1.2 Number and algebra - AHL content

For positive arguments and a valid base b, logarithms obey log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x − log_b y, and log_b(xᵏ) = k log_b x. These rules convert multiplication, division and powers into simpler additive relationships.

To solve 5ˣ = 17, take natural logs: x ln 5 = ln 17, so x = ln 17/ln 5. The same change-of-base idea works with a calculator that provides ln or log.

The product rule does not apply to a sum: log(a + b) is not log a + log b. Every logarithm argument must be positive, and the base must be positive and not equal to 1.

IB examination boundary: the logarithm base is $10$ or $e$, so use $\log$ or $\ln$ consistently. Example: $\ln(3x)-\ln x=\ln3$ for $x>0$; the domain condition cannot be discarded even if the simplified expression is constant. The laws apply only when every original logarithm argument is positive.

A rational exponent is defined so that a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ when the expression is real. The denominator n determines the root; the numerator m determines the power.

8^(2/3) = (∛8)² = 2² = 4. Rewriting the expression as a root first makes the order of operations visible and avoids treating m/n as an ordinary multiplier.

For even n, a negative base may not produce a real value. Keep the domain in mind, and remember that a^(−m/n) is the reciprocal of a^(m/n), not a negative result.

For a geometric series with first term a and common ratio r, the infinite sum is S∞ = a/(1 − r) only when |r| < 1. That condition means successive terms approach zero, so adding more terms changes the total by smaller and smaller amounts.

The series 12 + 6 + 3 + … has a = 12 and r = 1/2, so S∞ = 12/(1 − 1/2) = 24. By contrast, 12 + 24 + 48 + … has |r| > 1 and diverges.

Do not use the formula merely because a series is geometric. If r = 1 the terms never decrease, and if |r| > 1 they grow or fail to vanish; neither case has a finite sum to infinity.

Write a complex number as z = a + bi, where i² = −1. The real part a gives the horizontal coordinate and the imaginary part b gives the vertical coordinate, so z can be plotted as (a,b) on an Argand diagram.

Add and subtract complex numbers component by component. For multiplication, expand normally and replace i² with −1: (2 + i)(3 − i) = 7 + i. The conjugate of a + bi is a − bi, and z z̄ = a² + b² is real.

i is not a variable whose square is positive; i² is defined as −1. Keep the real and imaginary parts separate when interpreting a result geometrically.

For division, multiply numerator and denominator by the denominator's conjugate: $(3+2i)/(1-i)=(1+5i)/2$. Powers follow repeated multiplication or technology in Cartesian form. A real quadratic with negative discriminant has a conjugate pair of roots; for $x^2-4x+13=0$, $x=[4\pm\sqrt{-36}]/2=2\pm3i$, plotted symmetrically about the real axis on an Argand diagram.

A complex number can be written in Cartesian form z = a + bi or polar form z = r(cos θ + i sin θ) = re^{iθ}, where r = |z| and θ is an argument. Cartesian form is natural for addition; polar or exponential form makes multiplication, division and powers easier.

For z₁ = 2e^{iπ/3} and z₂ = 3e^{iπ/6}, z₁z₂ = 6e^{iπ/2}. Multiplication multiplies moduli and adds arguments; converting back gives 6i.

Arguments differ by multiples of 2π, and the principal argument depends on the chosen range. When converting from Cartesian form, use the correct quadrant rather than relying on arctan(b/a) alone.

Integer powers follow De Moivre's pattern: $[r\operatorname{cis}\theta]^n=r^n\operatorname{cis}(n\theta)$, while division divides moduli and subtracts arguments. Geometrically, multiplication by $r\operatorname{cis}\theta$ scales by $r$ and rotates by $\theta$. Adding same-frequency sinusoids is vector addition of complex amplitudes, which determines the resultant amplitude and phase; finding complex roots is not required in this objective.

A matrix is a rectangular array of numbers. Its dimensions are rows × columns, and those dimensions are part of the meaning: they tell you whether an operation is defined.

Addition requires equal dimensions. For multiplication, the inner dimensions must match: an m×n matrix can multiply an n×p matrix, producing an m×p matrix. Each output entry is a row–column dot product, so matrix multiplication is generally not commutative.

For example, [[1,2],[3,4]] [5,6]^T = [17,39]^T because the first output is 1·5+2·6 and the second is 3·5+4·6. The result is 2×1, as the dimension rule predicts.

Do not multiply entries pairwise unless the question explicitly asks for a different operation. Check dimensions first, then keep the row–column order; reversing the factors may be impossible or may give a different matrix.

For A=\begin{pmatrix}a&b\c&d\end{pmatrix}, $\det A=ad-bc$ and, when this is non-zero, $A^{-1}=(ad-bc)^{-1}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$. Identity $I$ satisfies $AI=IA=A$ and zero matrix $O$ satisfies $A+O=A$. A system can be written $A\mathbf x=\mathbf b$ and solved as $\mathbf x=A^{-1}\mathbf b$ when $A$ is invertible; technology handles larger determinants/inverses, while hand work is limited to $2\times2$.

An eigenvector is a non-zero vector v whose direction a matrix A does not change: Av = λv. The scale factor λ is its eigenvalue.

Find possible λ by solving det(A−λI)=0. For each λ, solve (A−λI)v=0 to obtain the corresponding eigenvectors. The determinant equation identifies scales; the second equation identifies the directions that actually realise them.

For A = [[2,0],[0,3]], the characteristic equation is (2−λ)(3−λ)=0, so λ=2 or 3. The vectors [1,0]^T and [0,1]^T are the matching eigenvectors because A preserves each axis and scales it by its eigenvalue.

An eigenvector cannot be the zero vector, and an arbitrary column of A is not automatically an eigenvector. A repeated eigenvalue may have more than one independent eigenvector, so always solve the null-space equation rather than guessing from the entries.

If a $2\times2$ matrix has two distinct real eigenvalues with independent eigenvectors, put those eigenvectors into columns of $P$ and the eigenvalues into diagonal $D$. Then $A=PDP^{-1}$ and $A^n=PD^nP^{-1}$, where $D^n$ simply raises each eigenvalue to the nth power. This makes repeated population movement or predator–prey transformations interpretable through dominant eigenvalues and their directions.