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7 Cards in this Set
- Front
- Back
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The sum of two rational numbers will always be (rational, irrational).
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The sum of two rational numbers will always be rational.
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The sum of an irrational number and a rational number will always be (rational, irrational).
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The sum of an irrational number and a rational number will always be irrational.
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The product of two rational numbers will always be (rational, irrational).
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The product of two rational numbers will always be rational.
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The product of a non-zero rational number and an irrational number will always be (rational, irrational).
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The product of a non-zero rational number and an irrational number will always be irrational.
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Why will the sum or product of two rational numbers always be rational?
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If a/b and c/d are the two rational numbers then their sum is (ad+bc)/bd and their product is ac/bd. Both of these are rational because the numerator and denominator are always integers.
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Why will the sum of an irrational number and a rational number always be irrational?
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An irrational number by definition is a number with a decimal portion that does not terminate or repeat and therefore cannot be expressed as a fraction. If you add a number that does repeat or terminate, it will not get rid of the non-repeating, non-terminating part of the irrational number, and therefore the sum will still be irrational.
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Why will the product of a non-zero rational number and an irrational number always be irrational?
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Suppose a is rational (and non-zero) and x is irrational. Suppose ax is rational; ax=b where b is rational. Then, x=b/a and x would be rational. This is a contradiction, therefore the product of a non-zero rational number and an irrational number must always be irrational.
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