Let's Learn Mathematics

Binary Operations

Definition of Binary Operations

A binary operation is a mathematical operation that combines two elements (called operands) from a set to produce another element. Examples of binary operations $+$, $\times$, $\div$, $-$.

Lesson Video

Watch this comprehensive explanation on binary operations with worked examples.

Practice Exercise

Question 1

A binary operation $*$ is defined on the set $R$ of real numbers by $a*b = a^2 + b - 4ab.$ Evaluate $2 * -3.$

[SSSCE 1995 Q3]

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Question 2

A binary operation is defined by $a * b = a^2 - b^2 + ab$, where $a$ and $b$ are real numbers. Evaluate $\sqrt{2} * \sqrt{3}$.

[SSSCE 1994 Q28]

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Question 3

A binary operation $\Delta$ is defined on the set of real numbers by $a \Delta b = a^b$. Find the value of $3 \Delta {-2}$.

[SSSCE 1993 Q6]

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Question 4

A binary operation is defined on the set of real numbers by $$a \Delta b = (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})$$ Find the value of $x$ if $x \Delta 2 = 1$

[SSSCE 2000 Q5]

Watch the video below for a clearer understanding of the question.

Question 5

A binary operation $\otimes$ is defined on the set of real numbers by $a \otimes b = a + b\sqrt{2}$. Find $(2 \otimes -3) \otimes 5$

[SSSCE 2002 Q1]

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Question 6

A binary operation $*$ is defined on the set $R$ of real numbers by $$a * b = ab + 2$$ where $a, b \in R$. Find $x$, if $4 * (3 * x) = 64$

Solution:

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Closure Property

Closure Property

A set S is said to be closed under a binary operation $*$ if for every element \( a, b \in S \), the result \( a * b \) is also in S.

For instance, the set of integers, $\mathbb{Z}$ is closed under the addition $(+)$ because anytime you perform addition on two integers, the result is also an integer.

Practice Exercise

Question 1

A binary operation $\Delta$ is defined on the set $T = \{ 1, 2, 3, 4 \}$ by $a \Delta b = a + b - ab$, where $a, b \in T$.

($i$) Copy and complete the table below:

\[ \begin{array}{|c|c|c|c|c|} \hline \Delta & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 1 & 1 & 1 \\ \hline 2 & 1 & \ & -1 & -2 \\ \hline 3 & 1 & -1 & \ & \ \\ \hline 4 & 1 & -2 & \ & -8 \\ \hline \end{array} \]

($ii$) Is $T$ closed with respect to $\Delta$?

[SSSCE 1994 Section B Q1]

Solution:

Question 2

A binary operation $*$ is defined on the set $S = \{0, 1, 2, 3\}$ by $a * b = (a + b) \mod 4$, where $a, b \in S$.

Is $S$ closed under the operation $*$?

Solution:

For $S$ to be closed under $*$, we need $(a + b) \mod 4 \in S$ for all $a, b \in S$.

Since we are taking modulo 4, the result will always be in $\{0, 1, 2, 3\}$. For example: $3 * 3 = (3 + 3) \mod 4 = 6 \mod 4 = 2 \in S$.

Therefore, $S$ is closed under $*$.

Question 3

A binary operation $\circ$ is defined on the set of positive integers by $a \circ b = a^b$. Is the set closed under this operation?

Solution:

For any positive integers $a$ and $b$, $a^b$ is always a positive integer.

For example: $2 \circ 3 = 2^3 = 8$, which is a positive integer.

Therefore, the set of positive integers is closed under $\circ$.

Question 4

A binary operation $*$ is defined on the set of even integers by $a * b = a + b$. Is the set of even integers closed under $*$?

Solution:

The sum of two even integers is always even.

If $a = 2m$ and $b = 2n$ for integers $m$ and $n$, then $a + b = 2m + 2n = 2(m + n)$, which is even.

Therefore, the set of even integers is closed under addition.

Question 5

A binary operation $*$ is defined on the set of odd integers by $a * b = a + b$. Is the set of odd integers closed under $*$?

Solution:

The sum of two odd integers is even, not odd.

For example: $3 + 5 = 8$, which is even, not odd.

Therefore, the set of odd integers is not closed under addition.

