Allama Iqbal Open University
Course Code: 4431
Pre-Calculus / Business Mathematics
Assignment 1 Solved examples for understanding
Or
“Solved Assignment for Study & Exam Preparation Purposes”
1
,Q No:1(a)
For every one of the accompanying assertions, either
demonstrate it is valid or give counter guide to show it is
misleading;
1. The result of two sane numbers is consistently judicious
2. The result of two sane numbers is consistently silly.
3. The result of a non-zero sane and an unreasonable is
dependably silly.
4. The result of two nonsensical numbers is consistently
judicious.
Answer:
Let's analyze each of the statements and determine if they are true or false.
1.The result of two reasonable numbers is dependably
levelheaded.
True.
Proof:
Let 𝑎 and 𝑏 be two rational numbers. By definition, a judicious number
𝑝
can be communicated as the proportion of two whole numbers. So 𝑎 = 𝑞 1 and
1
𝑝2
𝑏= 𝑑2
where 𝑎 and 𝑑 are integers, and 𝑞 ≠ 0 𝑎𝑛𝑑 𝑞 ≠ 0.
The product of 𝒂 and 𝒃 is:
𝑝1 𝑝2 𝑝 1 ×𝑝 2
𝑎×𝑏= × =
𝑞1 𝑞2 𝑞 1 ×𝑞 2
Since 𝑝1 × 𝑝2 and 𝑞1 × 𝑞2 are both integers, the product is a ratio of two
integers, hence rational.
2. The product of two rational numbers is always irrational.
False.
2
, Counterexample:
1 3
Consider the rational number 𝑎 = 2 and 𝑏 = 4.
Their product is:
1 3 3
𝑎×𝑏= × =
2 4 8
3/8 is a normal number, so the result of two levelheaded numbers isn't
unreasonable 100% of the time.
3. The product of a non-zero rational number and an irrational
number is always irrational.
True.
Proof:
Allow r to be a non-zero objective number and x be a silly number. Assume, for
the sake of contradiction, that their product 𝑟 × 𝑥 is rational. Then we can write:
𝑟×𝑥=𝑞
where q is a rational number. Since r is non-zero, we can partition the two sides
by r:
𝑞
𝑥=𝑟
𝑞
Since q and 𝑟 are both rational, 𝑟 is rational. This implies that xxx is rational,
contradicting the assumption that 𝑥 is irrational. In this manner, the result of a
non-zero normal number and an unreasonable number is generally silly.
4. The product of two irrational numbers is always rational.
False.
Counterexample:
Consider the irrational numbers √2 and √3 .
Their product is:
√2 × √3 = √6
3
Course Code: 4431
Pre-Calculus / Business Mathematics
Assignment 1 Solved examples for understanding
Or
“Solved Assignment for Study & Exam Preparation Purposes”
1
,Q No:1(a)
For every one of the accompanying assertions, either
demonstrate it is valid or give counter guide to show it is
misleading;
1. The result of two sane numbers is consistently judicious
2. The result of two sane numbers is consistently silly.
3. The result of a non-zero sane and an unreasonable is
dependably silly.
4. The result of two nonsensical numbers is consistently
judicious.
Answer:
Let's analyze each of the statements and determine if they are true or false.
1.The result of two reasonable numbers is dependably
levelheaded.
True.
Proof:
Let 𝑎 and 𝑏 be two rational numbers. By definition, a judicious number
𝑝
can be communicated as the proportion of two whole numbers. So 𝑎 = 𝑞 1 and
1
𝑝2
𝑏= 𝑑2
where 𝑎 and 𝑑 are integers, and 𝑞 ≠ 0 𝑎𝑛𝑑 𝑞 ≠ 0.
The product of 𝒂 and 𝒃 is:
𝑝1 𝑝2 𝑝 1 ×𝑝 2
𝑎×𝑏= × =
𝑞1 𝑞2 𝑞 1 ×𝑞 2
Since 𝑝1 × 𝑝2 and 𝑞1 × 𝑞2 are both integers, the product is a ratio of two
integers, hence rational.
2. The product of two rational numbers is always irrational.
False.
2
, Counterexample:
1 3
Consider the rational number 𝑎 = 2 and 𝑏 = 4.
Their product is:
1 3 3
𝑎×𝑏= × =
2 4 8
3/8 is a normal number, so the result of two levelheaded numbers isn't
unreasonable 100% of the time.
3. The product of a non-zero rational number and an irrational
number is always irrational.
True.
Proof:
Allow r to be a non-zero objective number and x be a silly number. Assume, for
the sake of contradiction, that their product 𝑟 × 𝑥 is rational. Then we can write:
𝑟×𝑥=𝑞
where q is a rational number. Since r is non-zero, we can partition the two sides
by r:
𝑞
𝑥=𝑟
𝑞
Since q and 𝑟 are both rational, 𝑟 is rational. This implies that xxx is rational,
contradicting the assumption that 𝑥 is irrational. In this manner, the result of a
non-zero normal number and an unreasonable number is generally silly.
4. The product of two irrational numbers is always rational.
False.
Counterexample:
Consider the irrational numbers √2 and √3 .
Their product is:
√2 × √3 = √6
3
