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 ¾ = 100 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 5 2, 3.4, 3 10 2, π 6) 20, -2 5 3, 10 7, 3.9, 3 3 8 2) 6.5, 6 3, 16, 20, 6.3333 8 3 7) -4.5, -4 5 1, 7, 5 23, 4.182182 3) 1.25, 2, 1 8 3, 1.875, 1 5 3 8) 3 5 2,, -3.875, 8 29, 3.6 4) 8 5 1, 8.22, 8 9 1, 8.3, 8.35235246 14 2 9) 1.75,, 5, 1.9, 1 5 3 5) 4 8 5, 4.375, 4.3, 4 4 3, 4.161616 10) 5.838383, 5 8 3, 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, 2, and on a number line and justify their placement. -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 4 1 16-2.3-2 4 1 4 2-2 4 2.) 2 5 2 4 4 2.09 17 8 9 3.) 5 7 3.4 24 2.5-3 8 7 1.9 8 3
806.2.1 Quiz 1. Write the following rational numbers in ascending order: 3 20 6.5, 6 8, 16, 3, 6.3333 2. Write the following rational numbers in order from smallest to largest: 1.25, 2, 1 8 3, 1.875, 1 5 3 3. 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, 16, 4 23, 6.8, 9 + π 5 6