Complex number

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real number — Real parts and imaginary coefficients are real numbers.
• Coordinate plane — The complex plane uses ordered pairs.
• 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.