1 Number and algebra

Scientific notation writes a non-zero number as a × 10ⁿ, where 1 ≤ |a| < 10. The coefficient carries the significant digits; the integer exponent records how far the decimal point has moved and therefore the scale of the quantity.

To convert 0.00042, move the decimal point four places right to obtain 4.2, so 0.00042 = 4.2 × 10⁻⁴. For multiplication, multiply coefficients and add exponents: (3 × 10⁵)(2 × 10⁻³) = 6 × 10². For addition or subtraction, first rewrite the terms with the same power of ten.

The exponent is not the number of significant figures. A negative exponent describes a number between 0 and 1, and a coefficient such as 42 is not normalized scientific notation because its magnitude is greater than 10.

IB written-form boundary: calculator notation such as 5.2E30 is not acceptable in a final answer; write $5.2\times10^{30}$. After multiplying or dividing, renormalize the coefficient so its magnitude is at least 1 and less than 10. For example, $24\times10^7=2.4\times10^8$.

An arithmetic sequence changes by the same common difference d at every step. If the first term is a, the nth term is uₙ = a + (n − 1)d; the index n counts terms, so the first term uses n = 1.

For 7, 11, 15, …, d = 4. The 20th term is 7 + 19(4) = 83, and the sum of the first 20 terms is 20(7 + 83)/2 = 900. The same sum formula pairs the first and last terms, which is why the average term is (a + uₙ)/2.

A sequence lists terms; a series adds them. Check that the claimed common difference is constant before using an arithmetic formula, and do not replace n − 1 with n when finding a term.

Sigma form makes the finite series explicit: $S_n=\sum_{k=1}^{n}[a+(k-1)d]$. In a simple-interest model, equal interest added each period produces an arithmetic sequence. For principal $1000$ at 4% simple interest, the yearly balances are $1040,1080,1120,\ldots$ with common difference $40$; real data may require an approximate common difference rather than a perfect one.

A geometric sequence is multiplied by the same common ratio r at each step. With first term a, its nth term is uₙ = arⁿ⁻¹; a negative ratio alternates signs while a ratio between 0 and 1 produces decay in magnitude.

For 3, 6, 12, …, $a=3$ and $r=2$. The fifth term is $3\cdot2^4=48$ and the first five terms sum to $3(1-2^5)/(1-2)=93$. In a decay model with $0<r<1$, the same finite formula adds the first $n$ measured stages.

A sequence lists terms; a series sums them. This objective uses nth terms, finite sums and sigma notation, so do not replace $n-1$ by $n$ or import a sum-to-infinity formula from a neighbouring objective.

For a fixed principal P and periodic rate i, simple interest grows linearly: A = P(1 + in). Compound interest grows by repeatedly multiplying the current balance: A = P(1 + i)ⁿ. Here i must be the rate per compounding period and n the number of those periods.

At 5% nominal annual interest compounded quarterly, the periodic rate is 0.05/4 and two years contains eight periods. A $1000 balance is therefore 1000(1 + 0.05/4)⁸, not 1000(1.05)².

Match the rate and number of periods before calculating. A quoted annual rate may be nominal or effective; deposits or withdrawals at different times require an annuity model rather than the one-payment formula.

Annual depreciation at rate $r$ uses $V_n=P(1-r)^n$. Inflation changes real value: if a nominal investment grows by factor $1+i$ while prices grow by $1+j$, its real-value factor per year is $(1+i)/(1+j)$. Always convert a nominal annual rate to the matching yearly, half-yearly, quarterly or monthly period before using technology or a financial package.

Integer exponents describe repeated multiplication and its inverse operations: aᵐaⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, and a⁰ = 1 for a ≠ 0. A logarithm reverses exponentiation: log_b(x) = y exactly when bʸ = x.

To solve 3·2ˣ = 24, divide by 3 to get 2ˣ = 8 and hence x = 3. For a non-integer result, take logarithms: x = ln(8/3)/ln 2.

The logarithm requires b > 0, b ≠ 1 and x > 0. Apply exponent laws only to products, quotients and powers; log(a + b) is not log a + log b.

An approximation replaces an exact value with a nearby value that is easier to report or use. Its accuracy is part of the statement: rounding to a given unit defines an interval, while significant figures communicate relative precision.

A length reported as 8.4 cm to the nearest 0.1 cm represents 8.35 ≤ L < 8.45. If the measured value is 8.37 cm, using 8.4 cm introduces an absolute error of 0.03 cm and a percentage error of about 0.36%. Keep full calculator precision until the final rounding.

The required precision depends on the context. A calculated result of 3.2 containers cannot be reported as 3.2 containers in practice: four whole containers are needed. Do not confuse a rounding bound with the exact value, and do not round intermediate values unnecessarily.

For an amortization or annuity calculation, identify present value, future value, periodic payment, periodic interest rate and number of periods, then enter them with the financial solver's sign convention. In IB examinations, payments occur at the end of each period.

For monthly deposits of $200 over 24 months at a monthly rate of 0.004, enter$N=24$,$I=0.4$percent per period,$PV=0$and$PMT=-200$, then solve for the future value. Check that the balance exceeds the total deposits because interest has accumulated.

Match the annual rate to the payment period and keep cash inflows and outflows with opposite signs. Knowing an annuity formula may help understanding, but the formula itself is not examined; beginning-of-period annuity-due settings are outside the stated examination convention.

A graphing calculator can locate numerical roots, intersections and solutions of systems, but it does not decide which solution is meaningful. Enter the equation in a form the calculator can interpret, set an appropriate domain or window, and record the required precision.

For x² − 5x + 6 = 0, graph y = x² − 5x + 6 or use the solver. The roots are x = 2 and x = 3; substituting either value gives zero, confirming the output.

A missed root can come from a poor window or initial guess, and transformed equations can introduce extraneous solutions. Always check the original equation and state domain restrictions or units.

For a system of up to three linear equations, enter all equations with consistent variable order and use the simultaneous-equation solver; examinations use systems with a unique solution. Example: $x+y=7$ and $2x-y=2$ gives $(x,y)=(3,4)$, verified in both equations. A root or zero of a polynomial is a value making the polynomial equal to zero.

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.