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The product of a non zero rational and an irrational number is
- a)Always irrational
- b)Always rational
- c)Rational or irrational
- d)One
Correct answer is option 'A'. Can you explain this answer?
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The product of a non zero rational and an irrational number isa)Always...
The product will always be irrational...
let us have an example--}
2( rational number) *√2(irrational number)=2√2(irrational)..
let us have an example--}
2( rational number) *√2(irrational number)=2√2(irrational)..
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The product of a non zero rational and an irrational number isa)Always...
Proof that the product of a non-zero rational and an irrational number is always irrational:
Assume that we have a non-zero rational number a and an irrational number b. We want to prove that their product ab is irrational.
1. Definition of rational and irrational numbers:
- A rational number is a number that can be expressed as a ratio of two integers, i.e., a/b, where a and b are integers and b is not equal to zero.
- An irrational number is a number that cannot be expressed as a ratio of two integers. It is a non-repeating, non-terminating decimal.
2. Assume that ab is rational:
- If ab is rational, then it can be expressed as a ratio of two integers, i.e., ab = p/q, where p and q are integers and q is not equal to zero.
- We can rearrange the equation to get b = (p/q)/a = p/(qa), which means that b is a rational number, which contradicts our assumption that b is irrational.
3. Therefore, ab must be irrational:
- Since the assumption that ab is rational leads to a contradiction, we can conclude that ab is irrational.
- This is true for any non-zero rational number a and any irrational number b.
Therefore, the product of a non-zero rational and an irrational number is always irrational (option A).
Assume that we have a non-zero rational number a and an irrational number b. We want to prove that their product ab is irrational.
1. Definition of rational and irrational numbers:
- A rational number is a number that can be expressed as a ratio of two integers, i.e., a/b, where a and b are integers and b is not equal to zero.
- An irrational number is a number that cannot be expressed as a ratio of two integers. It is a non-repeating, non-terminating decimal.
2. Assume that ab is rational:
- If ab is rational, then it can be expressed as a ratio of two integers, i.e., ab = p/q, where p and q are integers and q is not equal to zero.
- We can rearrange the equation to get b = (p/q)/a = p/(qa), which means that b is a rational number, which contradicts our assumption that b is irrational.
3. Therefore, ab must be irrational:
- Since the assumption that ab is rational leads to a contradiction, we can conclude that ab is irrational.
- This is true for any non-zero rational number a and any irrational number b.
Therefore, the product of a non-zero rational and an irrational number is always irrational (option A).
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The product of a non zero rational and an irrational number isa)Always irrationalb)Always rationalc)Rational or irrationald)OneCorrect answer is option 'A'. Can you explain this answer?
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