Maths AA terms: I

I

Identityalso ≡

An equality that is true for every value of the variable for which both sides are defined, such as $(x + 1)^2 \equiv x^2 + 2x + 1$. The symbol $\equiv$ means "is identically equal to".

Introduced in 1.6 Simple deductive proof

Identity function

The function $I(x) = x$, which maps every input to itself. Composing a function with its inverse, in either order, gives the identity function.

Introduced in 2.5 Composite functions

Image (of a point)also image

The point that a given point is moved to by a transformation. Under $y = f(x - a) + b$, the image of $(x, y)$ is $(x + a, y + b)$

Imaginary partHL only · also Im(z)

For $z = a + b\mathrm{i}$ with $a, b \in \mathbb{R}$, the real number $b$, written $\mathrm{Im}(z)$

Imaginary unit iHL only · also i

The number $\mathrm{i}$ defined by $\mathrm{i}^2 = -1$, so that $\mathrm{i} = \sqrt{-1}$

Implicit differentiationHL only

Differentiating both sides of an equation connecting $x$ and $y$ with respect to $x$, treating $y$ as a function of $x$ (so that, for example, $\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}$

Implicit equationHL only · also implicit relation, implicit function, explicit function

An equation connecting $x$ and $y$ that is not written in the form $y = f(x)$, such as $x^2 + y^2 = 25$

Improper algebraic fractionHL only · also improper fraction

An algebraic fraction whose numerator has a degree greater than or equal to that of its denominator, such as $\frac{x^2 + 1}{x^2 - 4}$. Dividing first writes it as a polynomial plus a proper fraction.

Introduced in 1.11 Partial fractions

Included angle

The angle between two given sides of a triangle, at the vertex where those two sides meet.

Introduced in 3.2 Trigonometry in triangles

Increasing functionalso increasing, strictly increasing

A function is increasing on an interval if larger inputs give larger outputs there: $x_1 < x_2$ implies $f(x_1) < f(x_2)$

Indefinite integral

The family of all antiderivatives of $f(x)$, written $\int f(x)\,dx = F(x) + C$, where $F'(x) = f(x)$

Independent eventsalso independence

Events for which the occurrence of one does not change the probability of the other: $P(A \cap B) = P(A)P(B)$, or equivalently $P(A \mid B) = P(A)$

Independent variablealso explanatory variable, dependent variable, response variable

In bivariate data, the variable $x$ that is controlled or used to make predictions, plotted on the horizontal axis. The dependent variable $y$ is the one predicted, plotted on the vertical axis.

Introduced in 4.4 Linear correlation and regression

Indeterminate formHL only · also 0/0, ∞/∞, indeterminate forms

An expression such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$

Inductive hypothesisHL only · also induction assumption

The assumption, made inside a proof by induction, that the statement $P(n)$ is true for one particular value $n = k$.

Introduced in 1.15 Proof by induction, contradiction and counterexample

Inductive stepHL only

The part of a proof by induction that uses the assumption that $P(k)$ is true to show that $P(k+1)$ is true.

Introduced in 1.15 Proof by induction, contradiction and counterexample

Inflation

A general rise in prices over time, usually given as an annual percentage rate. Inflation reduces what a fixed sum of money can buy.

Introduced in 1.4 Financial applications of geometric sequences

Initial conditionHL only

A known value of the solution at one point, such as $y = 2$ when $x = 0$ (written $y(0) = 2$), used to find a particular solution.

Introduced in 5.18 First-order differential equations

Inner functionalso inside function

In a composite function $f(g(x))$, the function $g$ that is applied to $x$ first. The function $f$, applied to the output of $g$

Instantaneous rate of change

The rate of change of a function at a single value of its input: the limit of the average rate of change over smaller and smaller intervals, equal to the derivative and to the gradient of the tangent.

Introduced in 5.1 Limits and the derivative

Integrand

The function being integrated: in $\int f(x)\,dx$ or $\int_a^b f(x)\,dx$, the integrand is $f(x)$

Integrating factorHL only

The function $e^{\int P(x)\,dx}$ that multiplies $\frac{dy}{dx} + P(x)y = Q(x)$

Integration by inspectionalso reverse chain rule

Integrating by recognising the integrand as the result of the chain rule, $k\,g'(x)f(g(x))$, guessing the antiderivative (here $k\,F(g(x))$ where $F' = f$

Integration by partsHL only

A method that reverses the product rule, used to integrate a product: $\int u\frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx$

Integration by substitutionHL only · also u-substitution, change of variable

A method that reverses the chain rule: a new variable $u = g(x)$ replaces part of the integrand, $dx$ is replaced using $\frac{du}{dx}$

Interquartile rangealso IQR

The difference between the upper and lower quartiles, $\mathrm{IQR} = Q_3 - Q_1$: the spread of the middle half of the data.

Introduced in 4.2 Presenting data

Intersectionalso A ∩ B, and

The event $A \cap B$ ("$A$ and $B$"), containing the outcomes that are in both $A$ and $B$.

Introduced in 4.6 Venn diagrams, tree diagrams and combined events

Interval notationalso interval

A way of writing a set of real numbers between two endpoints. Square brackets include an endpoint and round brackets exclude it: $[2, 5)$ means $2 \le x < 5$.

Introduced in 2.2 Functions, domain, range and inverse

Interval of convergenceHL only · also interval of validity

The set of values of $x$ for which a power series converges. A Maclaurin series can be used to evaluate or approximate its function only for $x$ in this interval, for example $-1 < x \leq 1$ for $\ln(1 + x)$

Invariant pointalso invariant points

A point that is mapped to itself by a transformation, so it lies on both the original graph and its image.

Introduced in 2.11 Transformations of graphs

Inverse functionalso f⁻¹

The function $f^{-1}$ that undoes $f$: if $f(a) = b$ then $f^{-1}(b) = a$

Inverse normal calculationalso inverse normal

Finding the value $k$ of a normally distributed variable $X$ that gives a stated probability, such as $P(X < k) = 0.9$. It reverses a normal probability calculation: the area is known and the boundary is found.

Introduced in 4.9 The normal distribution

Inverse proportionalso inversely proportional, inverse variation

A relationship between two positive quantities in which their product is constant, $xy = k$, so that $y = \frac{k}{x}$. Doubling one quantity halves the other.

Introduced in

Inverse trigonometric functionsHL only · also arcsin, arccos, arctan

The functions $\arcsin x$, $\arccos x$ and $\arctan x$, which return the angle with a given sine, cosine or tangent. Each is the inverse of sine, cosine or tangent restricted to a domain on which it is one-to-one.

Introduced in 3.9 Reciprocal and inverse trigonometric functions

Irrational numberHL only · also irrational

A real number that cannot be written as $\frac{p}{q}$ with $p, q \in \mathbb{Z}$ and $q \ne 0$

IsoclineHL only

A curve on which a differential equation $\frac{dy}{dx} = f(x, y)$ gives the same gradient $k$ everywhere, so all slope-field segments on it are parallel; its equation is