← Further Maths syllabus

EnglandFurther Maths

Core Pure: complex numbers

4 dot points across 4 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you add, multiply and divide complex numbers, and how are they represented on the Argand diagram?

The arithmetic of complex numbers, the complex conjugate and division, the Argand diagram, and solving quadratic, cubic and quartic equations with complex roots that occur in conjugate pairs.
A focused answer to the OCR A-Level Further Mathematics A content on the arithmetic of complex numbers and the Argand diagram, covering addition, subtraction and multiplication, the complex conjugate and division by multiplying by the conjugate, plotting on the Argand diagram, and solving polynomial equations whose complex roots occur in conjugate pairs.
11 min answer →

How does de Moivre's theorem give powers of complex numbers and let you derive trigonometric identities?

De Moivre's theorem for integer and rational powers, using it to find powers of complex numbers, and applying it with the binomial theorem to derive multiple-angle identities and to express powers of sine and cosine.
A focused answer to the OCR A-Level Further Mathematics A content on de Moivre's theorem, covering its statement for integer powers, using it to compute powers of complex numbers, deriving multiple-angle identities such as cos 3 theta and sin 3 theta with the binomial theorem, and expressing powers of cosine and sine in terms of multiple angles using z plus and minus its reciprocal.
12 min answer →

How do you write a complex number in modulus-argument and exponential form, and how do these forms make multiplication and division easy?

The modulus and argument of a complex number, modulus-argument form, the exponential form re^(i theta), and the multiplication and division rules in which moduli multiply or divide and arguments add or subtract.
A focused answer to the OCR A-Level Further Mathematics A content on the modulus-argument and exponential forms of a complex number, covering the modulus and argument and finding them with the correct quadrant, modulus-argument form, the exponential form re^(i theta), and the rules that moduli multiply or divide while arguments add or subtract.
11 min answer →

How do you find the nth roots of a complex number, and how do conditions on z describe loci on the Argand diagram?

The nth roots of unity and of a general complex number, their geometric arrangement as a regular polygon, and loci on the Argand diagram defined by modulus and argument conditions (circles, perpendicular bisectors, half-lines and regions).
A focused answer to the OCR A-Level Further Mathematics A content on roots of unity and loci, covering the nth roots of unity and of a general complex number and their arrangement as a regular polygon on a circle, and loci on the Argand diagram from modulus and argument conditions, including circles, perpendicular bisectors, half-lines and shaded regions.
12 min answer →