1 Number and algebra

Normalize a number into scientific notation.

Move the decimal point until the coefficient is at least 1 and less than 10; the number of places becomes the integer exponent.

Worked example
0.00052 = 5.2 × 10⁻⁴ because the decimal moves four places right.

Worked example
Write 7.1 × 10⁵ as an ordinary number. 710000; a positive exponent moves the decimal right.

Common boundary
E notation on a calculator is a display format, not the required written form.

Use the common difference to model an arithmetic sequence.

An arithmetic sequence changes by the same additive amount each step: aₙ=a₁+(n−1)d, and a finite sum is n/2[2a₁+(n−1)d].

Worked example
For 4, 7, 10, the difference is 3, so a₆=4+5(3)=19.

Worked example
What stays constant? the difference, not the ratio.

Common boundary
Do not use a geometric formula when the change is additive.

Complete finite-sum and sigma example: for $4,7,10,\ldots$, $a_1=4$ and $d=3$, so $a_n=4+3(n-1)$. For six terms, $S_6=\frac{6}{2}[2(4)+5(3)]=69$. The same sum is $\sum_{k=1}^{6}(3k+1)=69$. Use $a_n$ for one term and $S_n$ or $\Sigma$ for a total.

Use the common ratio to model geometric growth.

A geometric sequence multiplies by the same ratio: aₙ=a₁rⁿ⁻¹; finite sums use repeated multiplication.

Worked example
For 5, 10, 20, r=2, so a₅=5·2⁴=80.

Worked example
Is 3, 6, 12 arithmetic or geometric? geometric, because each term is multiplied by 2.

Common boundary
Equal increases do not make a sequence geometric; check multiplication.

Complete finite-series example: for $3,6,12,24,48$, $a_1=3$, $r=2$ and $n=5$. $S_5=\frac{a_1(1-r^n)}{1-r}=\frac{3(1-2^5)}{1-2}=93$, equivalently $\sum_{k=1}^{5}3\cdot2^{k-1}=93$. This finite formula is valid for $r\ne1$; if $r=1$, $S_n=na_1$.

Model compound interest or depreciation with repeated multiplication.

A percentage change each period is geometric: A=P(1±r)ⁿ, with the sign chosen for growth or depreciation.

Worked example
$1000 at 5$1102.50.

Worked example
Why is 5% added twice not 10% of the original? the second 5% applies to the new balance.

Common boundary
Compound percentage change is not simple linear addition.

Compounding-frequency example: $1200$ dollars at a nominal annual rate of $6\%$ compounded monthly for two years gives A=1200(1+0.06/12)^{24}=\1352.59$to the nearest cent. For annual depreciation replace the growth factor by$1-r$. To compare purchasing power with annual inflation$i$, divide the nominal value after$n$years by$(1+i)^n$; nominal growth alone does not guarantee a gain in real value.

Use exponent laws and logarithms to solve exponential equations.

Exponent laws simplify powers; a logarithm reverses exponentiation, so aˣ=b is equivalent to x=logₐb for a>0,a≠1.

Worked example
2ˣ=16 gives x=4; 10ˣ=3 gives x=log(3).

Worked example
What operation undoes a power? logarithm, with a valid positive base and argument.

Common boundary
A logarithm is not an ordinary division or a power of ten only.

Core integer exponent laws, for non-zero $a$ where required, are $a^ma^n=a^{m+n}$, $a^m/a^n=a^{m-n}$, $(a^m)^n=a^{mn}$, $a^0=1$ and $a^{-n}=1/a^n$. For example, $5^3\cdot5^{-6}=5^{-3}=1/125$. Logarithms reverse exponentiation: $10^x=b\iff x=\log_{10}b$ and $e^x=b\iff x=\ln b$, with $b>0$.

Write a deductive proof as linked equalities.

A proof starts from a known expression and uses justified equalities until it reaches the target; each line preserves truth.

Worked example
To show (n+1)²−n²=2n+1, expand to n²+2n+1−n², then simplify.

Worked example
Which line needs justification? every transformation, such as expansion or cancellation, must preserve equality.

Common boundary
A few confirming examples do not prove a statement for all allowed values.

Apply rational exponent and logarithm laws consistently.

Fractional powers represent roots and powers, while log laws convert products, quotients and powers into sums, differences and coefficients.

Worked example
x^(3/2)=(√x)³ for x≥0; log(ab)=log a+log b for positive a,b.

Worked example
When can you split log(ab)? both arguments must be in the valid domain.

Common boundary
Do not apply log laws to sums or ignore domain restrictions.

Use all three log laws only for positive arguments: $\log_a(xy)=\log_ax+\log_ay$, $\log_a(x/y)=\log_ax-\log_ay$, and $\log_a(x^m)=m\log_ax$. Change base with $\log_ax=\ln x/\ln a$. Example: $2^{x-1}=10$ gives $(x-1)\ln2=\ln10$, so $x=1+\ln10/\ln2\approx4.322$. Never split $\log(x+y)$.

Test convergence before summing an infinite geometric series.

An infinite geometric series has a finite sum only when |r|<1; then S∞=a₁/(1−r).

Worked example
3+1.5+0.75+… converges because r=0.5, giving S∞=6.

Worked example
Does 4+8+16+… have a finite sum? no; |r|=2≥1.

Common boundary
A pattern continuing forever is not automatically summable.

Expand a power with binomial coefficients.

For a non-negative integer n, (a+b)ⁿ is the sum of terms whose coefficients are nCr and whose powers of a and b add to n.

Worked example
(x+2)²=x²+4x+4; coefficients 1,2,1 come from Pascal’s triangle.

Worked example
What should the powers add to? n in every term.

Common boundary
Do not omit the middle term or change the coefficient pattern.

Full binomial theorem: $(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$ for $n\in\mathbb{N}$, where $\binom{n}{r}=n!/[r!(n-r)!]$. Example: $(2x-1)^3=\binom30(2x)^3+\binom31(2x)^2(-1)+\binom32(2x)(-1)^2+\binom33(-1)^3=8x^3-12x^2+6x-1$. Each term's total power is $n$.

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.