Shunya / Maths Explainers

Shunya · Maths lab

Maths explainers

Six ideas that are easier to see than to read. Watch a fraction turn into a repeating decimal, drag a complex number around the Argand plane, count possibilities with a tree and Pascal's triangle, grow an arithmetic or geometric series, fill in a Venn diagram, and solve a linear programming problem by sliding a line across a feasible region.

Lessons

Pick your class, follow the steps in the lab, and check what you see against the working under the graph.

Class 9 — Rational and irrational numbers

Concept: a rational number is a fraction p/q; its decimal either stops (terminates) or repeats a block forever. An irrational number, such as √2, has a decimal that neither stops nor repeats.

Open Number systems and press 1/8: the remainder reaches 0 and the decimal stops (0.125).
Press 1/7: the remainders 1, 3, 2, 6, 4, 5 come round again, so 142857 repeats forever.
Try 1/6, 7/13 and 1/97. Which denominators terminate? Only those made of 2s and 5s.
Switch to √n on the number line and slide the zoom: √2 is always trapped between two decimals but never lands on one.
Press √16 (a perfect square): the trap collapses to the whole number 4.

Takeaway: p/q gives a terminating or repeating decimal; a square root of a non-perfect-square never does.

Class 9 — Repeating decimals as fractions and the square-root spiral

Concept: multiplying a repeating decimal by a power of 10 and subtracting removes the endless tail, which leaves a fraction. The square-root spiral shows how to mark √2, √3, √5 … on a line using right triangles.

Open Decimal → fraction and press 0.333…: 10x − x = 3 gives x = 1/3.
Press 0.1666… and follow the two shifts (10x and 100x) and the subtraction to reach 1/6.
Press 0.999… and see that it equals exactly 1.
Open Root spiral and raise the number of triangles: each hypotenuse is √k, and √4, √9, √16 land on whole numbers.
Check one triangle by hand: (√5)² + 1² = 6, so the next hypotenuse is √6.

Takeaway: every repeating decimal is a fraction, and Pythagoras builds every square root on a ruler.

Class 10 — Arithmetic progressions

Concept: an AP has a constant difference d between neighbouring terms. The nth term is a + (n − 1)d and the sum of n terms is n/2 · [2a + (n − 1)d].

Open Sequences & series, choose Arithmetic (AP) and press 2, 5, 8, ….
Read the nth term and the sum in the working, then drag the last dot to change n.
Tick the bars: the total area of the bars is the sum Sₙ.
Move Which term equals… to 20 and read term number 7; try 21 and see it is not in the progression.
Press 10, 8, 6, …: a negative difference gives a decreasing line.

Takeaway: the terms of an AP lie on a straight line, and Sₙ = n/2 · (first + last).

Class 11 — Complex numbers on the Argand plane

Concept: z = a + bi is the point (a, b). Its modulus |z| is its distance from the origin, its argument is its angle, and its conjugate z̄ is its mirror image in the real axis. Adding is the parallelogram rule; multiplying adds the angles and multiplies the moduli.

Open Complex numbers, press 3 + 4i and read |z| = 5 and the argument from the working.
Tick show the conjugate and check that z·z̄ = |z|² = 25.
Open Add & subtract, drag z and w and watch z + w as the diagonal of the parallelogram.
Open Multiply & divide and press multiply by i (w = i): z turns a quarter turn.
Press multiply by z̄ to see that the product is real.

Takeaway: modulus, argument and conjugate describe z; multiplying by w turns and stretches z.

Class 11 — Roots of complex numbers and De Moivre's theorem

Concept: zⁿ has modulus |z|ⁿ and argument n times the argument of z. The n nth roots of z lie at equal angles 360° ÷ n apart on a circle, forming a regular polygon.

Open Roots & powers and press roots of unity (z = 1) with n = 3: three points form an equilateral triangle.
Increase n to 6 and check that the roots still add up to 0.
Press roots of 16i, set n = 4 and read the four roots, 2 ∠ 22.5° and three more.
Switch to powers of z and drag z inside and outside the unit circle to see the powers spiral in or out.
Read the angle between neighbouring roots in the cards.

Takeaway: the nth roots of z are equally spaced on a circle of radius |z|^(1/n).

Class 11 — Counting: the fundamental principle, permutations and combinations

Concept: if a job has stages that can be done in a, b and c ways, the whole job can be done in a × b × c ways. Choosing r items from n gives P(n, r) = n!/(n − r)! ordered arrangements or C(n, r) = P(n, r) ÷ r! unordered groups.

Open Counting and set 3 shirts, 2 trousers and 2 pairs of shoes: count the leaves of the tree (12).
Slide the path number to follow one outfit from root to leaf.
Open Permutations & combinations, set n = 5 and r = 3 and list all 60 permutations, then switch to the 10 combinations.
Check that P = C × r! in the cards.
Open Arrangements of a word, press BANANA and read 6! ÷ (3! 2!) = 60.

