Number and algebra

In an alternating-current circuit, impedances are modelled as complex numbers measured in ohms (Ω). Two components have impedances \(Z_1 = 3 + 4i\) and \(Z_2 = 2 - 5i\).
When components are connected in series, the total impedance is \(Z_S = Z_1 + Z_2\). When they are connected in parallel, the total impedance \(Z_P\) satisfies \(\frac{1}{Z_P} = \frac{1}{Z_1} + \frac{1}{Z_2}\).

A café sells coffee, tea and cake. The numbers sold on Monday, Tuesday and Wednesday of one week are given by the matrix
\(Q = \begin{pmatrix} 12 & 8 & 15 \\ 10 & 14 & 9 \\ 20 & 11 & 13 \end{pmatrix}\)
where the rows are Monday, Tuesday and Wednesday, and the columns are coffee, tea and cake. In that week the prices, in dollars, are \(P = \begin{pmatrix} 2.80 \\ 2.20 \\ 3.50 \end{pmatrix}\).

An engineer models two components in an alternating-current circuit using complex impedances, measured in ohms (Ω):
\(Z_1 = 4 + 3i\) and \(Z_2 = 2 - 6i\).
When the components are connected in series, the total impedance is \(Z_S = Z_1 + Z_2\). When they are connected in parallel, the total impedance \(Z_P\) satisfies \(\frac{1}{Z_P} = \frac{1}{Z_1} + \frac{1}{Z_2}\).

A graphic designer uses the transformation with matrix \(T = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}\) to distort a company logo drawn on a coordinate grid, where units are centimetres. A point with position vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) is mapped to \(T\begin{pmatrix} x \\ y \end{pmatrix}\).

In an AC circuit, two components are connected in series. Their impedances are \(Z_1=3+4i\) ohms and \(Z_2=5-2i\) ohms. The total impedance is \(Z=Z_1+Z_2\).

A voltage in an electronic circuit is represented by the complex number \(V_1=120e^{i\pi/3}\) volts. A second voltage is \(V_2=50-50i\) volts.

A bakery buys flour, butter and sugar at prices (in euros per kg) given by the matrix \(P=\begin{pmatrix}2.5&4.0&1.8\end{pmatrix}\). One batch of recipe A needs 3 kg flour, 1 kg butter and 4 kg sugar. One batch of recipe B needs 2 kg flour, 2 kg butter and 1 kg sugar. These are the columns of \(Q=\begin{pmatrix}3&2\\1&2\\4&1\end{pmatrix}\).

A taxi company charges a fixed fee plus a constant rate per kilometre. A 6 km journey costs 14.40 euros and an 11 km journey costs 24.40 euros. Let the fixed fee be \(f\) euros and the rate be \(r\) euros per km.

Two components in an audio filter are connected in series. Their impedances, in ohms (Ω), are Z₁ = 8 + 6i and Z₂ = 3 − 4i. The total impedance is Z = Z₁ + Z₂.

Two farms plant wheat, barley and maize. The areas planted, in hectares, are given by the matrix A, where each row is a farm (Farm 1, Farm 2) and the columns are wheat, barley, maize:
A = \(\begin{pmatrix} 12 & 8 & 5 \\ 9 & 14 & 6 \end{pmatrix}\)
The expected income per hectare, in euros, is wheat 1450, barley 1180 and maize 1620, written as the column matrix p = \(\begin{pmatrix} 1450 \\ 1180 \\ 1620 \end{pmatrix}\).

A musical tone is represented by the phasor z = 7(cos(2π/5) + i sin(2π/5)), where the modulus is the amplitude.

A garden centre sells seed mixes X and Y at constant prices of €x and €y per packet. Three packets of X and two packets of Y cost €19. Five packets of X and four packets of Y cost €34. This is written as M\(\begin{pmatrix} x \\ y \end{pmatrix}\) = \(\begin{pmatrix} 19 \\ 34 \end{pmatrix}\), where M = \(\begin{pmatrix} 3 & 2 \\ 5 & 4 \end{pmatrix}\).

