Engineering Tools for AI

學習資料

數學符號說明

整理常用數學符號的讀法、意義、LaTeX 指令、範例與 AI 提問想法的實用參考。

基礎運算

Plus

讀法
plus
LaTeX

意義

Adds two quantities.

範例

3+5=83+5=8

基礎運算

Minus

讀法
minus
LaTeX

意義

Subtracts one quantity from another, or marks a negative value.

範例

9−4=59-4=5

基礎運算

Multiplication sign

讀法
times
LaTeX

意義

Multiplies two quantities. In algebra, multiplication is often written with a dot or omitted.

範例

7×8=567\times 8=56

基礎運算

Division sign

讀法
divided by
LaTeX

意義

Divides one quantity by another nonzero quantity.

範例

12÷4=312\div 4=3

基礎運算

Fraction

讀法
a over b
LaTeX

意義

Represents division or a ratio with a numerator and denominator.

範例

36=12\frac{3}{6}=\frac{1}{2}

基礎運算

Plus-minus

讀法
plus or minus
LaTeX

意義

Indicates that both the positive and negative alternatives are being considered.

範例

10±3=13 or 710\pm 3=13\text{ or }7

基礎運算

Square root

讀法
square root of x
LaTeX

意義

The nonnegative value whose square is x, unless a different convention is stated.

範例

9=3\sqrt{9}=3

關係

Equals

讀法
equals
LaTeX

意義

States that two expressions have the same value or represent the same object.

範例

2+3=52+3=5

關係

Not equal

讀法
is not equal to
LaTeX

意義

Shows that two expressions do not have the same value.

範例

4≠54\ne 5

關係

Approximately equal

讀法
approximately equal to
LaTeX

意義

Indicates a rounded or approximate value rather than exact equality.

範例

π≈3.14\pi\approx 3.14

關係

Less than or equal to

讀法
less than or equal to
LaTeX

意義

Allows equality while comparing two quantities.

範例

x≤10x\le 10

關係

Greater than or equal to

讀法
greater than or equal to
LaTeX

意義

Compares two quantities while allowing equality.

範例

x≥0x\ge 0

關係

Proportional to

讀法
is proportional to
LaTeX

意義

Shows that one quantity changes by a constant multiple of another.

範例

y∝xy\propto x

集合論

Empty set

讀法
empty set
LaTeX

意義

The set with no elements.

範例

{1}∩{2}=∅\{1\}\cap\{2\}=\emptyset

集合論

Element of

讀法
is an element of
LaTeX

意義

Shows that a value belongs to a set.

範例

3∈{1,2,3}3\in\{1,2,3\}

集合論

Not an element of

讀法
is not an element of
LaTeX

意義

Shows that a value does not belong to a set.

範例

4∉{1,2,3}4\notin\{1,2,3\}

集合論

Proper subset

讀法
is a proper subset of
LaTeX

意義

Usually indicates that every element of one set is in another, and the two sets are not equal.

範例

{1,2}⊂{1,2,3}\{1,2\}\subset\{1,2,3\}

集合論

Subset or equal

讀法
is a subset of or equal to
LaTeX

意義

Allows two sets to be equal while stating subset inclusion.

範例

{1,2}⊆{1,2}\{1,2\}\subseteq\{1,2\}

集合論

Union

讀法
union
LaTeX

意義

Combines all elements that are in either set.

範例

{1,2}∪{2,3}={1,2,3}\{1,2\}\cup\{2,3\}=\{1,2,3\}

集合論

Intersection

讀法
intersection
LaTeX

意義

Keeps only the elements shared by both sets.

範例

{1,2}∩{2,3}={2}\{1,2\}\cap\{2,3\}=\{2\}

邏輯

Universal quantifier

讀法
for all
LaTeX

意義

States that a claim applies to every object in a domain.

範例

∀x∈R, x2≥0\forall x\in\mathbb{R},\ x^2\ge 0

邏輯

Existential quantifier

讀法
there exists
LaTeX

意義

States that at least one object satisfying a condition exists.

範例

∃x∈R:x2=4\exists x\in\mathbb{R}: x^2=4

邏輯

Negation

讀法
not
LaTeX

意義

Reverses the truth value of a statement.

範例

¬P\neg P

邏輯

Logical and

讀法
and
LaTeX

意義

True only when both connected statements are true.

範例

P∧QP\land Q

邏輯

Logical or

讀法
or
LaTeX

意義

True when at least one connected statement is true.

範例

P∨QP\lor Q

邏輯

Implication

讀法
implies
LaTeX

意義

Connects a condition to a conclusion.

範例

x>2⇒x2>4x>2\Rightarrow x^2>4

邏輯

If and only if

讀法
if and only if
LaTeX

意義

States that two statements imply each other.

範例

x2=1⇔x=±1x^2=1\Leftrightarrow x=\pm 1

數集

Natural numbers

讀法
the natural numbers
LaTeX

意義

The set used for counting numbers. Whether 0 is included depends on convention.

