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Why Is the Square Root of Pi an Irrational Number?

square root of pi: Symbol and value of pi against a galaxy background

Have you ever been asked what the square root of pi is? Disclaimer: The quick answer is that there is no precise answer! Because all square roots of irrational numbers are irrational numbers, the square root of pi is also an irrational number. However, that doesn't mean we can't approximate the answer. Just like we approximate the value of pi to be 3.14, we can approximate the square root of pi to be 1.77.

Defining Square Roots

Multiplying a number by itself is called squaring, or finding the square of a number. For example, 9 squared is equal to 81 because 9 x 9 = 81. The square root is just the inverse of squaring. Instead of multiplying a number by itself, it is finding the number that equals the original number when multiplied against itself. For example, the square root of 9 is 3, because 3 x 3 = 9.

The square of 9 is 81.

The square root of 9 is 3.

Square roots are often seen when solving quadratic functions or running calculations for particular theorems like Pythagorean's Theorem. It is expressed using the symbol square root of pi: Square root symbol, or the abbreviation sqrt, but it can also be expressed using exponentials. The square root of a number is the same as raising that number to an exponent of 1/2. In other words:

Square root formula

What Are Irrational Numbers, Anyway?

Irrational numbers are numbers that cannot be expressed as a fraction of a/b, where the numerator and denominator a and b are non-zero integers. In other words, irrational numbers can never be represented in decimal form without the digits going on forever.

The digits of pi are an example of this. Pi is one of the most famous irrational numbers. Other irrational numbers include the golden ratio phi or Euler's Number E (the base for natural logarithms).

Expanding Our Math Horizons

Asking to find the square root of pi is often thought of as a nonsensical question. But, famous mathematicians have expanded our understanding of math by asking transcendental questions such as these.

It pushes us to think beyond numbers as whole numbers or decimals that we typically use when we work with polygons or coefficients in algebraic expressions.

If these kinds of questions that go beyond algebraic numbers interest you, just look up what gauss or Gaussian numbers are, or why we even have a category for complex and imaginary numbers. Until then, just say that the square root of pi is an irrational number that approximates to be around 1.77.

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