What does a negative sign outside a square mean?
Signed arithmetic supports complex numbers and vector components.
Compare reasoning and find a refresher
−(3²)=−9, while (−3)²=9. Parentheses identify the quantity being squared.
Real numbers, complex coordinates, vectors and matrices
A school-entry bridge from fraction and coordinate arithmetic to justified rational powers, complex constructions, directed quantities and labeled matrix operations. Written proofs and coordinate drawings require human review; this unit preserves the existing algebra, geometry and linear algebra courses.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation
Check your starting point
Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
How do you name a point in the plane?
Complex numbers and directed vectors use two coordinates.
Compare reasoning and find a refresher
An ordered pair (x,y) gives horizontal then vertical position; reversing the order generally changes the point.
What does a unit tell you about a quantity?
A component, a speed and a displacement need different interpretations.
Compare reasoning and find a refresher
A velocity component in metres per second multiplied by a duration in seconds gives displacement in metres. A dimensionless multiplier preserves the velocity unit.
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Number structures: a mixed construction and reasoning portfolio
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence
Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Why fractional powers mean roots
Lesson details and objectives
Derive positive-base rational powers by preserving integer exponent laws.
- Derive positive-base rational powers by preserving integer exponent laws.
- Interpret negative exponents as reciprocals and verify a computed power.
- Distinguish principal even roots from all solutions of an equation.
6 practice tasks.
Rewrite radicals without losing the domain
Lesson details and objectives
Rewrite products, quotients and nested rational powers on explicitly stated domains.
- Rewrite products, quotients and nested rational powers on explicitly stated domains.
- Explain why the principal square root of x² is the absolute value of x.
- Diagnose exponent-law failures caused by negative bases or division by zero.
6 practice tasks.
Which number types survive arithmetic?
Lesson details and objectives
Prove closure of rational numbers under addition and multiplication using integer fractions.
- Prove closure of rational numbers under addition and multiplication using integer fractions.
- Prove rational-plus-irrational and nonzero-rational-times-irrational statements by contradiction.
- Construct counterexamples to unrestricted claims about irrational sums and products.
6 practice tasks.
Calculate with a real and an imaginary coordinate
Lesson details and objectives
Add, subtract and multiply complex numbers by distributivity and i²=−1.
- Add, subtract and multiply complex numbers by distributivity and i²=−1.
- Represent real and imaginary coordinates without treating i as a variable to solve for.
- Relate conjugation to an algebraic sign change and verify a conjugate-product identity.
6 practice tasks.
Use conjugates to divide and measure
Lesson details and objectives
Compute complex quotients by a nonzero conjugate multiplier and verify by multiplication.
- Compute complex quotients by a nonzero conjugate multiplier and verify by multiplication.
- Derive modulus from the conjugate product and the distance formula.
- Explain why the reciprocal exists for every nonzero complex number.
6 practice tasks.
Draw complex arithmetic and midpoints
Lesson details and objectives
Construct complex sums, differences and conjugates on equal-scale coordinate axes.
- Construct complex sums, differences and conjugates on equal-scale coordinate axes.
- Calculate a midpoint as an average and a distance as a difference modulus.
- Explain multiplication by i as a quarter-turn, with rectangular and polar interpretations.
6 practice tasks.
Why every quadratic has two complex roots
Lesson details and objectives
Prove the real-coefficient quadratic case of the Fundamental Theorem of Algebra, counting multiplicity.
- Prove the real-coefficient quadratic case of the Fundamental Theorem of Algebra, counting multiplicity.
- Verify complex factorizations by expansion and substitution.
- Distinguish a theorem about complex roots from real-axis graph crossings.
6 practice tasks.
Turn endpoints into a directed quantity
Lesson details and objectives
Compute displacement components from ordered endpoints and preserve units.
- Compute displacement components from ordered endpoints and preserve units.
- Distinguish a vector from its magnitude and represent equal vectors at different positions.
- Convert a simple magnitude-direction description to components and interpret velocity.
6 practice tasks.
Add arrows and find the actual resultant
Lesson details and objectives
Construct vector addition by components, end-to-end arrows and a parallelogram.
- Construct vector addition by components, end-to-end arrows and a parallelogram.
- Find magnitude and direction of a sum supplied in magnitude-direction form.
- Construct subtraction as addition of the opposite and as a correctly directed tip-to-tip arrow.
6 practice tasks.
Scale, reverse, and handle the zero vector
Lesson details and objectives
Construct positive, negative and zero scalar multiples with exact endpoint descriptions.
- Construct positive, negative and zero scalar multiples with exact endpoint descriptions.
- Prove the magnitude factor is the absolute value of the scalar.
- Relate scalar operations to a displacement model without assigning zero a direction.
6 practice tasks.
Keep the labels when operating on matrices
Lesson details and objectives
Encode a small contextual table as a matrix with ordered row and column labels.
- Encode a small contextual table as a matrix with ordered row and column labels.
- Add, subtract and scale matching matrices while interpreting individual entries and units.
- Reject operations with incompatible dimensions, labels or units.
6 practice tasks.
Which number laws survive for matrices?
Lesson details and objectives
Compute dimension-compatible products and demonstrate noncommutativity with a small example.
- Compute dimension-compatible products and demonstrate noncommutativity with a small example.
- Justify associativity and both distributive laws using entrywise finite sums.
- Distinguish zero and identity matrices and verify the 2×2 determinant criterion for an inverse.
6 practice tasks.