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# Unary Operation|Definition & Meaning

## Definition

Any **operation** with only **one input** is called a **unary operation,** e.g., **square root** (one number), **transpose** (one matrix), etc. This is in contrast to binary operations with two inputs, such as addition (two addends), division (dividend and divisor), exponentiation (base and power), etc. **Unary operators** are especially important in **programming languages.**

**Unary operations** in mathematics are the type of operations that have only **one input** and **one output.** In other words, such operators just **transform** a number into another one through some kind of **mapping** which may be **one-to-one** or **one-to-many. **

Some examples of **unary operators** include **additive** or **multiplicative inverse, absolute operator, factorial, trigonometric operators,** etc. The following figure depicts the basic concept.

**Figure 1: Unary Operation**

## Explanation of Unary Operation

The concept of **unary operation** is so simple that it is very **self-explanatory.** In this article, we will try to develop an **understanding** of the key concepts related to such **operators** through **examples.**

This will not only help us construct an **in-depth** intuitive foundation but also program our minds for solving **unary operation-related** problems simultaneously.

There are a lot of **unary operations** in use in all areas of science, technology, engineering, and **mathematics.**

For the sake of structure, we have classified them broadly into **three categories,** namely **numeric** unary operations, **matrix** unary operations, and **programming** unary operations. Some of the most **commonly used** unary operations are **listed below:**

**1. Numeric Unary Operations**

- Absolute Value
- Multiplicative Inverse
- Additive Inverse or Negation
- Trigonometric Operations
- Factorial
- Square Root, Cube Root, etc.
- Square, cube, etc.
- Hyperbolic Operations
- Logarithms

**2. Matrix Unary Operations**

- Matrix Transpose
- Adjoint of a Matrix
- Determinant of a Matrix
- Inverse of a Matrix

**3. Programming Unary Operations**

- 1s Complement
- 2s Complement
- Increment
- Decrement
- Logical Negation

## Examples of Unary Operations

Now that we have summarized a comprehensive list of unary operations that are commonly used in the field, we can dive deeper into the understanding of these operators through examples. Lets consider them one by one.

**1. Numeric Unary Operations**

As listed earlier, there are numerous numeric operators. The following diagram shows some of these:

**Figure 2: Numeric Unary Operations**

**(a) Absolute Value:** This operator simply **removes** the **negative sign.** For example, the absolute value of **+10** is **+10,** and the absolute value of **-10** is also **+10.**

**(b) Multiplicative Inverse:** This operator is also called a **reciprocal** operator. It converts the number into its **multiplicative inverse.** This means that if a number is **multiplied** by its multiplicative inverse, the **result** would be 1. For example, the multiplicative inverse of **10** is **1/10,** and that of **20** is **1/20.**

**(c) Additive Inverse or Negation: **This operator is also called the **negation** operator. It converts the number into its **additive inverse**, meaning that if a number is **added** into its additive inverse, the **result** would be 0. For example, the additive inverse of **10** is **-10,** and that of **20** is **-20.**

**(d) Trigonometric Operations:** Trigonometric **ratios** are most **common** in **geometry.** They can be defined as the **unary operations** on an **angle.** There are six such ratios by definition, including **sine, cosine, tangent,** and their inverses, namely **secant, co-secant,** and **co-tangent.** They operate on angles only. For example, the sine of **30 degrees** is equal to **0.5.**

**(e) Factorial:** Factorial of a positive number is defined as the **multiplication** of that number with all **positive numbers** that are **smaller** than it. For example factorial of the number **4** is equal to **4 x 3 x 2 x 1**, which is equal to **24.**

**(f) Square Root:** Square root is a **unary operation** that calculates such a number that the **square** of that number returns the original one. For example, the **square root** of **4** is **2**.

**(g) Square:** Square is a **unary operation** that returns the **multiplication** of the number by **itself.** For example, the square of **2** is equal to **2×2**, which equals **4**.

**(h) Logarithms:** Logarithms are used in **solving exponents.** They are defined as the exponent that raises a base to achieve the input number. For example, the **logarithm of 100 on base 10** equals **2**. This means that 10, raised to the power of 2, will result in 100.

**2. Matrix Unary Operations**

**Matrices** have their own set of **unary operations.** The following figure **summarizes** them:

**Figure 3: Matrix Unary Operations**

**Matrix Transpose:** Transpose of a matrix is obtained by **interchanging** its **rows** and **columns.**

**Adjoint of a Matrix:** Adjoint of a 2×2 matrix is defined as the matrix obtained by **interchanging** the **values** on the **main diagonal** while multiplying **off-diagonal** values with -1.

**Determinant of a Matrix:** The determinant of a 2×2 matrix is a **unary operation** that is defined as the **multiple of diagonal** entries **minus** the **multiple of non-diagonal** entries.

**Inverse of a Matrix:** Inverse of a 2×2 matrix is a **unary operation** defined as the **ratio** of the **adjoint** of the matrix to the **determinant** of the same matrix.

**3. Programming Unary Operations**

**Programming** languages help us build usable **software** on a higher level. However, they are **fundamentally** just some **mathematical operations.** Some unary operations are commonly used in this domain, which are **summarized** in the figure below:

**Figure 4: Programming Unary Operations**

**(a) 1s Complement:** 1’s complement is used for **subtraction** purposes in the domain of the **binary** number system. It is a **unary** operation. To find the one’s complement of a given binary number, we simply **invert all values.** For example, to find the **1’s complement** of a binary number **(1010101),** we invert it, which **returns (0101010),** which is the 1’s complement of the original number.

**(b) 2s Complement:** 2’s complement is also used for **subtraction** purposes in the domain of the binary number system. It is a unary operation. To find the one’s complement of a given binary number, we simply **invert all values** and **add** **1** to the 1’s complement. For example, to find the 2’s complement of a binary number **(1010101),** we invert it, which **returns** (0101010) + 1 = **(0101011),** which is the 2’s complement of the original number.

**(c) Increment:** Increment operation is a **unary operation** that **adds one** to the input number. For example, **incrementing** a number 4 would return **4+1 = 5** as the output

**(d) Decrement:** Decrement operation is a **unary operation** that **subtracts one** from the input number. For example, **decrementing** a number 4 would return **4-1 = 3** as the output

## Numerical Examples of Unary Operations

**Example 1**

Find the **absolute value** of -100.

**Solution**

Absolute value of **-100 = -1(-100) = 100.**

**Example 2**

Find the **additive inverse** of 21.

**Solution**

Additive inverse of **21 = -1(21) = -21.**

**Example 3**

Find the **multiplicative inverse** of 5.

**Solution**

Multiplicative **inverse of 21 = -1/5.**

*All images were created with GeoGebra.*