Multiplying two complex numbers

• the composite 4 does not divide P(4) ×17737
• the prime 2 divides P(2) ×2262
• at n = 0, 3 times the class-0 total less 2ⁿ is what the other roots contribute ×98
• (3 + 4i)^1 is not a real number ×60
• and it has length 5^1 ×60
• the search for 6 agrees with the prime-factor rule at 1 terms ×47
• the coefficient of x^0 at power 1 ×40
• P(0) − ρ^0 is twice the real part of σ^0 ×31
• the derivative of z^3 − 1 is 3z^2: coefficient 0 vanishes ×27
• the root at step 0 first returns to 1 after 1 multiplications ×26
• root 0 raised to the 5 is 1 ×24
• the coefficient of x^0 in the product is the one in xⁿ − 1 ×16
• and that number is the Möbius function of 1 ×12
• and the average over the 1th roots of ∏(1 + ζᵏ) gives the same ×12
• the count is the necklace count exactly when 1 is odd ×12
• the odd-divisor sum for 1 is a multiple of 1 ×12
• the primitive roots of order 1 add to a real number ×12
• the subsets of 1…1 with a total divisible by 1, listed, match the formula ×12
• the product 1,4,6,4,1 was found by the search ×10
• row 1: the terms written out add to the entry ×8
• the factor belonging to the roots of order 1 has degree φ(1) ×8
• the gaps at n = 6 repeat those at n = 0 ×8
• the product 1,-1,-1,1 was found by the search ×8
• the real part of (3 + 4i)^1 is not a multiple of 5, so the fraction is in lowest terms ×8
• 1 + ζ^0 has length 2|cos(0π/3)| ×7
• and it is the coefficients of index 0 mod 3 added one by one ×7
• the curve the image of the circle of radius 0.4 never passes exactly through the origin ×7
• the image of radius 0.4 turns once for each root inside it ×7
• the image of the circle of radius 0.4 closes up after a whole number of turns ×7
• the product 1,2,1 was found by the search ×7
• the walk along the image of the circle of radius 0.4 is fine enough to see which way it turned ×7
• the share is exactly 1/2 precisely when the die's polynomial vanishes at the other roots ×6
• the throws with a total divisible by 2, counted through the roots ×6
• the roots of the 1-th derivative lie in the hull of the ones before ×5
• at degree 4 nothing sits between measure 1 and the smallest found ×4
• exactly one root of p′ between the 1-th and 2-th roots ×4
• the circle of radius 0.4 has 0 roots inside and turns 0 times ×4
• the class-0 total found through the roots is real ×4
• the curve the image of radius 0.4 never passes exactly through the origin ×4
• the degree-1 cyclotomic products, multiplied out, are exactly the polynomials found ×4
• the image of radius 0.4 closes up after a whole number of turns ×4
• the walk along the image of radius 0.4 is fine enough to see which way it turned ×4
• x-polynomial x⁴ + 2x³ + 2x² + 2x + 1 has every root in the disc and is a product of cyclotomic polynomials ×4
• x-polynomial x⁴ − 4x³ + 6x² − 4x + 1 has every root in the disc and is a product of cyclotomic polynomials ×4
• side 1 stays outside the ellipse and touches it at its midpoint (equilateral) ×3
• side 1 stays outside the ellipse and touches it at its midpoint (obtuse) ×3
• side 1 stays outside the ellipse and touches it at its midpoint (scalene) ×3
• the product -1,-1,1,1 was found by the search ×3
• the product -1,3,-3,1 was found by the search ×3
• far out, the image turns once for each power of z, which is 3 times ×2
• the product -1,0,1 was found by the search ×2
• the product -1,2,0,-2,1 was found by the search ×2
• the product 1,-2,1 was found by the search ×2
• the product 1,0,-2,0,1 was found by the search ×2
• the roots agree on p₁ at c = -3 ×2
• the roots agree on p₁ at c = 1 ×2
• the roots agree on p₂ at c = -3 ×2
• the roots agree on p₂ at c = 1 ×2
• the roots agree on p₃ at c = -3 ×2
• the roots agree on p₃ at c = 1 ×2
• x-polynomial x³ + 2x² + 2x + 1 has every root in the disc and is a product of cyclotomic polynomials ×2
• x-polynomial x³ − 3x² + 3x − 1 has every root in the disc and is a product of cyclotomic polynomials ×2
• 271441 is 521 squared ×1
• a polynomial of degree 3 has 3 roots ×1
• all 12 roots add to zero ×1
• and |σ|²ρ = 1, the constant term ×1
• and close in it does not go round the origin at all ×1
• and has no imaginary part left ×1
• and it crosses between them ×1
• and it divides its own Perrin number ×1
• and its coefficients are whole numbers ×1
• and its measure is 1.17628 ×1
• and mod 4 it does outgrow anything mod 3 allows ×1
• and multiply to the constant term, with a sign that follows the degree ×1
• and one inside, its reciprocal ×1
• and so is the necklace sum ×1
• and the angles add ×1
• and the degrees of the factors add to n, because every root has exactly one order ×1