A light aircraft flies in a region of constant wind. Using a complex-plane model, a velocity is written as \(x+yi\), where \(x\) is the eastward component and \(y\) the northward component, both in km h⁻¹.
The velocity of the aircraft relative to the air is \(v_a = 210+120i\) and the velocity of the wind is \(w = -30+45i\).

In a digital synthesizer, the complex amplitude of a tone is \(A = 4e^{\frac{2\pi}{3}i}\). A filter multiplies the amplitude by \(0.5e^{-\frac{\pi}{4}i}\), producing the output amplitude \(B\).

An audio effect sends a stereo signal through repeated echoes. The left and right amplitudes \((L_n, R_n)\) after \(n\) echoes satisfy
\(\begin{pmatrix}L_{n+1}\\R_{n+1}\end{pmatrix}=\mathbf{M}\begin{pmatrix}L_n\\R_n\end{pmatrix}\), where \(\mathbf{M}=\begin{pmatrix}0.6&0.3\\0.2&0.5\end{pmatrix}\).

In equal temperament, the frequency of the note \(n\) semitones above the note A4 (440 Hz) is \(f=440\times2^{\frac{n}{12}}\) Hz.

An archaeologist records positions in a trench as complex numbers, with the real axis pointing east, the imaginary axis pointing north and the datum peg at 0, distances in metres. A corner stone is found at z = 6 + 8i.

A cycling shop sells helmets at €h each and locks at €l each. A customer pays €89 for 3 helmets and 2 locks. Another pays €124 for 4 helmets and 3 locks.

A cycling shop sells an electric bike for €3000 on a loan at a nominal annual interest rate of 6%, compounded monthly. The loan is repaid in 24 equal payments made at the end of each month.

The amount of data a data centre stores each day forms a geometric sequence. On day 1 it stores 40 TB and each day it stores 5% more than the day before.

In a treasure-hunting game, the map is modelled as the complex plane with the player at the origin, the real axis pointing east and the imaginary axis pointing north (units: metres). A treasure chest is at \(z = 4 - 7i\).

A sprite in a game is at the point \(z_0 = 2 + 3i\) in the complex plane (units: cm). A special effect moves every sprite to a new position by multiplying its coordinate by \(m = 1.5\,e^{0.7i}\).

In a puzzle game, a pair of scores \((p, q)\) is encoded as the vector \(\mathbf{c} = \mathbf{A}\begin{pmatrix}p\\q\end{pmatrix}\) where \(\mathbf{A} = \begin{pmatrix}2&3\\1&k\end{pmatrix}\) and \(k\) is a constant.

A space probe's antenna signal is represented by the complex number z = 5 − 12i (volts, as a phasor).

The displacement of a bridge sensor's vibration is modelled by the complex number z = 7e^2πi/5 (in mm).

A town sets up a bridge-repair fund with €25 000 invested at a nominal annual interest rate of 3.2%, compounded quarterly.

A space probe is 2.37 × 10⁸ km from Earth. A radio signal travels at 3.00 × 10⁸ m s⁻¹.

In an aviation weather model, a wind velocity is represented by the complex number \(w=-9+12i\) knots, where the real axis points east and the imaginary axis points north.

Two pure tones of the same frequency reach a microphone. Their pressure amplitudes and phases (in millipascals and radians) are represented by the complex numbers \(z_1=2e^{i\pi/3}\) and \(z_2=3e^{i\pi/6}\). The combined tone is represented by \(z_1+z_2\).

A farmer applies fertiliser to three fields. The matrix \(M\) gives the nutrient application (kg per hectare) for nitrogen, phosphorus and potassium in fields A, B and C:
\(M=\begin{pmatrix}120&40&60\\90&50&80\\150&30&40\end{pmatrix}\) (rows: fields A, B, C; columns: N, P, K).
The areas of the fields in hectares are given by \(H=\begin{pmatrix}12&8&5\end{pmatrix}\). The prices of N, P and K are €0.85, €1.90 and €0.70 per kg respectively, written as the column matrix \(p=\begin{pmatrix}0.85\\1.90\\0.70\end{pmatrix}\).