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Commutative Property

Commutative Property

A binary operation $*$ is said to be commutative if:

\[ a * b = b * a \quad \forall a,b \in S \]

Practice Exercise

Question 1

The operation $\Delta$ is defined on the set of non-zero rational numbers by $a \Delta b = \dfrac{a + b}{ab}$.
Determine whether $\Delta$ is commutative.

[SSSCE 1998 Section B Q1a]

Solution:

Question 2

A binary operation $\Delta$ is defined by $a \Delta b = a + b - 2b^2$ Show whether the binary operation $∆$ is commutative.

Solution:

Question 3

A binary operation $*$ is defined on the set of $R$, real numbers by $p*q = p^2 + q^2 - 2pq$, show whether the operation is commutative.

Solution:

Question 4

A binary operation $\triangle$ is defined on the set $R$ of real numbers by $$a \triangle b = \dfrac{ab}{a + b}$$
where $a, b \in R$

Determine whether the operation is commutative.

Solution:

Question 5

A binary operation $*$ on the set $R$ of real numbers by $$a * b = ab + 2$$ where $a, b \in R$. Show whether the operation is commutative.

Solution:

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Associative Property

Associative Property

A binary operation * is said to be associative if, $ \forall a,b,c \in S $:

\[ (a * b) * c = a * (b * c) \]

Practice Exercise

Question 1

The operation $\Delta$ is defined on the set of non-zero rational numbers by $a \Delta b = \dfrac{a + b}{ab}$.
Determine whether $\Delta$ is associative.

[SSSCE 1998 Section B Q1b]

Solution:

Question 2

A binary operation $∗$ is defined on the set of real numbers by $𝑎 ∗ 𝑏= 2𝑎 + 𝑎𝑏$. Show that ∗ is associative.

Solution:

Question 3

A binary operation $∇$ is defined on the set of real numbers by $𝑎∇𝑏=𝑎^2+2𝑎𝑏+𝑏^2$. Show that $∇$ is associative.

Solution:

Question 4

A binary operation $*$ is defined on the set $R$ of real numbers by $p*q = p^2 + q^2 - 2pq$, show whether the operation is associative.

Solution:

Question 5

A binary operation $\triangle$ is defined on the set $R$ of real numbers by $$a \triangle b = \dfrac{ab}{a + b}$$
where $a, b \in R$

Determine whether the operation is associative.

Solution:

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Distributive Property

Distributive Property

The operation * distributes over ○ if:

\[ a * (b ○ c) = (a * b) ○ (a * c) \] \[ (a ○ b) * c = (a * c) ○ (b * c) \]

Practice Exercise

Question 1

A binary operation $∗$ is defined on the set of real numbers by $𝑎 ∗ 𝑏= 2𝑎 + 𝑎𝑏$. Show that ∗ is distributive.

Solution:

Question 2

A binary operation $∇$ is defined on the set of real numbers by $𝑎∇𝑏=𝑎^2+2𝑎𝑏+𝑏^2$. Show that $∇$ is distributive.

Solution:

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Identity Element

An identity element (or neutral element) for a binary operation * on set S is an element \( e \in S \) such that:

\[ a * e = e * a = a \quad \forall a \in S \]

Additive Identity:

For addition, the identity element is 0:

\[ a + 0 = 0 + a = a \]

Multiplicative Identity:

For multiplication, the identity element is 1:

\[ a \times 1 = 1 \times a = a \]

Practice Exercise

Question 1

Find the identity element for $*$ defined by \( a * b = a + b - ab \) on ℝ.

Solution:

Let e be the identity: \[ a * e = a \] \[ a + e - ae = a \] \[ e - ae = 0 \] \[ e(1 - a) = 0 \]

This must hold for all a, so e = 0.

Verify: \( a * 0 = a + 0 - a \times 0 = a \)

Question 2

Does the operation \( a * b = a + b + 1 \) on $ℤ$ have an identity?

Solution:

Let e be the identity:

\[ a * e = a \] \[ a + e + 1 = a \] \[ e + 1 = 0 \] \[ e = -1 \]

Verify: \( a * -1 = a + (-1) + 1 = a \)

Also \( -1 * a = -1 + a + 1 = a \)

Thus, -1 is the identity element.

Question 3

A binary operation $\triangle$ is defined on the set $R$ of real numbers by $$ m \triangle n = m + n + 10 $$

Find the identity element.