• and the derivative's roots have the same average as the roots ×1
• and the middle coefficient lies strictly between −2 and 2 ×1
• and the two off the circle multiply to 1 ×1
• and their imaginary parts cancel ×1
• and their imaginary parts cancel, because the coefficients are real ×1
• and whole ones ×1
• and Σz³ is −3c ×1
• at a place where the discriminant is passing through zero ×1
• at most eight powers are labelled ×1
• between 1 and 8 dice are thrown ×1
• between 2 and 8 moduli from 2 to 12 are compared ×1
• between 20 and 300 polynomials a degree ×1
• between 8 and 400 powers are drawn ×1
• between two and five whole-number constant terms ×1
• between two and six powers are chained ×1
• each example lands in the group it was chosen for ×1
• each hull is no larger than the one before ×1
• each pairing's value is a root of the resolvent cubic, computed from the coefficients alone ×1
• each root found really is a root ×1
• each step of the division gives a whole number ×1
• eight of its ten roots are on the unit circle ×1
• every root found is a root of unity ×1
• every root found really is a root ×1
• every root is on the unit circle ×1
• every root of p′ is real ×1
• every root of p′ lies in the hull of the roots (cluster) ×1
• every root of p′ lies in the hull of the roots (line) ×1
• every root of p′ lies in the hull of the roots (scatter) ×1
• in both coordinates ×1
• Lehmer's polynomial has four roots on the upper half of the circle ×1
• mod 3 no class is ever more than 2/3 away from a third of 2ⁿ ×1
• mod 4 the gap is no larger than what 1 + i and 1 − i can contribute ×1
• no power of (3 + 4i)/5 up to the 60th returns to 1 ×1
• no power of a root of Lehmer's polynomial up to the 80th returns to 1 ×1
• no sampled root is as far as 1 from every root of the derivative ×1
• one fewer root of p′ than of p ×1
• one is outside ×1
• one real root and a complex pair ×1
• one to three of the named root sets ×1
• one to three of the named triangles ×1
• outside the hull the pushes never cancel ×1
• quartic is read only by the pairings view ×1
• row 1: the identity and the roots agree on p₁ ×1
• row 2: the identity and the roots agree on p₂ ×1
• row 3: the identity and the roots agree on p₃ ×1
• row 4: the identity and the roots agree on p₄ ×1
• row 5: the identity and the roots agree on p₅ ×1
• row 6: the identity and the roots agree on p₆ ×1
• row 7: the identity and the roots agree on p₇ ×1
• row 8: the identity and the roots agree on p₈ ×1
• set is read only by the vanish view ×1
• six roots, then five, four, three, two and one ×1
• subsets are listed for sets of at most 14 numbers ×1
• the 1-th powers of the roots add to p₁, found from the coefficients alone ×1
• the 2-th powers of the roots add to p₂, found from the coefficients alone ×1
• the 24 permutations fall into 6 classes of 4 by their effect on the pairings ×1
• the 3-th powers of the roots add to p₃, found from the coefficients alone ×1
• the 4-th powers of the roots add to p₄, found from the coefficients alone ×1
• the bar chart shows a set of 3 to 9 numbers inside the table ×1
• the classes are remainders mod 3 or mod 4 ×1
• the classes are remainders on division by 2 to 6 ×1
• the classes between them hold all 2ⁿ of the coefficients' total ×1
• the complex pair lies inside the unit circle ×1
• the derivative of a degree-5 polynomial has 4 roots ×1
• the distances to the two roots of p′ add to the same total at every midpoint (equilateral) ×1
• the distances to the two roots of p′ add to the same total at every midpoint (obtuse) ×1
• the distances to the two roots of p′ add to the same total at every midpoint (scalene) ×1
• the division leaves no remainder ×1
• the ellipse is centred on the centroid ×1
• the equation is zⁿ = 1 for n between 4 and 20 ×1
• the factors multiply out to a polynomial of the right degree ×1
• the family passes from three real roots to one ×1
• the fifth derivative's one root is the average of the six ×1
• the fifth-root polynomial has real coefficients ×1
• the grid runs to between 21 and 81 ×1
• the lengths multiply ×1
• the loops sit inside their panel, below its header ×1
• the modulus is small enough for exact products in a double ×1
• the nearest polynomial outside the disc is well clear of the tolerance ×1
• the number of powers drawn is between 2 and 24 ×1
• the number of powers taken is between 1 and 24 ×1
• the number of roots of unity is between 2 and 24 ×1
• the one outside is real ×1
• the pairings that come out whole are the resolvent's rational roots ×1
