exploring real numbers objectives: (1)to classify numbers (2)to compare numbers
TRANSCRIPT
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Exploring Real Numbers
Objectives:
(1)To classify numbers
(2)To compare numbers
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Number Groups
• Natural Numbers– 1, 2, 3, …
• Whole Numbers– 0, 1, 2, …
• Integers– … -3, -2, -1, 0, 1, 2, 3, …
• Rational Numbers:– Integers, fractions, finite decimals, repeating decimals
• Irrational Numbers:– Infinite, non-repeating decimals
• Real Numbers:– All rational and irrational numbers
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How are repeating decimals rational?
We use the 10x – x rule.
Ex: x = 0.99999….
10x = 9.999999999…- x = - 0.999999999…
9x = 9
9 9
x = 1
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How are repeating decimals rational?
Wait a minute…
If x = 0.999… and x = 1, then ????
0.999… = 1
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Definitions
• Counterexample: An example that proves a statement false.
• Inequality: a mathematical sentence that compares the value of two expressions using an inequality symbol, such as <, >, or ≠
• Opposites: two numbers that have the same distance from zero
• Absolute Value: a number’s distance from zero
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Example #1: Classifying Numbers
a. - (17/31)* Rational
b. 23* Natural, Whole, Integers, Rational
c. 0* Whole, integers, rational
d. 4.581
* rational
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Example #2: Using Counterexamples
Is each statement true or false? If it is false, give a counterexample.
a. All whole number are rational numbers.
- True, the easiest way to turn a whole number into a fraction is to put it over 1.
b. The square of a number is always greater than the number.
- False: 0.52 = 0.25. 0.5 is our counterexample
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Example #3: Ordering Fractions
Write -3/8, -1/2, and -5/12 in order from least to greatest.
1st Step: write each fraction as a decimal
-3/8 = -0.375
-1/2 = -0.5
-5/12 = -0.41666…
2nd Step: order the decimals from least to greatest
-0.5, -0.41666…, -0.375
3rd Step: replace decimals with their fraction equivalents
-1/2, -5/12, -3/8
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Example #4: Finding Absolute Value
Find each absolute value:
.a 12 12
.b -5.6 5.6
.c 5 – 8 -3
3