Solution:

Question 4

The operation $*$ is defined on the set $R$ of real numbers by $$ a * b = a + b - 3ab $$ Find the identity element.

Solution:

Question 5

A binary operation $\triangle$ is defined on the set $R$ of real numbers by $$ x \triangle y = \dfrac{x + y}{1 + xy} $$ Find the identity element.

Solution:

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Inverse Element

Given a binary operation * on S with identity element e, the inverse of an element \( a \in S \) is an element \( a^{-1} \) such that:

\[ a * a^{-1} = a^{-1} * a = e \]

Additive Inverse:

For addition, the inverse of a is -a:

\[ a + (-a) = 0 \]

Multiplicative Inverse:

For multiplication, the inverse of a is $\frac{1}{a}$ ($a ≠ 0$):

\[ a \times \frac{1}{a} = 1 \]

                                
                                     Important Notes
                                
                                Not all elements may have inverses
An element may be its own inverse (e.g., -1 for multiplication)
The identity element is always invertible and is its own inverse

Practice Exercise

Question 1

Find inverses under * defined by \( a * b = a + b - ab \) (identity is 0).

Solution:

Question 2

For \( a * b = a + b + 1 \) (identity is -1), find the inverse of 5.

Solution:

Question 3

A binary operation $*$ is defined on the set $R$ of real numbers by $$ a * b = a + b + \dfrac{3}{4} $$. Find the inverse element.

Solution:

Question 4

The operation $*$ is defined on the set $R$ of real numbers by $$ a * b = 5ab $$. Find the inverse element under the operation.

Solution:

Question 5

A binary operation $\triangle$ is defined on the set $R$ of real numbers by $$ x \triangle y = \dfrac{x + y}{1 + xy} $$ Find the inverse element.

Solution:

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Practice Problems

Problem 1

Define * on ℤ by \( a * b = a + b + 2 \). Show that * is; $(i)$ commutative and $(ii)$associative.

Problem 2

Let * be defined on ℝ by \( a * b = \dfrac{a + b}{1 + ab} \). Find the identity element and the inverse of 2.

Identity:

Problem 3

Determine if the operation ○ defined by \( a ○ b = |a - b| \) on ℕ is associative.

Problem 4

Let $*$ be defined on $R$ by \( a * b = a + b - 2 \). Show that $*$ is commutative and find the identity element.

Problem 5

A binary operation $*$ is defined on $R$ by \( a * b = ab + a + b \). Determine whether the operation is associative.

Problem 6

Define a binary operation $∗$ on $Z$ by \( a * b = a + b + 5 \). Find the identity element and the inverse of $3$.

Problem 7

A binary operation $\triangle$ is defined on $R$ by \( x \triangle y = x + y + xy \). Show that the operation is commutative.

Problem 8

Let $*$ be defined on $R$ by \( a * b = \dfrac{a + b}{2} \). Is the set $R$ closed under this operation?

Problem 9

A binary operation $*$ is defined by \( a * b = a^2 + b^2 \). State whether $*$ is commutative and associative.

Problem 10

Define a binary operation $∇$ on $R$ by \( a ∇ b = a - b \). Test whether the operation is associative.

Problem 11

A binary operation $*$ is defined on $R$ by \( a * b = a + b + ab \). Find the identity element of the operation.

Problem 12

Let $*$ be defined on $R$ by \( a * b = a + b - ab \). Find the inverse of $4$ under the operation.

Problem 13

A binary operation $\circ$ is defined on $R$ by \( a \circ b = \dfrac{ab}{2} \). Determine whether an identity element exists.

Problem 14

Define a binary operation $*$ on $R$ by \( a * b = a^2b \). Is the operation commutative?

Problem 15

A binary operation $*$ is defined on $R$ by \( a * b = a + b + 1 \). Show that the inverse of $x$ is $-x - 2$.

Problem 16

Let $*$ be defined on $R$ by \( a * b = a - b + 2 \). Determine whether $*$ is commutative.

Problem 17

A binary operation $\triangle$ is defined on $R$ by \( x \triangle y = xy + x \). Test whether the operation is associative.

Problem 18

Define a binary operation $*$ on $R$ by \( a * b = \dfrac{a + b}{1 + ab} \). Find the identity element of the operation.

