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The statement e^{i \theta} = \cos \theta + i \sin \theta \quad is known as Euler's relation (and as Euler's formula) and is considered the first bridge between the fields of algebra and geometry, as it relates the exponential function to the trigonometric sine and cosine functions.
If you substitute \theta = \pi the relation simplifies to e^{i\pi} =-1 \quad, known as Euler's identity.


MathsNet imageLeonhard Euler (15 April 1707 – 18 September 1783) was a Swiss mathematician who made enormous contributions to a wide range of mathematics and physics including analytic geometry, trigonometry, geometry, calculus and number theory. Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it. He was the most prolific writer of mathematics of all time. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc. as functions rather than as chords as Ptolemy had done.
He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz's differential calculus and Newton's method of fluxions into mathematical analysis. He introduced beta and gamma functions, and integrating factors for differential equations. He studied continuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, the wave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies (1765).

The number e = 2.718281828459 is Euler's number, the base of the natural logarithm. Euler's identity, e^{i\pi} + 1 = 0 is also sometimes called Euler's equation.

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an object with both mass and size that cannot be taken to be a particle


the study of change; a major branch of mathematics that includes the study of limits, derivatives, rates of change, gradient, integrals, area, summation, and infinite series. Historically, it has been referred to as "the calculus of infinitesimals", or "infinitesimal calculus".
There are widespread applications in science, economics, and engineering.


The trigonometrical function defined as adjacent/hypotenuse in a right-angled triangle.

differential calculus

the study of how functions change when their inputs change.

exponential function

A function having variables as exponents.


A rule that connects one value in one set with one and only one value in another set.


In Statistics: the difference between the largest and smallest values in a data set; a simple measure of spread or variation
In Pure Maths: the values that y can take given an equation y=f(x) and a domain for x.


The trigonometrical function defined as opposite/hypotenuse in a right-angled triangle.


The study of the relationships between the angles and sides of triangles.


Equal to F x s, where F is the force in Newtons and s is the distance travelled and is measured in Joules.

Full Glossary List