1.1 Number and algebra - SL content

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.