -1, closer to the 0 3. Place –16/8, -0.5, and 0.4444... on a number line and justify their placement. -16/8 is equal to -2 -0.5 is halfway between 0 and -1 0.444... is close to 0.5 Now, let’s practice! ☺ 806.2.1 ∼ Locate Rational and Irrational numbers on a Number Line Practice Put the following sets of numbers in order on the number line below each set. 1.) 2.3 2 2.) 3.) 7 5 2 1 4 16 -2.3 -2 5 2 4 4 2.09 17 8 24 2.5 -3 7 8 1.9 3.4 1 4 4 2 - 4 2 9 8 3 806.2.1 Quiz 1. Write the following rational numbers in ascending order: 3 6.5, 6 8 , 20 16 , 3 , 6.3333… 2. Write the following rational numbers in order from smallest to largest: 1.25, 3. 2 3 3 , 1 8 , 1.875, 1 5 Compare the following rational numbers using the symbols < or >: 3 π, 5 , 0.827, .075, 1 4. Which of the following rational or irrational numbers belongs between the 5 and the 6 on the number line below? 35 9 , 5 23 16 , , 6.8, 4 9+ π 6