2WBB0Calculus I

2WBB0 · Topic 03

Complex numbers track

Polar form, powers, roots, and geometric loci for programmes assigned the complex branch.

What you need to be able to do

Convert between Cartesian and polar/exponential form.
Multiply, divide, power, and root complex numbers using modulus and argument.
Interpret absolute-value equations as geometric loci.

1. Cartesian, polar, and exponential form

Cartesian form is best for addition and geometric coordinates. Polar or exponential form is best for multiplication, division, powers, and roots.

The argument must match the correct quadrant. An arctangent alone can return an angle with the right tangent but the wrong point.

Exam method

Compute the modulus from the Pythagorean theorem.
Locate the quadrant from the signs of real and imaginary parts.
Choose an argument consistent with that quadrant.
Use Euler's formula to return to a+bi.

Cartesian, polar, and exponential form: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Cartesian, polar, and exponential formSelf-review

Worked example 1

4e^{5\pi i/6}

2. Powers and all roots

De Moivre's rule turns powers into multiplication of the argument. Root questions reverse the process but require every argument differing by a full turn before division by n.

Listing only the principal root loses most of the points. The n roots are equally spaced around a circle.

Exam method

Write the right-hand side in polar form.
Take the nth root of the modulus.
Use arguments (θ+2πk)/n for k=0 through n−1.
Convert to a+bi only if the question asks for it.

Powers and all roots: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Powers and all rootsSelf-review

Worked example 1

z^4+16=0

3. Loci in the complex plane

The modulus |z-a| is the distance from the point z to the point a. Equal-distance equations describe perpendicular bisectors; constant-distance equations describe circles.

Writing z=x+iy and squaring both sides is the reliable algebraic backup when the geometry is not obvious.

Exam method

Translate each modulus into a distance statement.
Recognize circle, perpendicular bisector, or ratio-of-distances geometry.
For algebra, substitute z=x+iy and compare squared moduli.
State and sketch the final line or circle with its centre and radius.

Loci in the complex plane: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Loci in the complex planeSelf-review

Worked example 1

z\in\mathbb C

Formulas used in this topic

$z=a+bi=re^{i\theta}=r(\cos\theta+i\sin\theta)$
$|z|=\sqrt{a^2+b^2},\qquad \arg(zw)=\arg z+\arg w$
$z^n=r^ne^{in\theta}$
$z_k=|w|^{1/n}e^{i(\arg w+2\pi k)/n},\quad k=0,\ldots,n-1$

2WBB0 · Topic 03

Complex numbers track

Polar form, powers, roots, and geometric loci for programmes assigned the complex branch.

What you need to be able to do

Convert between Cartesian and polar/exponential form.
Multiply, divide, power, and root complex numbers using modulus and argument.
Interpret absolute-value equations as geometric loci.

1. Cartesian, polar, and exponential form

Cartesian form is best for addition and geometric coordinates. Polar or exponential form is best for multiplication, division, powers, and roots.

The argument must match the correct quadrant. An arctangent alone can return an angle with the right tangent but the wrong point.

Exam method

Compute the modulus from the Pythagorean theorem.
Locate the quadrant from the signs of real and imaginary parts.
Choose an argument consistent with that quadrant.
Use Euler's formula to return to a+bi.

Cartesian, polar, and exponential form: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Cartesian, polar, and exponential formSelf-review

Worked example 1

4e^{5\pi i/6}

2. Powers and all roots

De Moivre's rule turns powers into multiplication of the argument. Root questions reverse the process but require every argument differing by a full turn before division by n.

Listing only the principal root loses most of the points. The n roots are equally spaced around a circle.

Exam method

Write the right-hand side in polar form.
Take the nth root of the modulus.
Use arguments (θ+2πk)/n for k=0 through n−1.
Convert to a+bi only if the question asks for it.

Powers and all roots: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Powers and all rootsSelf-review

Worked example 1

z^4+16=0

3. Loci in the complex plane

The modulus |z-a| is the distance from the point z to the point a. Equal-distance equations describe perpendicular bisectors; constant-distance equations describe circles.

Writing z=x+iy and squaring both sides is the reliable algebraic backup when the geometry is not obvious.

Exam method

Translate each modulus into a distance statement.
Recognize circle, perpendicular bisector, or ratio-of-distances geometry.
For algebra, substitute z=x+iy and compare squared moduli.
State and sketch the final line or circle with its centre and radius.

Loci in the complex plane: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Loci in the complex planeSelf-review

Worked example 1

z\in\mathbb C

Formulas used in this topic

$z=a+bi=re^{i\theta}=r(\cos\theta+i\sin\theta)$
$|z|=\sqrt{a^2+b^2},\qquad \arg(zw)=\arg z+\arg w$
$z^n=r^ne^{in\theta}$
$z_k=|w|^{1/n}e^{i(\arg w+2\pi k)/n},\quad k=0,\ldots,n-1$

2WBB0 · Topic 03

Complex numbers track

Polar form, powers, roots, and geometric loci for programmes assigned the complex branch.

What you need to be able to do

Convert between Cartesian and polar/exponential form.
Multiply, divide, power, and root complex numbers using modulus and argument.
Interpret absolute-value equations as geometric loci.

1. Cartesian, polar, and exponential form

Cartesian form is best for addition and geometric coordinates. Polar or exponential form is best for multiplication, division, powers, and roots.

The argument must match the correct quadrant. An arctangent alone can return an angle with the right tangent but the wrong point.

Exam method

Compute the modulus from the Pythagorean theorem.
Locate the quadrant from the signs of real and imaginary parts.
Choose an argument consistent with that quadrant.
Use Euler's formula to return to a+bi.

Cartesian, polar, and exponential form: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Cartesian, polar, and exponential formSelf-review

Worked example 1

4e^{5\pi i/6}

2. Powers and all roots

De Moivre's rule turns powers into multiplication of the argument. Root questions reverse the process but require every argument differing by a full turn before division by n.

Listing only the principal root loses most of the points. The n roots are equally spaced around a circle.

Exam method

Write the right-hand side in polar form.
Take the nth root of the modulus.
Use arguments (θ+2πk)/n for k=0 through n−1.
Convert to a+bi only if the question asks for it.

Powers and all roots: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Powers and all rootsSelf-review

Worked example 1

z^4+16=0

3. Loci in the complex plane

The modulus |z-a| is the distance from the point z to the point a. Equal-distance equations describe perpendicular bisectors; constant-distance equations describe circles.

Writing z=x+iy and squaring both sides is the reliable algebraic backup when the geometry is not obvious.

Exam method

Translate each modulus into a distance statement.
Recognize circle, perpendicular bisector, or ratio-of-distances geometry.
For algebra, substitute z=x+iy and compare squared moduli.
State and sketch the final line or circle with its centre and radius.

Loci in the complex plane: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded

Loci in the complex planeSelf-review

Worked example 1

z\in\mathbb C

Formulas used in this topic

$z=a+bi=re^{i\theta}=r(\cos\theta+i\sin\theta)$
$|z|=\sqrt{a^2+b^2},\qquad \arg(zw)=\arg z+\arg w$
$z^n=r^ne^{in\theta}$
$z_k=|w|^{1/n}e^{i(\arg w+2\pi k)/n},\quad k=0,\ldots,n-1$