Topic 1: Number and algebra

Number and algebra is the foundation topic of IB Mathematics: Analysis and Approaches. Almost every later topic leans on the skills introduced here, so fluency matters more than memorising isolated facts. At Standard Level you will work with arithmetic and geometric progressions, sigma notation, exponents and logarithms, the binomial theorem with positive integer powers, and simple systems of equations. At Higher Level the same ideas are pushed further: you add rigorous proof techniques such as induction, the binomial theorem with the general term, and a substantial block on complex numbers. This article walks through each strand, gives the key formulae, and shows how the pieces connect. Where a result is HL only it is flagged clearly so SL students know what they can skip.

Arithmetic sequences and series

An arithmetic sequence increases by a fixed common difference $d$ each term. If the first term is $u_1$, the $n$th term is $u_n = u_1 + (n-1)d$. The terms form a straight-line pattern: plotting $u_n$ against $n$ gives points on a line of gradient $d$.\n\nThe sum of the first $n$ terms (an arithmetic series) has two equivalent forms. Using the first and last term: $S_n = \\frac{n}{2}(u_1 + u_n)$. Using the first term and common difference: $S_n = \\frac{n}{2}\\left(2u_1 + (n-1)d\\right)$. Choose the first when you know both ends, the second when you know $d$.\n\nA classic exam move is to set up two equations from two given terms. For example if $u_3 = 11$ and $u_7 = 27$, subtract to get $4d = 16$, so $d = 4$ and $u_1 = 3$. Always state $u_1$ and $d$ before computing a sum. Watch for the difference between the number of terms $n$ and the value of the final term, a common slip.

Geometric sequences, series and sum to infinity

A geometric sequence multiplies by a fixed common ratio $r$ each term, so $u_n = u_1 r^{n-1}$. The ratio can be found by dividing any term by the one before it: $r = \\frac{u_{n+1}}{u_n}$.\n\nThe sum of the first $n$ terms is $S_n = \\frac{u_1(r^{n}-1)}{r-1} = \\frac{u_1(1-r^{n})}{1-r}$, valid for $r \\neq 1$. The two versions are identical; use whichever keeps the arithmetic positive.\n\nWhen the ratio satisfies $\\lvert r \\rvert \\lt 1$ the terms shrink towards zero and the series converges to a finite sum to infinity: $S_\\infty = \\frac{u_1}{1-r}$. This formula appears in compound interest, drug-dosage and bouncing-ball problems. If $\\lvert r \\rvert \\geq 1$ the series diverges and $S_\\infty$ does not exist, so always check the ratio first. A frequent application is the financial model $FV = PV\\left(1 + \\frac{r}{100k}\\right)^{kn}$ for compound interest, which is geometric growth in disguise.

Sigma notation

Sigma notation is a compact way to write a sum. The expression $\\sum_{k=1}^{n} a_k$ means add the terms $a_k$ as the index $k$ runs from the lower limit $1$ to the upper limit $n$. For instance $\\sum_{k=1}^{4} (2k+1) = 3 + 5 + 7 + 9 = 24$.\n\nArithmetic and geometric series both translate directly into sigma form. An arithmetic series is $\\sum_{k=1}^{n} \\left(u_1 + (k-1)d\\right)$ and a geometric series is $\\sum_{k=1}^{n} u_1 r^{k-1}$. Recognising which type a sigma expression represents lets you apply the closed-form sum formula instead of adding term by term.\n\nUseful manipulations: a constant factor can be pulled out, $\\sum c\\,a_k = c\\sum a_k$, and a sum splits over addition, $\\sum (a_k + b_k) = \\sum a_k + \\sum b_k$. Be careful with the limits: the number of terms in $\\sum_{k=m}^{n}$ is $n - m + 1$, not $n - m$.

Exponents and logarithms

The laws of exponents let you combine powers with the same base: $a^m \\times a^n = a^{m+n}$, $\\frac{a^m}{a^n} = a^{m-n}$, and $(a^m)^n = a^{mn}$. Special cases include $a^0 = 1$, $a^{-n} = \\frac{1}{a^{n}}$ and the fractional power $a^{1/n} = \\sqrt[n]{a}$, so $a^{m/n} = \\sqrt[n]{a^{m}}$.\n\nA logarithm is the inverse of exponentiation: $\\log_a b = x$ is exactly the statement $a^{x} = b$, for $a \\gt 0$, $a \\neq 1$ and $b \\gt 0$. The laws mirror the exponent laws: $\\log_a (xy) = \\log_a x + \\log_a y$, $\\log_a \\frac{x}{y} = \\log_a x - \\log_a y$, and $\\log_a x^{m} = m\\log_a x$.\n\nThe change-of-base rule $\\log_a x = \\frac{\\log_b x}{\\log_b a}$ lets you evaluate any logarithm on a calculator. The natural logarithm $\\ln x = \\log_e x$ and its partner $e^{x}$ are used to solve growth and decay equations: take logs of both sides to bring the unknown exponent down, for example solving $5^{x} = 20$ gives $x = \\frac{\\ln 20}{\\ln 5}$.

