Notes for Unit 1 Part A: Rational vs. Irrational Natural Number: Whole Number: Integer: Rational Number: Irrational Number: Rational Numbers All are Real Numbers Integers Whole Numbers Irrational Numbers Natural Numbers 1
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Lesson 2: Rational Numbers Rational numbers can always be set up as fractions. For Example: Whole number to Fraction Integer to Fraction Mixed Fraction to Improper Fraction We can convert these fractions to decimals using long division. For Example: 3
Lets try some more... Show that 18.423 is a rational number. 4
Show that is a rational number. Let x = repetend 100.. Multiply both sides by 10 n, where n is the number of digits in the. Here there are two digits in the repetend, so we multiply by 10 2, or When we multiply the equation by 10 2 on both sides, we end up with the bottom equation. Now, Subtract the top equation from the bottom equation. The key idea here is that the result is no longer an infinite repeating decimal; in this case, after the first decimal place the repeating parts subtract to zero. Now, Divide both sides by 99. x is a rational number because it can be represented as a ratio. 5
Learning Targets: I can apply the definitions and properties of rational and irrational numbers. I can name the two consecutive whole numbers that an irrational number falls between. 6
This is Rational because it is a perfect square. This is Irrational because it is not a perfect square. Irrational Numbers Converting to Decimals An irrational number is a number that is not rational. It is a real number that cannot be written as a simple fraction. When converting an irrational number to a decimal you have to decide which two whole numbers the irrational number falls between. For example: What two perfect square roots have the between them? and The two whole numbers that the falls between are 1 and 2. Because if you solve and your answers are 1 and 2 respectfully. You are able to conclude that the 's answer must fall between 1 and 2 since it falls between the perfect square roots of and. 7
More Examples: Determine which two whole numbers the square roots below fall between. Now lets compare decimals: Order these decimals from least to greatest. Order these rational and irrational numbers from least to greatest. 8
Comparing using >, <, or = 3.4 4.3 9.89 Choose one of the symbols below: < > = Lesson 4 Rounding & Truncating What kinds of rounding are there? Rounding is a type of approximating that substitutes a more precise value with a number that is easier to work with. There are three main kinds of rounding: rounding up (rounding to a higher value), rounding down (rounding to a lower value), and rounding to the nearest (rounding to the closer value). Example 1: Round 12.92753201 to the nearest a. whole number. b. tenth. c. thousandth. d. hundred thousandth. e. millionth. Example 2: The product of 0.1349 and 0.00024 is 0.000032376. What will a calculator with an 8 digit display and that rounds show for the answer? 9
Rounding Negative Numbers When rounding negative numbers you still use the same rules as the positive numbers. It might help to create a number line. Example 3: Round 81.423 a. to the nearest tenth. 81.5 81.4 81.3 b. to the nearest hundredth. 81.43 81.42 81.41 Truncating Truncating is cutting off a decimal after a particular decimal place. Example 4: Truncate the decimal 234.78923456 at the a. hundredths place. b. Tenths place. c. ten thousandths place. 10
Approximating with Irrational Numbers Find an interval of the given width containing a. One b. 0.1 c. One ten thousandth d. One Millionth Big Hint: Write the decimal approximation of pi to one more decimal place than the width. Graph the intervals of pi on the number line. Whole Number Interval 0 1 2 3 4 5 6 7 8 9 10 Tenth Hundredth 11
Calculating with Irrational Numbers without a Calculator Between what two whole number is the answer for this problem 5 + 2 + 2. 2 < 5 > 3 +1 < 2 > 2 +2 = 2 = 2 5 < 5 + 2 + 2 > 7 So the answer is between 5 and 7. Now, check if this is correct with a calculator. 12