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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.

## Summary/Background

Leonhard 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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## Glossary

### body

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

### calculus

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.

### cosine

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

### differential calculus

the study of how functions change when their inputs change.

### equation

A statement that two mathematical expressions are equal.

### exponential function

A function having variables as exponents.

### function

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

### identity

An equation which is true for all values of the variable.

### light

having negligible mass.

### logarithm

If y = ax then the logarithm to base a of y is x.

### range

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.

### sine

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

### trigonometry

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

### union

The union of two sets A and B is the set containing all the elements of A and B.

### work

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

## This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AQA A-Level (UK - Pre-2017)FP2Complex NumbersEuler relation-
CCEA A-Level (NI)FP2Complex NumbersEuler relation-
CIE A-Level (UK)P3Complex NumbersEuler relation-
Edexcel A-Level (UK - Pre-2017)FP2Complex NumbersEuler relation-
I.B. Higher Level1Complex NumbersEuler relation-
Methods (UK)M3Complex NumbersEuler relation-
OCR A-Level (UK - Pre-2017)FP3Complex NumbersEuler relation-
OCR-MEI A-Level (UK - Pre-2017)FP2Complex NumbersEuler relation-
WJEC A-Level (Wales)FP2Complex NumbersEuler relation-
Scottish (Highers + Advanced)AM2Complex NumbersEuler relation-
Universal (all site questions)CComplex NumbersEuler relation-
Edexcel A2 Further Maths 2017Core Pure MathsComplex NumbersEuler Relation-
AQA A2 Further Maths 2017Pure MathsFurther Complex NumbersEuler Relation-
OCR A2 Further Maths 2017Pure CoreFurther Complex NumbersEuler Relation-
OCR MEI A2 Further Maths 2017Core Pure BComplex NumbersEuler Relation-
Edexcel AS/A2 Further Maths 2017Core Pure MathsComplex NumbersEuler Relation-
AQA AS/A2 Further Maths 2017Pure MathsFurther Complex NumbersEuler Relation-