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Answers
ANSWER:
To Do:
- Find 2 irrational numbers for each of the following case, that satisfy them.
Solution:
1. Two Irrational numbers whose difference is an irrational number.
⇒ 1 + √8 & 1 - √8
Proof:
⇒ Difference of 1 + √8 & 1 - √8 = 1 + √8 - 1 + √8 =2√8 = Irrational Number.
(1 + √8 & 1 - √8 are irrational numbers)
2. Two Irrational numbers whose difference is a rational number.
⇒ √8 + 1 & √8 - 1
Proof:
⇒ Difference of √8 + 1 & √8 - 1 - √8 = √8 - 1 - √8 + 1 = 2 = Rational Number.
( √8 + 1 & √8 - 1 are irrational numbers)
3. Two Irrational numbers whose sum is an irrational number.
⇒ √8 + 1 & √8 - 1
Proof:
⇒ Sum of √8 + 1 & √8 - 1 = √8 - 1 + √8 - 1 = 2√8 = Irrational Number.
(√8 + 1 & √8 - 1 are irrational numbers)
4. Two Irrational numbers whose sum is a rational number.
⇒ 1 + √8 & 1 - √8
Proof:
⇒ Sum of 1 + √8 & 1 - √8 = 1 + √8 + 1 - √8 = 2 = Rational Number.
(1 + √8 & 1 - √8 are irrational numbers)
5. Two Irrational numbers whose product is an irrational number.
⇒ √2 & √3
Proof:
⇒ Product of √2 & √3 = √2 × √3 = √6 = Irrational Number.
(√2 & √3 are irrational numbers)
6. Two Irrational numbers whose product is a rational number.
⇒ √8 & √2
Proof:
⇒ Product of √8 & √2 = √8 × √2 = √16 = 4 = Rational Number.
(√8 & √2 are irrational numbers)
7. Two Irrational numbers whose quotient is an irrational number.
⇒ √6 & √2
Proof:
⇒ Quotient of √6 & √2 = √6/√2 = √3 = Irrational Number.
(√6 & √2 are irrational numbers)
8. Two Irrational numbers whose quotient is a rational number.
⇒ √8 & √2
Proof:
⇒ Quotient of √8 & √2 = √8/√2 = √4 = 2 = Rational Number.
(√8 & √2 are irrational numbers)
Learn More:
- Irrational Numbers: Numbers which cannot be represented in p/q forms.
- Eg: √2, √3, etc.
hope it helps !!
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