Surds, indices and systems of equations

A surd is an irrational root left in exact form, such as $\\sqrt{2}$ or $3\\sqrt{5}$. Simplify by extracting square factors: $\\sqrt{50} = \\sqrt{25 \\times 2} = 5\\sqrt{2}$. To rationalise a denominator, multiply top and bottom by a suitable surd or conjugate, for example $\\frac{1}{\\sqrt{3}} = \\frac{\\sqrt{3}}{3}$ and $\\frac{1}{2+\\sqrt{3}} = 2 - \\sqrt{3}$ after multiplying by $2 - \\sqrt{3}$. Keeping surds exact avoids rounding error in non-calculator papers.\n\nSystems of linear equations are solved by elimination, substitution, or for larger systems on the calculator. SL works with two equations in two unknowns; HL extends to three equations in three unknowns, where you may meet a unique solution, no solution (inconsistent), or infinitely many solutions (dependent). Always interpret the result geometrically: two lines meeting, being parallel, or coinciding; three planes intersecting at a point, a line, or not at all.

The binomial theorem

The binomial theorem expands $(a+b)^{n}$ for a positive integer $n$ without multiplying out every bracket. The full expansion is $(a+b)^{n} = \\sum_{r=0}^{n} \\binom{n}{r} a^{n-r} b^{r}$, where the binomial coefficient $\\binom{n}{r} = \\frac{n!}{r!(n-r)!}$ counts combinations and matches a row of Pascal's triangle.\n\nThe general term is $\\binom{n}{r} a^{n-r} b^{r}$, which is the key to finding one specific term without writing the whole expansion. To find, say, the term in $x^{4}$, set the power of $x$ in the general term equal to $4$, solve for $r$, then substitute back.\n\nAt SL you mainly expand fully or extract a named term for positive integer $n$. At HL the same general-term technique is used for harder coefficient problems and combined with the laws of indices when $a$ or $b$ themselves contain powers of $x$, such as expanding $\\left(2x - \\frac{1}{x}\\right)^{6}$ and finding the constant term.

Proof (SL and HL induction)

Proof asks you to argue that a statement holds in general, not just in examples. Direct proof starts from known facts and reasons to the conclusion. Proof by deduction and proof by counterexample (a single case that breaks a claim) appear at both levels.\n\nHigher Level adds proof by mathematical induction, the standard tool for statements about all positive integers $n$. There are three parts. Basis: show the statement is true for the smallest value, usually $n = 1$. Inductive step: assume it is true for $n = k$ (the inductive hypothesis) and use that to prove it for $n = k+1$. Conclusion: state that since it holds for $n=1$ and truth at $k$ forces truth at $k+1$, it holds for all integers $n \\geq 1$.\n\nInduction proves results such as $\\sum_{r=1}^{n} r = \\frac{n(n+1)}{2}$, divisibility claims, and formulae for sequences. HL also meets proof by contradiction, for example the classic proof that $\\sqrt{2}$ is irrational. Marks are awarded for clear structure and the explicit closing statement, so never omit the conclusion.

Complex numbers (HL only)

A complex number has the form $z = a + b\\,i$ where $i^{2} = -1$, with real part $a$ and imaginary part $b$. They are added and multiplied like binomials, replacing $i^{2}$ by $-1$. The complex conjugate $z^{*} = a - b\\,i$ is used to divide: multiply numerator and denominator by the conjugate of the denominator.\n\nGeometrically, $z$ is a point on the Argand diagram. Its modulus $\\lvert z \\rvert = \\sqrt{a^{2}+b^{2}}$ is the distance from the origin and its argument $\\arg z$ is the angle from the positive real axis. This gives the polar (modulus-argument) form $z = r(\\cos\\theta + i\\sin\\theta)$ and the exponential form $z = r e^{i\\theta}$, linked by Euler's formula $e^{i\\theta} = \\cos\\theta + i\\sin\\theta$.\n\nPolar form makes multiplication and powers easy: moduli multiply and arguments add. De Moivre's theorem states $\\left(r(\\cos\\theta + i\\sin\\theta)\\right)^{n} = r^{n}(\\cos n\\theta + i\\sin n\\theta)$, which is used to find powers and the $n$ distinct $n$th roots of a complex number, evenly spaced around a circle of radius $\\sqrt[n]{r}$.