Problem 19

A binary operation $*$ is defined on $R$ by \( a * b = a + b + ab^2 \). State whether the operation is commutative.

Problem 20

Let $*$ be defined on $R$ by \( a * b = ab - a - b \). Find the identity element and the inverse of $2$.

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Quick Test

This section contains 100 multiple choice questions. You have 60 minutes to complete it.

Each question has four options labeled A to D. Select the correct answer for each question.

60:00

Question 1

Which of the following best describes a binary operation on a set $S$?

A function from $S \times S$ to $S$

A function from $S$ to $S$

A function from $S$ to $S \times S$

A relation on $S$

Question 2

If $a * b = a + b$ on the set of real numbers, what is $3 * 5$?

8

8

15

2

Question 3

Which property states that $a * b = b * a$ for all $a, b \in S$?

Commutative

Associative

Closure

Distributive

Question 4

Which of the following operations is not commutative on $R$?

Addition

Multiplication

Subtraction

Addition of integers

Question 5

A set $S$ is said to be closed under a binary operation $*$ if

$a * b \in S$ for all $a, b \in S$

$a + b \in S$

$a * b \notin S$

$a = b$

Question 6

If $a * b = a + b + 1$, what is $2 * 3$?

5

6

7

4

Question 7

Which property states that $(a * b) * c = a * (b * c)$?

Commutative

Associative

Closure

Identity

Question 8

If $a * b = ab$, what is the identity element?

0

1

-1

$a$

Question 9

An identity element $e$ of a binary operation $*$ satisfies

$a * e = e * a = a$

$a * e = 0$

$a * a = e$

$e * e = a$

Question 10

If $a * b = a + b$, what is the identity element?

0

1

-1

$a$

Question 11

The inverse of an element $a$ under a binary operation $*$ satisfies

$a * a^{-1} = e$

$a * a = e$

$a^{-1} * a^{-1} = e$

$a + a^{-1} = 1$

Question 12

Under addition on $R$, what is the inverse of $5$?

5

-5

1/5

0

Question 13

Which of the following sets is closed under multiplication?

Integers

Natural numbers excluding 1

Odd numbers under division

Whole numbers under subtraction

Question 14

Which operation has no identity element on $R$?

Subtraction

Addition

Multiplication

Addition of integers

Question 15

If $a * b = a + b + 2$, what is the identity element?

2

-2

0

1

Question 16

If $a * b = ab$, what is the inverse of $4$?

-4

$\dfrac{1}{4}$

0

4

Question 17

Which of the following is an example of a binary operation?

Addition on real numbers

Square root on real numbers

Logarithm

Differentiation

Question 18

If $*$ is commutative, which of the following is true?

$a * b = b * a$

$a * b = a$

$a * a = b$

$a * e = e$

Question 19

A binary operation must always satisfy which property?

Closure

Commutativity

Associativity

Distributivity

Question 20

If $a * b = a + b - ab$, what is $2 * 3$?

5

-1

1

6

Question 21

Let $*$ be defined on $R$ by \( a * b = a + b - 1 \). What is the identity element?

1

-1

0

2

Question 22

If $a * b = a + b - 1$, what is the inverse of $4$?

-3

-2

5

3

Question 23

Which of the following is not a binary operation on $R$?

Addition

Multiplication

Division by zero

Subtraction

Question 24

Let $*$ be defined by \( a * b = ab - a \). Find $2 * 3$.

6

4

3

1

Question 25

A binary operation $*$ on a set $S$ is associative if

$(a * b) * c = a * (b * c)$

$a * b = b * a$

$a * a = a$

$a * b \in S$

Question 26

Which of the following operations is associative on $R$?

Addition

Subtraction

Division

Taking square roots

Question 27

If $a * b = a + b + ab$, what is $1 * 2$?

3

5

4

6

Question 28

For $a * b = a + b + ab$, what is the identity element?

0

1

-1

2

Question 29

Which of the following operations is commutative but not associative?

$a * b = a + b^2$

$a * b = a + b$

$a * b = ab$

$a * b = a - b$

Question 30

Which element has an inverse under addition on $R$?

Every real number

Only positive numbers

Only zero

Only integers

Question 31

Let $*$ be defined on $R$ by \( a * b = a - b \). Does an identity element exist?

No

Yes, 0

Yes, 1

Yes, -1

Question 32

Which of the following is closed under subtraction?