A plane flies 1260 km from airport P to airport Q along a straight route and back again, in a steady wind that blows along the route. The outward flight takes 2.1 hours and the return flight takes 2.4 hours. Let \(v\) km h⁻¹ be the airspeed of the plane and \(w\) km h⁻¹ the wind speed.

The displacement of a warehouse robot from its charging point is modelled by complex numbers, in metres. Move one is \(z = 2 + 5i\) and move two is \(w = 3 - i\).

A delivery robot's target is located at the point represented by the complex number \(z = -4 + 7i\) on a floor plan, with distances in metres.

A small fashion label makes jackets, dresses and skirts. The fabric needed per garment, in metres, is shown in matrix F, with columns for wool and silk. In one week the label makes 12 jackets, 20 dresses and 30 skirts, written as the row matrix \(\mathbf{P} = (12\ \ 20\ \ 30)\).
\(\mathbf{F} = \begin{pmatrix}3.5&0.5\\2.0&1.5\\1.2&0.8\end{pmatrix}\)

The biomass of a fish stock in a protected bay is modelled by \(S(t) = 480 \times 0.92^t\) tonnes, where t is the number of years since monitoring began.

Archaeologists date a wooden beam from a settlement using the amount of carbon-14 it contains. The fraction of carbon-14 remaining after \(t\) years is \(F=\left(\tfrac12\right)^{t/5730}\).

A cyclist's training plan has her ride 40 km in week 1, and each week she rides 6 km more than the week before.

Two components in an electronic cooling circuit have complex impedances \(Z_1=(12+5i)\) ohms and \(Z_2=(8-3i)\) ohms.

Six standing stones at an ancient site are located at the points \(z_k=5e^{ik\pi/3}\), \(k=0,1,2,3,4,5\), in the complex plane. Distances are in metres and the origin is the centre of the site.

A beekeeper models a forager bee's flight in the complex plane, with the hive at the origin, the real axis pointing east and the imaginary axis pointing north. Distances are in metres. The bee's first leg is the displacement \(z=90+120i\). Its second leg is the displacement \(w=-60+80i\).

A hospital orders three items for each of three wards. Matrix A gives the number of boxes (gloves, masks, gel bottles) ordered by Wards 1, 2 and 3 (rows):
\(A=\begin{pmatrix}12&8&5\\20&15&10\\6&4&2\end{pmatrix}\)
Matrix B gives the price, in £, of one unit of each item (rows) from Supplier 1 and Supplier 2 (columns):
\(B=\begin{pmatrix}4.50&4.20\\6.00&6.40\\3.25&3.00\end{pmatrix}\)

In a hospital, patients are moved each day between three wards. The numbers of patients in the wards at the start of day n are given by \(\mathbf{p}_{n+1}=M\mathbf{p}_n\), where \(M=\begin{pmatrix}0.5&0.25&0.25\\0.25&0.5&0.25\\0.25&0.25&0.5\end{pmatrix}\) and the total number of patients is constant.

During a Formula 1 stint, the lap time t seconds is modelled by \(t=ax^2+bx+c\), where x is the age of the tyres in laps. The model passes through the points (1, 90.2), (5, 88.6) and (9, 89.8).

A warehouse robot is located by a complex number \(z\), where the real part is the distance east and the imaginary part is the distance north of its docking station, in metres. The robot is at \(z = 4 + 3i\).

A fashion studio orders scarves and hats for three of its shops. The number of each item ordered is given by the matrix \(\mathbf{A}\), where the rows are Shop 1, Shop 2 and Shop 3 and the columns are scarves and hats.
\(\mathbf{A}=\begin{pmatrix}12&8\\20&5\\15&10\end{pmatrix}\)
A scarf costs €18.50 and a hat costs €12.00 to make. The costs are written as the column matrix \(\mathbf{c}=\begin{pmatrix}18.5\\12\end{pmatrix}\).