Takeaway: multiply the ways for each stage; divide by r! when order does not matter.

Class 11 — Pascal's triangle and the binomial theorem

Concept: row n of Pascal's triangle gives C(n, 0), C(n, 1), …, C(n, n), the coefficients of (a + b)ⁿ. Each entry is the sum of the two above it and every row adds to 2ⁿ.

Open Pascal's triangle and set the row n = 5: 1 5 10 10 5 1.
Move the entry slider and see the two entries above it in the row before highlighted in teal.
Read (a + b)⁵ in the working and check that its coefficients are the row.
Check that the row adds to 2⁵ = 32.
Raise the rows to 12 and look for the symmetry C(n, k) = C(n, n − k).

Takeaway: C(n, k) = C(n − 1, k − 1) + C(n − 1, k), and the row sum is 2ⁿ.

Class 11 — Geometric progressions, AM–GM–HM and standard sums

Concept: in a GP each term is r times the one before, so aₙ = a·rⁿ⁻¹. If |r| < 1 the terms shrink and the sum settles on a/(1 − r). For two positive numbers, AM ≥ GM ≥ HM.

Choose Geometric (GP) and press 8, 4, 2, … (r = ½): the teal partial sums climb to the dashed line at 16.
Press 2, 4, 8, … (r = 2): the partial sums have no limit.
Open AM, GM, HM and drag the meeting point: the GM (height) never exceeds the AM (radius).
Set a = b and see all three means equal.
Open Standard sums, compare 1 + 2 + … + n with n(n + 1)/2, and see the cubes add up to the square of the sum.

Takeaway: a geometric series converges only when |r| < 1, and AM ≥ GM ≥ HM with equality when the numbers are equal.

Class 11 — Sets and Venn diagrams

Concept: a set is a collection of distinct members. |A ∪ B| = |A| + |B| − |A ∩ B|, because the overlap is counted twice. De Morgan's laws say (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

Open Sets & Venn diagrams and press Tea and coffee: 28 + 22 − 9 = 41 people drink at least one.
Use Shade this region to highlight A ∪ B, A ∩ B and A − B and read how many are in each.
Open Three sets, press Survey of 100 students and read the region counts.
Press An impossible one and see the red region with a negative count.
Open Sets of elements, press overlap and check De Morgan's laws in the working.

Takeaway: count the overlap once, and a set of n members has 2ⁿ subsets.

Class 12 — Linear programming

Concept: maximise or minimise a linear objective Z = ax + by subject to linear constraints. The constraints bound a feasible region, and the best value of Z occurs at one of its corner points.

Open Linear programming and press Furniture: max 5x + 4y to see four constraints and the teal feasible region.
Read the corner points and the value of Z at each in the working; the largest is the answer.
Slide the objective line from the worst corner to the best: the last corner it touches is the optimum.
Press Diet: min 4x + 3y and switch between maximise and minimise on the same region.
Press Unbounded and Infeasible to see the two special cases, and Two best points for many optimal solutions.

Takeaway: evaluate Z at every corner point; for an unbounded region check that the best value really exists.

The maths behind it

What each explainer covers

Number systems — a fraction turned into a decimal by long division (and why it terminates or repeats), a repeating decimal turned back into a fraction, √n squeezed between decimals on the number line, and the square-root spiral. Classes 9 and 10.

Complex numbers — a draggable z on the Argand plane with its conjugate, modulus and argument; sum and difference as a parallelogram; product and quotient as a turn and a stretch; nth roots and powers (De Moivre). Class 11.

Counting — the multiplication principle as a tree you can follow, permutations and combinations listed out, Pascal's triangle and the binomial coefficients, and the arrangements of the letters of a word. Class 11.

Sequences & series — arithmetic progressions with draggable terms and bars for the sum, geometric progressions with partial sums closing in on the sum to infinity, AM ≥ GM ≥ HM on a semicircle, and the standard sums and Fibonacci numbers. Classes 10 and 11.

Sets & Venn diagrams — two and three sets by counts (inclusion–exclusion, impossible numbers flagged) and by typed elements (union, intersection, difference, complement, subsets, De Morgan). Class 11.

Linear programming — type an objective and up to six constraints, see the feasible region and its corner points, slide the objective line to the best corner, and meet unbounded, infeasible and many-optima cases. Class 12.

Part of Shunya. See the syllabus map for the chapters these explainers map to. Complex numbers and counting can be checked against the Function & Graph Explorer and the Statistics & Probability lab.