Integers

Natural numbers

Positive integers

Whole numbers

Question 33

If $a * b = ab + 1$, what is $2 * 3$?

6

7

5

4

Question 34

Which property guarantees that a result of a binary operation stays within the set?

Closure

Identity

Inverse

Commutativity

Question 35

If $a * b = \dfrac{a + b}{2}$, what is $4 * 6$?

5

10

12

4

Question 36

The identity element under multiplication on $R$ is

1

0

-1

$a$

Question 37

Which real number has no inverse under multiplication?

0

1

-1

2

Question 38

If $*$ is a binary operation on $R$, which statement is always true?

$a * b$ is uniquely defined for all $a, b \in R$

$a * b = b * a$

$(a * b) * c = a * (b * c)$

An identity element exists

Question 39

Let $*$ be defined by \( a * b = a^2 + b^2 \). Is the operation commutative?

Yes

No

Only for integers

Only for positives

Question 40

Which of the following best describes an identity element?

An element that leaves all others unchanged

An element equal to zero

An element with no inverse

An element greater than all others

Question 41

Let $*$ be defined on $R$ by \( a * b = a + b + 3 \). What is the identity element?

3

-3

0

1

Question 42

If $a * b = a + b + 3$, what is the inverse of $5$?

-5

-11

2

-2

Question 43

Which of the following operations is not associative?

Subtraction

Addition

Multiplication

Addition on integers

Question 44

Let $*$ be defined by \( a * b = ab + b \). Find $2 * 3$.

6

7

9

12

Question 45

A binary operation has an inverse for every element only if

an identity element exists

the operation is commutative

the operation is distributive

the set is finite

Question 46

Which of the following elements has an inverse under multiplication on $R$?

0

2

Only 1

All real numbers

Question 47

Let $*$ be defined by \( a * b = a - b + 1 \). Find $3 * 1$.

3

2

1

0

Question 48

Which statement about identity elements is correct?

An identity element is unique

There can be more than one identity element

Every operation has an identity element

Identity elements depend on the element chosen

Question 49

Let $*$ be defined on $R$ by \( a * b = \dfrac{ab}{2} \). What is the identity element?

0

2

1

-2

Question 50

Which of the following operations always has an identity element?

Addition on real numbers

Subtraction on real numbers

Division on real numbers

Square root operation

Question 51

If $a * b = ab + a$, what is $2 * 4$?

6

10

8

12

Question 52

Which of the following operations is closed on the set of integers?

Subtraction

Division

Square root

Logarithm

Question 53

Let $*$ be defined by \( a * b = a + 2b \). Find $3 * 2$.

7

7

10

8

Question 54

Which of the following properties is not required for a binary operation?

Commutativity

Closure

Well-defined

Two operands

Question 55

If $a * b = \dfrac{a + b}{ab}$, is $*$ a binary operation on $R$?

No

Yes

Only for positive numbers

Only for integers

Question 56

Which statement about inverses is correct?

Inverses are unique when they exist

Every element has an inverse

Inverses depend on the order

Inverses exist without an identity

Question 57

Let $*$ be defined by \( a * b = a^2 + b \). Is the operation commutative?

No

No

Yes

Only for integers

Question 58

If $a * b = a + b + ab$, which structure is formed on $R$?

A group

A semigroup

A ring

Not closed

Question 59

Which of the following operations does not have inverses for all elements?

Multiplication on $R$

Addition on $R$

Addition on $Z$

Addition on $Q$

Question 60

Which condition must be satisfied for a binary operation to form a group?

Existence of identity and inverses

Commutativity only

Closure only

Distributivity

Question 61

Let $*$ be defined on $R$ by \( a * b = a + b - 4 \). What is the identity element?

4

4

-4

0

Question 62

If $a * b = a + b - 4$, what is the inverse of $7$?

1

-3

11

4

Question 63

Which of the following sets is not closed under addition?

Natural numbers excluding zero

Integers

Rational numbers

Real numbers

Question 64

Let $*$ be defined by \( a * b = a^2 + b^2 \). Find $1 * 2$.

3

5

4

6

Question 65

Which property ensures that a binary operation gives exactly one output?

Well-defined

Associative

Commutative

Inverse

Question 66

Which of the following operations is both commutative and associative?