Two fishing boats record their catches after a day at sea. Boat 1 used 3 nets of type X and 2 nets of type Y and landed 1260 kg of fish. Boat 2 used 5 nets of type X and 4 nets of type Y and landed 2300 kg. Let each net of type X catch \(x\) kg and each net of type Y catch \(y\) kg, and assume every net of the same type catches the same mass.

A rectangular fish tank has internal dimensions 1.85 m by 0.62 m by 0.48 m.

In a strategy game, positions on the map are stored as complex numbers, with the base at the origin. A scout camp is located at \(z=-6+8i\), where the real part is the distance east (in map units) and the imaginary part is the distance north.

A hospital pharmacy supplies three medicines to two wards. The matrix \(\mathbf{N}\) shows the number of boxes ordered per week, with rows for Ward A and Ward B and columns for medicines P, Q and R:
\(\mathbf{N}=\begin{pmatrix}12&8&5\\ 9&15&10\end{pmatrix}\)
The cost per box, in euros, is €4.50 for P, €7.20 for Q and €12.00 for R.

A puzzle game encodes each secret position \((x,y)\) into a pair of display numbers \((p,q)\) using the matrix
\(\mathbf{M}=\begin{pmatrix}4&3\\2&2\end{pmatrix},\qquad \begin{pmatrix}p\\q\end{pmatrix}=\mathbf{M}\begin{pmatrix}x\\y\end{pmatrix}.\)

The voltage of a pump motor controller is modelled by the complex number \(v=6e^{i\pi/6}\) (in volts). A second component has voltage \(w=2e^{i\pi/3}\) (in volts).

In a strategy game, a scout stands at the point represented by the complex number z = 6 + 8i, where distances are in metres from the base at the origin.

In a puzzle game, a magic portal sits at the point z = 4e^2πi/3 in the complex plane.

Chlorine is added to a water storage tank. The concentration of chlorine is modelled by C = 2.4 × 0.85^t mg per litre, where t is the time in hours after the chlorine is added.

In a role-playing game, the experience points (XP) needed to move from level n to level n + 1 form an arithmetic sequence. Moving from level 1 to level 2 needs 400 XP, and each further level-up needs 75 XP more than the one before.

Two components of a railway signalling circuit are connected in series. Their impedances are \(Z_1 = (12 + 5i)\) Ω and \(Z_2 = (3 - 9i)\) Ω. The circuit is supplied with a voltage of 24 V, which is represented by the real number 24.

Two radio signals from a telescope array are represented by the complex numbers \(z_1 = 7e^{i\pi/5}\) and \(z_2 = 2e^{-i\pi/3}\).

An observatory borrows €60 000 to renew a telescope mirror. The loan has a nominal annual interest rate of 5.4%, compounded monthly (that is 0.45% per month). The observatory makes a payment of €1100 at the end of each month.

A fund manager invests €8000 in an account that pays compound interest with the interest added each month. After 7 years the account contains €10 450.

Two radio signals arrive at a receiving antenna. Their amplitudes and phases, in millivolts, are represented by the complex numbers s₁ = 6 + 8i and s₂ = −2 + 3i. The combined signal is s = s₁ + s₂.

On a trek, a climber ascends 600 m on the first day. On each later day the ascent is 80 % of the previous day's ascent.

To protect a data link, pairs of numbers \(\begin{pmatrix}x\\y\end{pmatrix}\) are encoded by multiplying by the matrix \(E=\begin{pmatrix}3&2\\4&3\end{pmatrix}\).

The altitudes of the camps on a mountain form an arithmetic sequence. Camp 2 is at 4550 m and camp 6 is at 6070 m.

Two sonar signals received by a research vessel are represented by the complex numbers \(z_1=2+3i\) and \(z_2=4-i\).

In the plan of a rotating façade, a marker on the panel is at the point represented by \(z=-2+2\sqrt{3}\,i\), where lengths are in metres.

A logistics firm uses small vans carrying \(x\) crates each and large vans carrying \(y\) crates each. A load of 2 small vans and 1 large van carries 140 crates, and a load of 5 small vans and 3 large vans carries 380 crates.

