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MathidiaMathematics Academy
Complex numbers
Lesson 2 of 75
Easy practice

Cartesian form

Objective. Add, multiply and divide complex numbers.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Cartesian form is developed as connected advanced mathematics rather than a memorised recipe. Extend the number system and interpret arithmetic geometrically. Begin by identifying the objects, definitions, assumptions and domain restrictions that control the problem. Move between symbolic, graphical, numerical and contextual representations, explaining why each transformation preserves the required meaning. Compare efficient methods, retain exact values until approximation is justified, and use notation that makes every dependency visible. The final stage is to test limiting cases, dimensions, signs and alternative representations so the conclusion is both rigorous and useful.

Worked example

Find |3+4i|.

-1124578+4
Number line from -1 to 8 with a jump of 4 forwards from 3
  1. 1Translate the problem into precise notation, identify known quantities and constraints, and choose a representation that exposes the mathematical structure.
  2. 2Select the governing theorem, definition or model; state why its conditions hold before substituting or transforming any expression.
  3. 3Carry out the algebra, geometry or computation in justified stages, retaining exact forms and tracking domains, signs, dimensions and parameters.
  4. 4Verify the result through substitution, differentiation, an independent representation or a limiting case. The checked conclusion is 5.

Answer: 5

Concept mastery

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Practice

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Common mistake. A common error in cartesian form is to apply a familiar formula without confirming its hypotheses, domain or orientation. This can create an algebraically polished answer that is mathematically invalid. State the controlling condition first, preserve exact notation, and test the result against the original model rather than only the final line.

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