Why is 3.141592 an irrational number?
it is irrational it is pi
original value is = 3.1415926535897932384626433832795028841971 693993751058209749445923078164062862089986280348253421170679
Thank you!
of 3.14 lol
The number 3.141592 is an approximation of a mathematical constant called pi (π). Pi is an irrational number, which means it cannot be expressed as a simple fraction or a ratio of two whole numbers. Here's why pi is irrational:
To determine why pi is irrational, we need to understand the concept of rational and irrational numbers. A rational number can be expressed as a fraction, where the numerator and denominator are both integers. For example, 7/3 is a rational number because it can be written as a fraction of two integers.
On the other hand, an irrational number cannot be expressed as a fraction. Instead, it has an infinite and non-repeating decimal representation. Pi is one such number.
Several proofs demonstrate that pi is irrational. The most well-known proof was discovered by Johann Lambert in the 18th century and later refined by other mathematicians. The proof uses the idea of contradiction, known as reductio ad absurdum, to show that assuming pi is rational leads to a contradiction.
The proof begins by assuming that pi can be expressed as a fraction, say a/b, where a and b are integers with no common factors. By squaring both sides of this assumption, we get (pi^2) = (a^2)/(b^2). Rearranging the equation, we find that pi^2 is equal to (a^2)/(b^2), or pi^2 = (a^2 * b^-2).
From this equation, we can conclude that pi^2 is rational since it can be expressed as the ratio of two integers—namely, (a^2) and (b^2). However, this contradicts our initial assumption that pi is irrational. Therefore, our initial assumption must be false, and pi cannot be rational.
Hence, pi is an irrational number, and its decimal representation goes on infinitely without any repeating pattern. The approximation 3.141592 is commonly used to represent pi, but it is important to remember that pi cannot be precisely expressed as a finite decimal or a fraction.
those are the first 7 digits of π (pi), which is irrational
the number you have written is rational ... because it has an end