A logistics company has two warehouses, A and B. Each week 20% of the stock in A is moved to B and 30% of the stock in B is moved to A. The proportions in (A, B) change according to \(\binom{a_{n+1}}{b_{n+1}}=\mathbf{T}\binom{a_n}{b_n}\), where \(\mathbf{T}=\begin{pmatrix}0.8&0.3\\0.2&0.7\end{pmatrix}\).

The cooling fan in a data centre has a complex impedance Z₁ = 6 + 8i ohms. It is connected in series with a second component of impedance Z₂ = 4 − 3i ohms.

On an archaeological site, the position of a post-hole relative to a datum point is represented by the complex number z = 6e^iπ/3 metres, where the real axis points east and the imaginary axis points north. A second feature is at w = 2e^iπ/6 metres from the same datum.

In a cycle computer circuit, the voltage is V = (7 − i) volts and the impedance is Z = (2 + 3i) ohms. The current is I = V/Z.

A data centre adds storage capacity every month. In month 1 it adds 40 TB and in each later month it adds 8% more than in the previous month, so the amounts added form a geometric sequence.

The tip of a robot arm is at position z = 0.45 + 0.60i metres in the complex plane, where the base of the arm is at the origin, the real axis is horizontal and the imaginary axis is vertical.

A boutique sells scarves and belts. A customer pays €112 for 3 scarves and 2 belts. Another customer pays €94 for 1 scarf and 4 belts. Let a scarf cost €s and a belt cost €b. The situation is modelled by
A(s, b)^T = (112, 94)^T, where A = [[3, 2], [1, 4]] (rows listed in turn).

A camera calibration for a robot maps each point (x, y) of the robot's frame, in centimetres, to image coordinates using the matrix M = [[3, −1], [2, 4]] (rows listed in turn), so that the image is M(x, y)^T.

The capacity of the battery in a robot vacuum cleaner is measured when fully charged. At the first charge it is 4500 mAh, and each later full charge gives a capacity that is 98.5% of the previous one. So the capacities form a geometric sequence u₁ = 4500, u₂ = 4500 × 0.985, …

A ground station tracks a space probe. In a plane through the station, the probe's position is represented by the complex number \(z=4.2+3.1i\), where the real and imaginary parts are in thousands of kilometres.

In a motion-capture study, a marker on an athlete's knee is represented by the complex number \(z_1=6e^{i\pi/6}\) (in cm). When the athlete's leg moves, the marker's new position is \(z_2=wz_1\), where \(w=0.8e^{i\pi/4}\).

A rail ticket machine records its sales. On Monday it sold 3 single tickets and 2 day-return tickets for a total of €21. On Tuesday it sold 5 single tickets and 4 day-return tickets for a total of €37. Let \(s\) and \(r\) be the prices in euros of a single and a day-return ticket. The system can be written as \(A\begin{pmatrix}s\\r\end{pmatrix}=\begin{pmatrix}21\\37\end{pmatrix}\) where \(A=\begin{pmatrix}3&2\\5&4\end{pmatrix}\).

A sports club buys 12 training bibs and 8 cones for €62. A second order, of 5 bibs and 9 cones, costs €40. Let a bib cost €\(x\) and a cone cost €\(y\).

Two antennas on a space probe produce signals represented by the complex numbers z₁ = 3 + 4i and z₂ = 1 − 2i, where i² = −1.

A sensor on a bridge deck records a vibration as the complex number z = −5 + 12i.

A coffee roaster buys two kinds of green bean, Arabica and Robusta. One mixed batch uses 2 kg of Arabica and 3 kg of Robusta and costs €39.50. A second batch uses 4 kg of Arabica and 1 kg of Robusta and costs €47.50. Let a and r be the prices in € per kg of Arabica and Robusta.

A coffee shop sells bags of light roast and dark roast. On Friday it sells 40 bags of light roast and 25 of dark roast. On Saturday it sells 55 bags of light roast and 30 of dark roast. A bag of light roast costs €6.50 and a bag of dark roast costs €8.20.