Formulas used in this lab

Number systems Class 9

A rational number: $\dfrac{p}{q},\qquad p, q \text{ integers},\ q \ne 0$
Terminating decimal: $q = 2^m \cdot 5^n \ \Rightarrow\ \dfrac{p}{q} \text{ terminates}$
Repeating decimal to a fraction: $0.\overline{ab} = \dfrac{ab}{99},\qquad 0.\overline{a} = \dfrac{a}{9}$
Rationalising: $\dfrac{1}{a + b\sqrt{c}} = \dfrac{a - b\sqrt{c}}{a^2 - b^2 c}$
Laws of surds: $\sqrt{a}\sqrt{b} = \sqrt{ab},\qquad (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$

Arithmetic progressions Class 10

nth term: $a_n = a + (n - 1)d$
Sum of n terms: $S_n = \dfrac{n}{2}\,[2a + (n - 1)d]$
Sum from the last term l: $S_n = \dfrac{n}{2}\,(a + l)$
A term from the sums: $a_n = S_n - S_{n-1}$
Three numbers in AP: $2b = a + c$

Complex numbers Class 11

The imaginary unit: $i^2 = -1,\qquad i^4 = 1$
Modulus and conjugate: $|z| = \sqrt{a^2 + b^2},\qquad z\bar{z} = |z|^2$
Polar form: $z = r(\cos\theta + i\sin\theta),\qquad r = |z|$
Product and quotient: $|z_1z_2| = |z_1||z_2|,\qquad \arg(z_1z_2) = \arg z_1 + \arg z_2$
Quotient by the conjugate: $\dfrac{z_1}{z_2} = \dfrac{z_1\bar{z_2}}{|z_2|^2}$
Triangle inequality: $|z_1 + z_2| \le |z_1| + |z_2|$

Roots of complex numbers Class 11

De Moivre's theorem: $[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)$
The nth roots of z (k = 0, 1, …, n − 1): modulus and angle: $r^{1/n},\quad \dfrac{\theta + 2\pi k}{n}$
Roots of unity: $\omega = e^{2\pi i/n},\qquad 1 + \omega + \dots + \omega^{n-1} = 0$
Cube roots of unity: $\omega^3 = 1,\qquad 1 + \omega + \omega^2 = 0$

Counting Class 11

Multiplication principle: $m \times n \times p \text{ ways}$
Permutations: ${}^nP_r = \dfrac{n!}{(n - r)!}$
Combinations: ${}^nC_r = \dfrac{n!}{r!\,(n - r)!}$
Link between them: ${}^nP_r = {}^nC_r \cdot r!,\qquad {}^nC_r = {}^nC_{n-r}$
Pascal's rule: ${}^nC_r = {}^{n-1}C_{r-1} + {}^{n-1}C_r$
Arrangements with repeats: $\dfrac{n!}{p!\,q!\,r!}$
In a circle: $(n - 1)!$

Binomial theorem Class 11

Expansion: $(a + b)^n = \sum_{r=0}^{n} {}^nC_r\, a^{n-r}b^r$
General term: $T_{r+1} = {}^nC_r\, a^{n-r}b^r$
Number of terms: $n + 1$
Sum of the coefficients: ${}^nC_0 + {}^nC_1 + \dots + {}^nC_n = 2^n$
Alternating sum: ${}^nC_0 - {}^nC_1 + {}^nC_2 - \dots = 0$

Geometric progressions and means Class 11

nth term: $a_n = ar^{n-1}$
Sum of n terms: $S_n = \dfrac{a(1 - r^n)}{1 - r},\qquad r \ne 1$
Sum to infinity: $S_\infty = \dfrac{a}{1 - r},\qquad |r| \lt 1$
The three means: $AM = \dfrac{a + b}{2},\quad GM = \sqrt{ab},\quad HM = \dfrac{2ab}{a + b}$
Order and link: $AM \ge GM \ge HM,\qquad GM^2 = AM \cdot HM$

Standard sums Class 11

Natural numbers: $1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}$
Squares: $1^2 + 2^2 + \dots + n^2 = \dfrac{n(n + 1)(2n + 1)}{6}$
Cubes: $1^3 + 2^3 + \dots + n^3 = \left[\dfrac{n(n + 1)}{2}\right]^2$
Fibonacci numbers: $F_n = F_{n-1} + F_{n-2},\qquad F_1 + \dots + F_n = F_{n+2} - 1$

Sets Class 11

Two sets: $|A \cup B| = |A| + |B| - |A \cap B|$
Three sets: $|A \cup B \cup C| = \sum |A| - \sum |A \cap B| + |A \cap B \cap C|$
De Morgan's laws: $(A \cup B)' = A' \cap B',\qquad (A \cap B)' = A' \cup B'$
Difference and complement: $A - B = A \cap B',\qquad A' = U - A$
Number of subsets: $|P(A)| = 2^n$

Linear programming Class 12

Objective function: $Z = ax + by$
Constraints: $a_1x + b_1y \le c_1,\quad a_2x + b_2y \ge c_2,\quad x, y \ge 0$
Corner point theorem: the best Z is at: $\text{a corner point}$
Two optimal corners: the optimum is: $\text{the whole edge}$
Unbounded region: check that: $Z \gt M \text{ meets it}$