• the permutations fixing every pairing are the identity and the three double swaps ×1
• the point drawn is a root of Lehmer's polynomial ×1
• the point is on the unit circle ×1
• the point whose powers are drawn is one the family names ×1
• the polygon drawn has between 3 and 16 corners ×1
• the polygon has 5 to 16 sides ×1
• the polynomial has a degree ×1
• the polynomial has a degree between 1 and 6 ×1
• the polynomial is monic, so its power sums are whole numbers ×1
• the polynomials searched have degree 2 to 4 ×1
• the power drawn is between 2 and 16 ×1
• the power sum is still an exact whole number ×1
• the product -1,0,0,0,1 was found by the search ×1
• the product -1,1 was found by the search ×1
• the product 1,1 was found by the search ×1
• the pushes cancel at each root of p′ ×1
• the quartic has no whole-number root, so its pairings are not trivially rational ×1
• the quartic is one the family names ×1
• the radii drawn are between 2 and 5 positive numbers no larger than 4 ×1
• the real root is the plastic number, 1.3247… ×1
• the remainders are taken on division by 2 to 12 ×1
• the ring outside the hull was checked ×1
• the root set is one the family names ×1
• the roots add to minus the next coefficient over the leading one ×1
• the roots are distinct ×1
• the roots sum to zero ×1
• the search stays small enough to finish ×1
• the sequence is drawn to between 12 and 40 terms ×1
• the sequence settles on the nearest whole number to ρⁿ well inside the range ×1
• the seven-term sum splits into turned regular polygons ×1
• the seventh root returns to 1 at the seventh power ×1
• the sign of the discriminant says how many roots are real ×1
• the six-term sum is minimal and is no union of turned polygons ×1
• the smallest measure at degree 10 is Lehmer's polynomial ×1
• the sum is one the family names ×1
• the sum is zero, exactly, modulo the cyclotomic polynomial ×1
• the sums take all three of the values that function takes ×1
• the sweep meets both cases ×1
• the sweep takes between 20 and 800 samples ×1
• the swept family is a depressed cubic with a negative linear term ×1
• the table of sums runs to between 6 and 16 ×1
• the table runs to at most 12 powers ×1
• the table runs to between 3 and 10 powers ×1
• the table runs to between 6 and 16 ×1
• the third derivative's three roots still span a triangle ×1
• the totals account for every throw ×1
• the two chains drawn are for moduli in the table ×1
• the vanish view takes its n from the named sum ×1
• the view is one the family draws ×1
• three to seven real roots ×1
• with an even number of sides some diagonals meet three or more at a point ×1
• with an odd number of sides no three diagonals meet, so there are C(n, 4) crossing points ×1
• with nothing imaginary left over ×1
• x-polynomial x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x² + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x² + 2x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x² + x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x² − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x² − 2x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x² − x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x³ + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x³ + x² + x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x³ + x² − x − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x³ − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x³ − x² + x − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x³ − x² − x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ + 2x² + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ + 2x³ − 2x − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ + x² + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ + x³ + 2x² + x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ + x³ + x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ + x³ + x² + x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ + x³ − x − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ − 2x² + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ − 2x³ + 2x − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ − x² + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ − x³ + 2x² − x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ − x³ + x − 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ − x³ + x² − x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• x-polynomial x⁴ − x³ − x + 1 has every root in the disc and is a product of cyclotomic polynomials ×1
• Σz is 0 for every member ×1
• Σz² is 6 for every member ×1

The whole library · What the figures prove