Addition on real numbers

Subtraction on real numbers

Division on real numbers

Taking square roots

Question 67

If $a * b = ab - 1$, what is $3 * 2$?

5

5

6

4

Question 68

Let $*$ be defined on $R$ by \( a * b = ab - 1 \). Does an identity element exist?

Yes

No

Only for integers

Only for positive numbers

Question 69

The identity element of the operation \( a * b = ab - 1 \) is

2

1

0

-1

Question 70

Which of the following statements is always true for inverses?

An inverse is unique

Every element has an inverse

Inverses are optional

Inverses exist without identity

Question 71

Let $*$ be defined by \( a * b = a + 2b + 1 \). Find $2 * 3$.

7

9

11

6

Question 72

Which of the following operations is not closed on $R$?

Division

Addition

Multiplication

Subtraction

Question 73

Let $*$ be defined by \( a * b = \dfrac{a + b}{a - b} \). Is $*$ a binary operation on $R$?

No

Yes

Only for integers

Only for positives

Question 74

Which of the following is required for a set to form a group?

Closure, identity and inverses

Commutativity only

Distributivity

Order of elements

Question 75

If $a * b = a^2b$, is the operation commutative?

No

Yes

Only for $a=b$

Only for integers

Question 76

Which element is its own inverse under addition?

0

1

-1

2

Question 77

Let $*$ be defined by \( a * b = ab + a + b \). What is the identity element?

0

1

-1

2

Question 78

Which statement correctly describes a binary operation?

It combines two elements of a set to give one element of the same set

It combines one element to give two outputs

It maps elements to a different set

It must be commutative

Question 79

Which operation forms an abelian group on $R$?

Addition

Multiplication

Subtraction

Division

Question 80

Why is division on $R$ not a binary operation?

Division by zero is undefined

It is not commutative

It has no identity element

It is not associative

Question 81

Let $*$ be defined on $R$ by \( a * b = a + b - 6 \). What is the identity element?

6

-6

0

3

Question 82

If $a * b = a + b - 6$, what is the inverse of $10$?

2

-4

16

6

Question 83

Which of the following is an example of a closed set under addition?

Integers

Natural numbers

Positive integers

Whole numbers excluding zero

Question 84

Let $*$ be defined by \( a * b = ab + a \). Find $3 * 2$.

9

6

8

12

Question 85

Which property ensures that the order of grouping does not affect the result?

Associativity

Commutativity

Closure

Identity

Question 86

Which of the following operations is commutative on $R$?

Addition

Subtraction

Division

Taking logarithms

Question 87

If $a * b = a - b + 4$, what is $5 * 3$?

6

2

4

12

Question 88

Which statement about identity elements is true?

An identity element leaves every element unchanged

Every operation has more than one identity

Identity depends on the element chosen

Identity exists only in groups

Question 89

Let $*$ be defined on $R$ by \( a * b = ab \). What is the inverse of $5$?

\( \dfrac{1}{5} \)

-5

0

1

Question 90

Which of the following does not have an inverse under multiplication?

0

1

-2

3

Question 91

Let $*$ be defined by \( a * b = a + 3b \). Find $1 * 4$.

13

7

12

4

Question 92

Which of the following operations is not closed on the integers?

Division

Addition

Subtraction

Multiplication

Question 93

Let $*$ be defined by \( a * b = \dfrac{a + b}{2} \). What is $6 * 10$?

8

16

5

3

Question 94

Which property ensures that a result belongs to the same set?

Closure

Commutativity

Associativity

Inverse

Question 95

If $a * b = a^2 + b$, is the operation commutative?

No

Yes

Only for integers

Only for positives

Question 96

Which structure requires closure, associativity, identity and inverses?

Group

Set

Ring

Function

Question 97

Which of the following operations forms a group on $R$?

Addition

Multiplication

Subtraction

Division

Question 98

Why does subtraction fail to form a group on $R$?

It is not associative and has no identity

It is not closed

It has no inverses

It is commutative

Question 99

Which statement best summarizes a binary operation?

A rule combining two elements of a set to produce one element of the same set

A rule combining one element to give two outputs

A mapping between two different sets

A relation without rules

Question 100

Which of the following is the main reason division on $R$ is not a binary operation?

Division by zero is undefined

It is not commutative

It has no inverses

It is not associative