1.2 Number and algebra - AHL content

Count arrangements by deciding whether order matters.

Use the multiplication principle for sequential choices, permutations when order matters, and combinations when it does not.

Worked example
Choosing president and secretary from 5 uses 5P2=20; choosing a 2-person team uses 5C2=10.

Worked example
Does swapping two team members make a new team? no; use combinations.

Common boundary
Do not use permutations when the selected group has no roles.

Extended binomial form: for rational $n$, $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots$. When $n$ is negative or fractional this is an infinite expansion valid for $|x|<1$. For example, $(1+x)^{-1}=1-x+x^2-x^3+\cdots$. This objective excludes circular arrangements, identical-object permutations and proof of the theorem.

Decompose a proper rational expression into partial fractions.

For distinct linear factors, write unknown numerators over each factor, clear denominators, then solve coefficients.

Worked example
(3x+5)/(x(x+2))=A/x+B/(x+2); substituting x=0 gives A=5/2.

Worked example
What condition must hold first? numerator degree is lower than denominator degree.

Common boundary
Partial fractions is not polynomial division unless the fraction is improper.

Complete the decomposition: $\frac{3x+5}{x(x+2)}=\frac{A}{x}+\frac{B}{x+2}$ gives $3x+5=A(x+2)+Bx$. Setting $x=0$ gives $A=5/2$; setting $x=-2$ gives $-1=-2B$, so $B=1/2$. Therefore $\frac{3x+5}{x(x+2)}=\frac{5}{2x}+\frac{1}{2(x+2)}$. Recombine the fractions to check the numerator.

Represent a complex number and read its geometry.

Write z=a+bi with i²=−1; Re(z)=a, Im(z)=b, modulus is distance from origin and conjugation reflects across the real axis.

Worked example
For z=3−4i, |z|=5 and its conjugate is 3+4i.

Worked example
What does conjugation change? the sign of the imaginary part only.

Common boundary
The modulus is not the imaginary part.

Argand and argument example: $z=3-4i$ is the point $(3,-4)$ on the Argand diagram. Its modulus is $|z|=\sqrt{3^2+(-4)^2}=5$ and its principal argument is $\arg z=\tan^{-1}(-4/3)\approx-53.13^\circ$ because the point lies in quadrant IV. Always use the quadrant, not inverse tangent alone, to select the argument.

Convert complex numbers between Cartesian and polar form.

Polar form z=r(cosθ+i sinθ) separates magnitude r from direction θ; products multiply magnitudes and add arguments.

Worked example
(2 cis 30°)(3 cis 20°)=6 cis 50°.

Worked example
What happens to arguments when multiplying? add them, accounting for the chosen branch.

Common boundary
Do not add Cartesian coordinates when multiplying complex numbers.

Cartesian-to-polar/Euler example: for $z=1+i$, $r=\sqrt{1^2+1^2}=\sqrt2$ and $\theta=\pi/4$, so $z=\sqrt2\,\mathrm{cis}(\pi/4)=\sqrt2e^{i\pi/4}$. Products multiply moduli and add arguments; quotients divide moduli and subtract arguments. Vector addition is performed most directly in Cartesian form.

Use De Moivre to find powers and roots.

De Moivre gives (r cisθ)ⁿ=rⁿ cis(nθ); roots share magnitude r^(1/n) and differ by 2π/n in argument.

Worked example
The cube roots of 8 include arguments 0, 120° and 240°, all with modulus 2.

Worked example
Why are there n roots? rotating by 2π/n gives distinct solutions.

Common boundary
Taking only the principal root misses the other roots.

Two required extensions: if a polynomial $P$ has real coefficients and $P(c)=0$, then $P(\overline c)=\overline{P(c)}=0$, so non-real roots occur in conjugate pairs. For positive integers, prove De Moivre by induction: the $n=1$ case is immediate; assuming $(\mathrm{cis}\,\theta)^k=\mathrm{cis}(k\theta)$, multiply by $\mathrm{cis}\,\theta$ and use angle addition to obtain $\mathrm{cis}((k+1)\theta)$. This completes the inductive step.

Choose induction or contradiction to prove a claim.

Induction proves a statement for all integers by a base case plus an implication from k to k+1; contradiction assumes the claim is false and derives impossibility.

Worked example
For 1+2+…+n=n(n+1)/2, verify n=1, then assume k and prove k+1.

Worked example
What must induction include besides the algebra? a valid base case and the inductive link.

Common boundary
Checking several values is not induction.

Contradiction example: assume $\sqrt2=p/q$ in lowest terms. Then $p^2=2q^2$, so $p$ is even; writing $p=2k$ shows $q$ is also even, contradicting lowest terms. Counterexample example: the claim 'there are no positive integer solutions to $x^2+y^2=10$' is false because $(x,y)=(1,3)$ gives $1^2+3^2=10$. State exactly which universal claim the example violates.

Classify and solve a linear system.

Row reduction or technology reveals whether equations meet at one point, share a line/plane, or conflict; classify before reporting a solution.

Worked example
x+y=2 and 2x+2y=4 have infinitely many solutions because the equations are dependent.

Worked example
What signals no solution? a contradictory row such as 0=1.

Common boundary
Three equations do not guarantee a unique solution.

Three-equation example: solve $x+y+z=6$, $2x-y+z=3$, and $x+2y-z=2$. Subtracting the first equation from the second gives $x-2y=-3$; subtracting it from the third gives $y-2z=-4$. Hence $x=2y-3$, $y=2z-4$, and substitution into the first equation gives $7z=21$. Therefore $(x,y,z)=(1,2,3)$, a unique solution; substitution checks all three equations.