Flashcards for:
Irrational Numbers
What misconception might one have about repeating decimals in irrational numbers?
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One might think that repeating decimals mean no digit or pattern ever appears again, which is incorrect for irrational patterns.
What is a rational number?
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A rational number is any number that can be written as a ratio of two integers, like a fraction.
Can all numbers be written as a fraction?
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No, not all numbers can be written as a fraction. Numbers that cannot are called irrational numbers.
What is an example of a rational number?
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Examples of rational numbers include fractions such as 1/2 or 2/3.
Why are some numbers called irrational?
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Numbers are called irrational because they cannot be accurately expressed as a fraction or decimal with a repeating pattern.
Is pi a rational or irrational number?
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Pi is an irrational number.
Can pi be represented accurately by a fraction like 22/7?
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No, fractions like 22/7 are just approximations of pi, not its exact value.
What happens to the decimal digits of pi?
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The decimal digits of pi go on forever and do not repeat in a pattern.
Do decimal digits of irrational numbers ever repeat?
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No, they do not repeat in a periodic pattern like rational numbers do.
What is an example of a decimal representation of a rational number?
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The decimal representation of a rational number like 22/7 has a repeating sequence of digits, such as '142857'.
Why are irrational numbers considered 'mysterious'?
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Irrational numbers are 'mysterious' because their decimal representations never end and don't follow a repeating pattern, making them unpredictable.
What does it mean for an irrational number on the number line?
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An irrational number will never precisely line up with a mark on the number line, even if you zoom in infinitely.
Are there more irrational numbers or rational numbers?
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There are more irrational numbers than rational numbers.
How are irrational numbers different from rational numbers in terms of decimal representation?
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Irrational numbers have non-repeating, endless decimals, unlike rational numbers which have repeating decimal patterns.
What is the significance of the number line regarding irrational numbers?
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On the number line, subdividing forever will never make an irrational number align exactly with a mark.
How does one approximate an irrational number like pi in decimal form?
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Irrational numbers like pi can be approximated as decimals such as 3.14 or 3.14159, but these are not exact values.
What does 'non-repeating decimal' mean for an irrational number?
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Non-repeating decimal means that the digits continue indefinitely without any sequence repeating.
Can you have repeating patterns within the non-repeating decimals of an irrational number?
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Yes, you can have repeating patterns like 123 appear, but the entire sequence does not form a repeating cycle.
How is a rational number's decimal pattern determined?
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A rational number's decimal either terminates or enters into a repeating cycle of digits.
Why are approximations like 355/113 used for pi?
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Approximations like 355/113 provide a close but not exact value of pi, useful for calculations.
What misconception might one have about repeating decimals in irrational numbers?
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One might think that repeating decimals mean no digit or pattern ever appears again, which is incorrect for irrational patterns.
What is a rational number?
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A rational number is any number that can be written as a ratio of two integers, like a fraction.