Notations

Some reoccurring notations that IB assumes to be prior knowledge are listed below. Interpreting and using correct notations are assessed in all papers and in the internal assessment.

Contents

Interval notation

Interval notation is used only in HL papers, but it can be helpful to know in SL for more effective communication on the IA and exams.

x∈[0,1]x \in \left[0, 1\right] means 0≤x≤10 \leq x \leq 1

x∈]0,1]x \in \left]0, 1\right] means 0<x≤10 < x \leq 1

x∈[0,1[x \in \left[0, 1\right[ means 0≤x<10 \leq x < 1

x∈]0,1[x \in \left]0, 1\right[ means 0<x<10 < x < 1

Intervals using ±∞\pm \infty (positive or negative infinity) must use the exclusive notation, eg, [0,∞[\left[0, \infty\right[ or ]−∞,4[\left] -\infty, 4\right[.

[−3,−1]∪[2,∞[\left[-3, -1\right] \cup \left[2, \infty\right[ is the way to specify two non-overlapping intervals.

Sets of numbers

Here are some common sets.

symbolnamedescriptionexamples
Z\mathbb{Z}integersmultiples of 11−5,0,7-5, 0, 7
Z+\mathbb{Z}^+positive integersintegers greater than 0010,45,10110, 45, 101
Z−\mathbb{Z}^-negative integersintegers less than 00−273,−5-273, -5
N\mathbb{N}natural numbers00 and positive integers0,1010, 101
Q\mathbb{Q}rational numbersratios of integers7,457, \frac 45
R\mathbb{R}real numbersnumbers on number line3,π,−2+23, \pi, -2+\sqrt 2

HL students should also recognize C\mathbb{C} for complex numbers.

It is worth pointing out that ∞\infty and −∞-\infty belong to none of the above sets. The real numbers is all numbers on the open interval, ie x∈R{x \in \mathbb R} means x∈]−∞,∞[{x \in \left]-\infty, \infty\right[}.

Where is that bracket?

Generally speaking, an operator takes the first term as its argument.

Examples

  •  d dx2x+5=( d dx2x)+5\displaystyle \frac{\d}{\d x} 2x + 5 = \left(\frac{\d}{\d x} 2x\right) + 5
  • 1n∑xx2−μ2=(1n∑x(x2))−μ2\displaystyle \frac 1n \sum_{x} x^2 - \mu^2 = \left(\frac 1n \sum_{x} \left(x^2\right)\right) - \mu^2, formula for variance (HL)
  • ln⁡sin⁡x+5=(ln⁡(sin⁡x))+5\ln \sin x + 5 = (\ln (\sin x)) + 5, this is super rare, but similar notations have appeared on exams.

If you want to apply an operator to an expression with addition or subtraction at the top level, be sure to use parentheses.

iff

iff,   ⟺  \iff, means if and only if. A iff BA \text{ iff } B means both if AA then BB, but also if BB then AA. So A  ⟺  BA \iff B means they are both true or both false.

In contrast, A  ⟹  BA \implies B means if AA is true then BB is also true, but it does not necessarily mean BB is true means AA is true.