In a telecommunications link, two signal components are represented by the complex numbers \(z = 5 + 2i\) and \(w = 3 - 4i\).

A mountain is estimated to be 6800 m high. Its true height is 6749 m.

The acidity of a food sample is measured by its pH, where \(\text{pH}=-\log_{10}[\text{H}^+]\) and \([\text{H}^+]\) is the hydrogen ion concentration in mol dm\(^{-3}\).

The zeros of the transfer function of a telecoms filter are the solutions of \(z^2-8z+41=0\).

A seismometer near a volcano records a tremor signal represented by the complex number z = −4 + 7i (units: µm/s).

A theatre sells three ticket types (stalls, circle, balcony) for each of three performances. The numbers of tickets sold are given by the matrix N with rows for the performances and columns for the ticket types: row 1 is (120, 80, 40), row 2 is (150, 90, 60), row 3 is (100, 70, 50). The ticket prices in euros are (45, 30, 18) for stalls, circle and balcony respectively.

Two cargo ships are loaded with crates of type P and type Q. Ship 1 carries 3 crates of type P and k crates of type Q. Ship 2 carries 2 crates of type P and 5 crates of type Q. Let a type P crate have mass x tonnes and a type Q crate have mass y tonnes. The matrix of crate numbers is M with rows (3, k) and (2, 5).

After an eruption, the thickness of the ash layer, T cm, at a distance d km from the vent is modelled by T = Ae^−kd. The layer is 40 cm thick at 2 km and 10 cm thick at 6 km.

A theatre turntable is centred at the origin of an Argand diagram whose axes are measured in metres. A prop sits at the point represented by \(z = 3 + i\). The turntable rotates anticlockwise through \(60°\) about the origin, which is equivalent to multiplying z by \(e^{i\pi/3}\).

A port charges x euros per standard container and y euros per refrigerated container. A ship unloading 4 standard and 3 refrigerated containers is charged €1200 in total. A second ship unloading 2 standard and 5 refrigerated containers is charged €1300 in total.

A volcano monitoring station uses a filter whose characteristic equation is \(z^3 - 5z^2 + 17z - 13 = 0\). One solution is \(z = 2 + 3i\).

Marine growth on a section of a ship's hull has mass \(m = 4.2e^{0.18t}\) kg, where t is the time in days since the hull was cleaned.

The centre of a revolving theatre stage is the origin of an Argand diagram, with distances in metres. A prop is at the point represented by z = 3 + 4i. The stage then turns anticlockwise through 40° about its centre, which moves the prop to the point represented by w = ze^iθ, where θ is the angle of turn in radians.

A theatre sells stalls tickets at €s each and balcony tickets at €b each. At the first show, 120 stalls and 80 balcony tickets were sold, taking €5280. At the second show, 90 stalls and 150 balcony tickets were sold, taking €5580.

The cargo hold of a ship is a cuboid measured as 38.6 m by 17.4 m by 9.2 m.

An archaeological team lays out a grid with a fixed datum peg at the origin. The east axis is the real axis and the north axis is the imaginary axis, with distances in metres. A marker P is at the point represented by \(z = 7 + 4i\).

A cycling team buys energy bars and energy gels. Let b be the price of one bar and g the price of one gel, both in euros. An order of 6 bars and 4 gels costs €17.40. An order of 3 bars and 5 gels costs €13.65.

In a data centre, a cable multiplies a signal by a complex gain \(g\). The input signal is represented by \(2 - i\) and the output signal by \(7 + 4i\), so that output = \(g \times\) input.

A cycling club needs £10 000 in 6 years to replace its track bikes. It invests a lump sum in an account paying a nominal annual interest rate of 3.0%, compounded quarterly.

On an architect's site plan, two displacement vectors are represented by the complex numbers a = 4 + 3i and b = 2 − i (units: metres).

A freight depot loads two types of crate. A type A crate has mass x kg and a type B crate has mass y kg. Load 1 consists of 3 crates of type A and 2 of type B, with total mass 1450 kg. Load 2 consists of 4 crates of type A and 5 of type B, with total mass 2750 kg.