FOUNDATIONS 09
Mathematical Notation
Mathematical notation is a compact language. The aim is not only to recognise symbols, but to read the structure and meaning of a mathematical statement.
The main idea
A small change in notation can produce a completely different mathematical object. For example, \(x^2\), \(x_2\), \(f(x)\), \(\Delta x\) and \(dx\) all contain \(x\), but they do not mean the same thing.
\[\boxed{\text{symbol}\longrightarrow\text{structure}\longrightarrow\text{meaning}}\]Good notation lets complicated ideas be expressed precisely and compactly.
1. Variables and constants
Letters such as \(x\), \(y\) and \(t\) often represent quantities that may vary. A constant has a fixed value within the problem.
The letter itself does not determine the meaning: \(t\) often represents time only because we define it that way.
\(t=\text{time}\)
\(g=9.81\text{ m/s}^2\)
2. Equality, approximation and inequality
Relationship symbols say how two mathematical quantities are connected.
equal
approximately equal
less than
greater than or equal
3. Brackets show structure
Brackets tell us which quantities belong together. They can change the value of an expression.
but
\(2x+3\) is different
4. Superscripts
A superscript commonly represents a power:
\[x^3=x\cdot x\cdot x.\]But superscripts can later label derivatives, inverses or components, so their meaning must always be read from context.
\(x^{-1}\): reciprocal of \(x\)
\(f^{-1}\): inverse function, not \(1/f\)
5. Subscripts
Subscripts usually label related quantities rather than indicating multiplication.
\[x_1,\quad x_2,\quad x_3.\]For example, \(x_1\) may mean the first value of \(x\), while \(x_2\) means the second.
\(x_2=5\)
\(\Delta x=x_2-x_1=3\)
6. Fractions group numerator and denominator
The fraction bar means division, but it also acts like brackets around the entire numerator and denominator.
\[\frac{x+1}{x-1}.\]This means \((x+1)\div(x-1)\), not \(x+1\div x-1\).
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denominator: \(x-1\)
restriction: \(x\ne1\)
7. Sets and membership
Braces can describe a set. The symbol \(\in\) means “is an element of”, while \(\subset\) describes containment of one set inside another.
\(2\in A\)
\(5\notin A\)
\(\mathbb N\subset\mathbb R\)
8. Greek letters
Greek letters provide additional mathematical symbols. They are not special operations by themselves; their meaning comes from their definition.
9. Function notation
\(f(x)\), read “\(f\) of \(x\)”, means the output of function \(f\) when the input is \(x\). The brackets do not mean multiplication.
input \(x=3\)
\(f(3)=3^2+1=10\)
10. Finite-change notation
\(\Delta x\) means a finite change in \(x\):
\[\Delta x=x_2-x_1.\]Similarly, \(\Delta y=y_2-y_1\). The ratio \(\Delta y/\Delta x\) is an average rate of change.
\(\Delta x=x_2-x_1\)
\(\frac{\Delta y}{\Delta x}=\text{average rate}\)
11. Differential notation
Later in calculus we meet \(dx\), \(dy\) and
\[\frac{dy}{dx}.\]These are related to change, but \(dx\) is not simply another spelling of \(\Delta x\). Their precise meaning develops from limits and differentials.
\(dx\): differential notation
\(\frac{dy}{dx}\): derivative notation
12. Common calculus symbols
It is useful to recognise the notation that will appear later, even before studying each idea fully.
limit
differentiate with respect to x
integral
partial derivative symbol
summation
infinity
13. Read mathematics in layers
Complicated expressions become easier when read from their internal structure outward.
↓ add 1
\(x+1\)
↓ square
\((x+1)^2\)
↓ divide by 3
\(\frac{(x+1)^2}{3}\)
14. Context determines meaning
The same-looking symbol can mean different things in different settings. Mathematical notation must therefore be read together with its definition.
\(e\): a variable, or Euler's number
\(P\): a point, probability, or matrix
meaning comes from context
Notation is mathematical language. Read symbols together with their structure and definitions. Superscripts, subscripts, brackets, fractions and function notation each carry different information, while symbols such as \(\Delta x\), \(dx\), \(dy/dx\), \(\int\) and \(\partial\) prepare the language used throughout calculus.
Where the foundations lead
\[\boxed{\text{numbers}\rightarrow\text{algebra}\rightarrow\text{graphs}\rightarrow\text{change}\rightarrow\text{functions}\rightarrow\text{calculus}}\]The next section develops functions in detail: the objects whose changing behaviour calculus is designed to study.