Contents

# Property|Definition & Meaning

## Definition

Defining **characteristics** or **qualities** that allow us to **identify** a particular thing and **distinguish** it from others. For example, **symmetry** and **dimensionality.** A **circle** is perfectly **symmetrical** around any line passing through its center, while a **trapezium** has no line of **symmetry.** Similarly, a circle is a **two-dimensional** shape, whereas a sphere is **three-dimensional** (has depth).

A **property** of an **object** refers to a **characteristic** or **attribute** that describes or defines the object. It can be a physical property such as **size, shape, color, or texture**, or it can be a functional property such as **weight, capacity, or electrical conductivity.**

**Figure 1: Example of Property**

## Mathematical Property

A **property** can also be a **mathematical property,** such as **symmetry, associativity,** or **distributive,** which can be used to describe and define **mathematical objects.** The properties of an object are used to **distinguish** it from other objects and to describe its **unique characteristics.**

**Figure 2: Qualities of Mathematical Property**

A **mathematical property** is a **characteristic** of a mathematical object or concept that **stays consistent** and **unchanged** regardless of the context in which it is used. It is a **fundamental aspect** of mathematics that provides a basis for **mathematical reasoning** and **problem-solving.**

### Significance of Mathematical Property

Properties play a **critical role** in many areas of **science, technology, engineering, and mathematics**. These properties **define, identify,** and **describe** mathematical objects and **concepts** and are used in many areas of **mathematics,** including **arithmetic, algebra, calculus, geometry,** and **others.**

Understanding these properties is essential for **solving mathematical problems** and for making **mathematical arguments.**

## Examples of Mathematical Properties

### Properties in Arithmetic

**Equality Property**

One of the most **basic** mathematical **properties** is the concept of **equality.** For example, the **property** of **equality** states that if two expressions are equal, then they can be used **interchangeably** in any mathematical calculation.

This property is often used in **arithmetic** and **algebra,** where it is **critical** to determine whether two expressions are equal in order to perform operations such as **division, multiplication, subtraction, and addition**.

**Inverse Operations **

Another important property in mathematics is the concept of **inverse operations**. Inverse operations are any two operations that **undo each other**, much like how **subtraction undoes addition** so they are inverse operations. This is true because **subtracting a number from some other number** is the **same** as **adding the opposite of that number** to it.

Similarly, **multiplication** and **division** are** inverse operations**, as dividing a number x by another number y is the same as multiplying x by the reciprocal of y.

**Congruence Property**

In **geometry,** the concept of **congruence** is an important property. **Congruence** refers to the **similarity** of two **geometric figures** in **size** and **shape.** Two figures are **congruent** if they have the same **shape** and **size** and can be **superimposed** on each other exactly.

This **property** is widely used in **geometry** to determine the properties of **lines, angles, triangles,** and other **geometric** **figures.**

### Properties of a Geometric Shape

Let us consider the example of a **unit sphere** and a **unit cube** for the sake of this section.

**Figure 3: Geometric Properties**

A geometric shape can be characterized and described with the help of its mathematical properties. Following are some of the chief properties among them:

**Perimeter:**The sum**of all edge lengths or side lengths**of a geometric shape is called the**perimeter.**For example, the maximum perimeter of the**unit sphere**is 2$\pi$.**Surface Area:**The**space occupied by a 2D shape**in a 2D cartesian plane is called the**area.**For example, the maximum surface area of a**unit sphere**is 4$\pi$.**Length:**The measured**distance**between two**extreme vertices**of a geometric shape along the**first dimension**is called its**length. Unit cube**has a length of 1.**Width:**The measured**distance**between two**extreme vertices**of a geometric shape along the**second dimension**is called its**width.****Unit cube**has a width of 1.**Height:**The measured**distance**between two**extreme vertices**of a geometric shape along the**third dimension**is called its**height.****Unit cube**has a height of 1.**Faces:**The**flat surfaces**(all points have the same**surface normal)**of a geometric shape are called their**faces.****Unit cube**has 6 faces. The**unit sphere**has no faces.**Vertices:**The**points**where**edges meet**are called**vertices**of a geometric shape.**Unit cube**has 8 vertices.**Unit sphere**has no vertices.**Edges:**The**lines**along which the**faces**of a geometric shape**meet**are called its**edges.****Unit cube**has 12 edges. The**unit sphere**has no edges.**Volume:**The**space occupied**by a**3D shape**in the 3D cartesian plane is called**volume.**For example, the maximum perimeter of the**unit sphere**is 4$\pi$/3.

### Properties of Set Theory

Five properties are commonly used in **set theory.** These are:

**Commutative**property: If the**order**of inputs**does not change**the**result**of a mathematical operator, then it’s called a**commutative**property.**Associative**property: If the**grouping**of inputs**does not change**the**final results**of a mathematical operator, then it’s called an**associative**property.**Distributive**property: If the result of an operator is**invariant**to the fact that this operator is applied to a**group**or an**individual**input, then it’s called**distributive**property.**Identity**property: The**existence**of a number that is**unaffected**by a mathematical**operator**is called**identity**property.**Inverse**property: The**existence**of a**pair**of numbers such that when a mathematical**operator**is**applied**to the pair, results in the**identity**is called**inverse**property.

**Figure 4: Set Properties**

### Properties in Trigonometry

There are three most **fundamental trigonometric** properties known to us, which are often called **trigonometric identities.** These include:

\[ \sin^2\theta + \cos^2\theta = 1 \]

\[ 1 + \tan^2\theta = \sec^2\theta \]

\[ 1 + \cot^2\theta = \csc^2\theta \]

Where sin, cos, tan, csc, sec, and cot are the sine, cosine, tangent, cosecant, secant, and cotangent functions, respectively.

## Numerical Examples of Property

Now that we have good background knowledge of what the property is, we can solve some numerical problems.

### Example 1

List some **mathematical properties** of a **cube** with a side length of **2 meters**.

**Solution**

Using the geometrical properties discussed earlier, we can find that:

**Perimeter**= 12 x 2 = 24 meters**Surface Area**= 6 x 2 x 2 = 24 square meters**Length**= 2 meters**Width**= 2 meters**Height**= 2 meters**Faces**= 6**Vertices**= 8**Edges**= 12**Volume**= 2 x 2 x 2 = 8 cubic meters

### Example 2

**Prove** that $ \sec^2\theta + \csc^2\theta = \sec^2\theta \ \csc^2\theta $ **using mathematical properties.**

**Solution**

Consider the following **trigonometric identity:**

\[ \sin^2\theta + \cos^2\theta = 1 \]

**Substituting** $ \sin^2\theta = \dfrac{ 1 }{ \csc^2\theta } $ and $ \cos^2\theta = \dfrac{ 1 }{ \sec^2\theta } $ in **above equation:**

\[ \dfrac{ 1 }{ \csc^2\theta } + \dfrac{ 1 }{ \sec^2\theta } = 1 \]

\[ \dfrac{ \sec^2\theta + \csc^2\theta }{ \csc^2\theta \ \sec^2\theta} = 1 \]

\[ \sec^2\theta + \csc^2\theta = \csc^2\theta \ \sec^2\theta \]

Hence proved.

*All images were created with GeoGebra.*