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Concept
Mathematical constant
Summary
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and pi occurring in such diverse contexts as geometry, number theory, statistics, and calculus.
Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (pi). Other constants are notable more for historical reasons than for their mathematical properties. The more popular constants have been studied throughout the ages and computed to many decimal places.
All named mathematical constants are definable numbers, and usually are also computable numbers (Chaitin's constant being a significant exception).
These are constants which one is likely to encounter during pre-college education in many countries.
Pi
The constant pi (pi) has a natural definition in Euclidean geometry as the ratio between the circumference and diameter of a circle. It may be found in many other places in mathematics: for example, the Gaussian integral, the complex roots of unity, and Cauchy distributions in probability. However, its ubiquity is not limited to pure mathematics. It appears in many formulas in physics, and several physical constants are most naturally defined with pi or its reciprocal factored out. For example, the ground state wave function of the hydrogen atom is
where is the Bohr radius.
pi is an irrational number and a transcendental number.
The numeric value of pi is approximately 3.1415926536 . Memorizing increasingly precise digits of pi is a world record pursuit.
Imaginary unit
The imaginary unit or unit imaginary number, denoted as i, is a mathematical concept which extends the real number system to the complex number system The imaginary unit's core property is that i2 = −1.
Official source
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