The **dual numbers** , in mathematics and especially in complex analysis, extend the real numbers to the new element incorporating *ε* , with the property that *ε *^{2} = 0 (i.e., *ε* is nilpotente ). They have no relation to the solution of the 2nd degree equations with real coefficients. The fundamental applications are in geometry. They were introduced by the German geometer Edward Study. It has some uses in number theory.

## Summary

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- 1 Definition
- 2 Whole operations
- 3 Conjugates
- 1 Division

- 4 Source

## Definition

A dual number is w = l + mε, where ε ^{2} = 0 and both l and m are real, it has a certain analogy with an ordinary complex number: z = a + bi, where a and b are in R, i ^{2} = -1.

## Whole operations

The addition, subtraction and multiplication of dual numbers are defined through the formulas

- (l + mε) + (n + pε) = (l + n) + (m + p) ε
- (l + mε) – (n + pε) = (ln) + (mp) ε
- (l + mε) × (n + pε) = ln + (lp + mn) ε,

The last formula makes us see that the product of a dual w = l + mε for a z = n + pε if lp + mn = 0; whence if l ≠ 0 results m / l = -p / n

On the other hand the product of w = l + mε with w * = l – mε is a real number

w · w * = (l + mε) (l – mε) = a ^{2}

## Conjugated

The number w * = l-mε is called the **conjugate** of the number w = l + mε and conversely w * is the conjugate of w. The square root *l* of the product ww * (with the same value as (w + w *) / 2) is called the modulus of the number w and is denoted by | w |

- the sum w + w * = 2l of two conjugated numbers is a real number:
- the difference w – w * = 2mε is a
*pure imaginary,*that is, it is distinguished from ε only by a real factor m- - also w matches its sss conjugate is real
- fits (z + w) * = z * + w *; (zw) * = z * w *

### Division

The quotient (n + pε) 🙁 l + mε) = n / l + (- mn + lp) ε / l ^{2} , hence for the division to be feasible, the module l of w is required to be different from 0 ; however, as opposed to ordinary complexes, a dual number of null modulus may not be 0. case: w = pε