# Complex number

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A complex number a+b*i is represented by the ordered real pair (a,b), with addition, subtraction and multiplication defined using i²=−1.

mko-complex-number · definition · v0.10.0



## Mathematical statement

A complex number z=a+b*i is uniquely determined by its real part a and imaginary coefficient b and can be viewed as the ordered pair (a,b) in the complex plane. Addition and subtraction are componentwise; multiplication follows from distributivity and i²=−1.





## Representative formula (exact TeX)

z=C(a,b)





## Assumptions


- Exercises restrict real parts and imaginary coefficients to integers and give answers as ordered pairs.
- Component order is semantic data. Graph positions and unmarked scales supply no missing values; complex division is not included in this slice.





## Symbol scope


- z — complex number
- C — complex ordered-pair constructor





## Intuition, definition, and misconceptions

The numbers a+b*i and c+d*i are equal exactly when a=c and b=d.

The product has real part a*c−b*d and imaginary coefficient a*d+b*c; the negative sign comes from i²=−1.

A real number r embeds as (r,0). A complex number with a nonzero imaginary coefficient cannot equal any embedded real candidate.

Do not swap real and imaginary components or forget i²=−1 when multiplying.





## Adopted derivation guide

[Complex numbers: computing (2+3i)(4−5i) as an ordered pair and deciding equality](https://ocme.evemisslab.com/en/derivations/derivation-mko-complex-number-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) — Real parts and imaginary coefficients are real numbers.
- [Coordinate plane](https://ocme.evemisslab.com/en/mko/mko-coordinate-plane/) — The complex plane uses ordered pairs.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Complex addition, subtraction and multiplication combine real arithmetic.





## Verification boundary

This slice covers only integer-pair addition, subtraction, multiplication and real embedding; division, polar form and complex analysis are not covered.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add conjugation, modulus, division and polar form.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/complex-number.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-complex-number/index.md)