FP1.1 - Complex numbers

Definition of complex numbers in The meaning of conjugate, modulus, argument, real part, the form a + ib and imaginary part and equality of complex numbers should be known. rcos θ + irsin θ.

Use fp1.1.1 - definition of complex numbers to connect the rule to the data and decision in the question.

This matters because fp1.1.1 - definition of complex numbers determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.1 - definition of complex numbers to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.1 - Definition of complex numbers is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Sum, product and quotient of | z z | = | z | | z | 1 2 1 2 complex numbers.; Knowledge of the result arg(z z) = arg z + arg z is not 1 2 1 2 required.

Use fp1.1.2 - sum, product and quotient of complex numbers to connect the rule to the data and decision in the question.

This matters because fp1.1.2 - sum, product and quotient of complex numbers determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.2 - sum, product and quotient of complex numbers to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.2 - Sum, product and quotient of complex numbers is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Geometrical representation of complex numbers in the Argand diagram.; Geometrical representation of sums, products and quotients of complex numbers.

Use fp1.1.3 - argand diagrams and complex operations to connect the rule to the data and decision in the question.

This matters because fp1.1.3 - argand diagrams and complex operations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.3 - argand diagrams and complex operations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.3 - Argand diagrams and complex operations is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Complex solutions of quadratic equations with real coefficients.

Use fp1.1.4 - complex solutions of quadratic equations to connect the rule to the data and decision in the question.

This matters because fp1.1.4 - complex solutions of quadratic equations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.4 - complex solutions of quadratic equations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

Finding conjugate complex roots Knowledge that if z is a root of f(z) = 0 then z * is 1 1 and a real root of a cubic equation also a root. with integer coefficients.

Use fp1.1.5 - finding conjugate complex roots to connect the rule to the data and decision in the question.

This matters because fp1.1.5 - finding conjugate complex roots determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.5 - finding conjugate complex roots to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.5 - Finding conjugate complex roots is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Finding conjugate complex roots For example, and/or real roots of a quartic (i) f(x) = x4 − x3 − 5x2 + 7x + 10 equation with real coefficients.; Given that x = 2 + i is a root of f(x) = 0, use algebra to find the three other roots of f(x) = 0 (ii) g(x) = x4 − x3 + 6x2 + 14x − 20 Given g(1) = 0 and g(−2) = 0, use algebra to solve g(x) = 0 completely.

Use fp1.1.6 - finding conjugate complex roots to connect the rule to the data and decision in the question.

This matters because fp1.1.6 - finding conjugate complex roots determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp1.1.6 - finding conjugate complex roots to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP1.1.6 - Finding conjugate complex roots is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.