範例

n∈Nn\in\mathbb{N}

數集

Integers

讀法
the integers
LaTeX

意義

All whole numbers, including negative values, zero, and positive values.

範例

−3∈Z-3\in\mathbb{Z}

數集

Rational numbers

讀法
the rational numbers
LaTeX

意義

Numbers expressible as a ratio of two integers with nonzero denominator.

範例

23∈Q\frac{2}{3}\in\mathbb{Q}

數集

Real numbers

讀法
the real numbers
LaTeX

意義

Numbers represented on the continuous number line.

範例

2∈R\sqrt{2}\in\mathbb{R}

數集

Complex numbers

讀法
the complex numbers
LaTeX

意義

Numbers of the form a+bi, where i is the imaginary unit.

範例

3+2i∈C3+2i\in\mathbb{C}

數集

Pi

讀法
pi
LaTeX

意義

The constant ratio of a circle's circumference to its diameter.

範例

A=πr2A=\pi r^2

微積分與分析

Infinity

讀法
infinity
LaTeX

意義

A symbol for unbounded growth or an unending process, not an ordinary real number.

範例

lim⁡x→∞1x=0\lim_{x\to\infty}\frac{1}{x}=0

微積分與分析

Summation

讀法
sum
LaTeX

意義

Adds a sequence of terms over an index range.

範例

∑k=14k2=30\sum_{k=1}^{4}k^2=30

微積分與分析

Product notation

讀法
product
LaTeX

意義

Multiplies a sequence of terms over an index range.

範例

∏k=14k=24\prod_{k=1}^{4}k=24

微積分與分析

Integral

讀法
integral
LaTeX

意義

Represents accumulation, signed area, or an antiderivative depending on context.

範例

∫01x2 dx=13\int_0^1 x^2\,dx=\frac{1}{3}

微積分與分析

Partial derivative symbol

讀法
partial
LaTeX

意義

Used when differentiating a multivariable function with respect to one variable.

範例

∂f∂x\frac{\partial f}{\partial x}

微積分與分析

Nabla

讀法
nabla
LaTeX

意義

Used for gradient, divergence, curl, and related vector-calculus operations.

範例

∇f\nabla f

代數

Matrix

讀法
matrix
LaTeX

意義

A rectangular arrangement of entries used to represent linear maps, systems, and data.

範例

[1234]\begin{bmatrix}1&2\\3&4\end{bmatrix}

代數

Transpose

讀法
A transpose
LaTeX

意義

Swaps rows and columns of a matrix.

範例

[1234]T=[1324]\begin{bmatrix}1&2\\3&4\end{bmatrix}^T=\begin{bmatrix}1&3\\2&4\end{bmatrix}

代數

Inverse matrix

讀法
A inverse
LaTeX

意義

A matrix that multiplies with A to produce the identity matrix, when it exists.

範例

AA−1=IAA^{-1}=I

代數

Determinant

讀法
determinant of A
LaTeX

意義

A scalar value that captures scaling, orientation, and invertibility information of a square matrix.

範例

det⁡[abcd]=ad−bc\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc

幾何

Angle

讀法
angle
LaTeX

意義

Represents an angle formed by two rays or segments.

範例

∠ABC=60∘\angle ABC=60^\circ

幾何

Degree

讀法
degrees
LaTeX

意義

A unit for measuring angles, where a full turn is 360 degrees.

範例

90∘90^\circ

幾何

Perpendicular

讀法
is perpendicular to
LaTeX

意義

Shows that two lines or vectors meet at a right angle.

範例

AB⊥CDAB\perp CD

幾何

Parallel

讀法
is parallel to
LaTeX

意義

Shows that two lines or vectors have the same direction and do not meet in a plane.

範例

AB∥CDAB\parallel CD

括號與分隔符

Absolute value

讀法
absolute value of x
LaTeX

意義

The distance of a number from zero on the number line.

範例

∣−3∣=3|-3|=3

括號與分隔符

Floor

讀法
floor of x
LaTeX

意義

The greatest integer less than or equal to x.

範例

⌊3.8⌋=3\lfloor 3.8\rfloor=3

括號與分隔符

Ceiling

讀法
ceiling of x
LaTeX

意義

The smallest integer greater than or equal to x.

範例

⌈3.2⌉=4\lceil 3.2\rceil=4

縮寫與標記慣例

End of proof

讀法
QED
LaTeX

意義

Marks the end of a proof. Some texts use Q.E.D. or a filled square.

範例

∴x=1.□\therefore x=1.\quad\square

縮寫與標記慣例

Therefore

讀法
therefore
LaTeX

意義

Introduces a conclusion that follows from previous statements.

範例

a=b, b=c ∴a=ca=b,\ b=c\ \therefore a=c

縮寫與標記慣例

Because

讀法
because
LaTeX

意義

Introduces a reason or supporting statement.

範例

∵x>0\because x>0

延伸參考

以下連結僅作為延伸閱讀資料。本頁說明與範例依網站目的重新撰寫。