Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π =\(\frac{ c}{d}\). This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
Solution and Explanation
There is no contradiction. When we measure a length with scale or any other instrument,
we only obtain an approximate rational value. We never obtain an exact value.
For this reason, we may not realise that either c or d is irrational.
Therefore, the \(\frac{ c}{d}\) fraction is irrational.
Hence, π is irrational.
Top Questions on Operations on Real Numbers
- Let α and β be real numbers. Consider a \(3 \times 3\) matrix A such that \(A^2 = 3A + \alpha I\). If \(A^4 = 21A + \beta I\), then
- JEE Main - 2023
- Mathematics
- Operations on Real Numbers
- Let \(λ ≠ 0\) be a real number. Let α, β be the roots of the equation \(14x^2 – 31x + 3λ = 0\) and α, γ be the roots of the equation \(35x^2 – 53x + 4λ = 0\). Then \(\frac{3\alpha}{\beta}\) and \(\frac{4\alpha}{\lambda}\) are the roots of the equation
- JEE Main - 2023
- Mathematics
- Operations on Real Numbers
For real number a, b (a > b > 0), let
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi\)
and
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi\)
Then the value of (a – b)2 is equal to _____.- JEE Main - 2022
- Mathematics
- Operations on Real Numbers
- Let for some real numbers \(α\) and \(β\), \(a = α – iβ\). If the system of equations \(4ix + (1 + i) y = 0\) and \(8\bigg(cosx\frac{2π}{3}+i\;sin\frac{2π}{3}\bigg)x+\bar ay=0 \)has more than one solution, then \(\frac{α}{β}\) is equal to
- JEE Main - 2022
- Mathematics
- Operations on Real Numbers
- Let A = [1, 2, 3, 4, 5], m be the number of relations such as 4x ≤ 5y XRY and n be the minimum number of elements to be added from A × A to make a symmetric relation. Then the value of n + m.
- JEE Main
- Mathematics
- Operations on Real Numbers
Questions Asked in CBSE Class IX exam
- (i) The kind of person the doctor is (money, possessions)
(ii) The kind of person he wants to be (appearance, ambition)- CBSE Class IX
- The snake and the mirror
- What are the main features of the mechanical teachers and the schoolrooms that Margie and Tommy have in the story?
- CBSE Class IX
- The fun they had
- The text you read is a travelogue where the author, Vikram Seth, talks about his visit to two sacred places in Kathmandu. Imagine that you were with Vikram Seth on his visit to Pashupatinath temple, and you were noting down all that you saw and did there, so that you could write a travelogue later.
Record in point form • what you see when you reach the Pashupatinath temple • what you see happening inside the temple • what you do when inside the temple • what you see outside the temple • what your impressions are about the place.- CBSE Class IX
- Kathmandu
When 3.0g of carbon is burnt in 8.00g oxygen, 11.00g of carbon dioxide is produced. What mass of carbon dioxide will be formed when 3.00g of carbon is burnt in 50.0g of oxygen? Which law of chemical combination will govern your answer?
- CBSE Class IX
- Laws of Chemical Combinations
- Factorise : (i) x3 – 2x 2 – x + 2 (ii) x 3 – 3x 2 – 9x – 5 (iii) x 3 + 13x 2 + 32x + 20 (iv) 2y 3 + y 2 – 2y – 1.
- CBSE Class IX
- Factorisation of Polynomials