Essential Rules for Dividing Integers with Solved Examples
The set of integers is made up of numbers, including natural numbers, additive inverses and zero. When we subtract a small number from a larger number, we get a whole number. However, there are no whole numbers to represent 3-8. In order to describe such differences, we created integers.
Division Rules of Integers
As we are familiar with natural and whole numbers, we can apply the properties of the division of integers.
The Division of Integers Rules:
Rule 1: The quotient of two positive integers is always a positive integer.
Rule 2: The quotient value of two negative integers is always a positive integer.
Rule 3: The quotient of one positive integer and one negative integer is always negative.
One thing to remember is that when dividing, you should always divide without signs, but once you have the solution of the integer, give the sign according to the sign given in the problem.
What are the Properties of Integers?
Integers have a few properties that regulate their operations. Many equations can be solved using these principles or properties. To refresh your memory, integers are any positive or negative number, including zero. The properties of these integers will assist in quickly simplifying and answering a series of integer operations.
All properties and identities for addition, subtraction, multiplication and division of numbers apply to all integers as well. Integers are the set of positive, zero and negative numbers represented by the letter Z.
Integers have five main operational properties, which are as follows:
Closure Property
Identity Property
Properties of Integers Under Division:
Let's look at the properties of integer division.
Property 1:
When one integer '\[x\]' is divided by another integer '\[y\],' the result is that the integer '\[x\]' is divided into '\[y\]' equal parts.
If '$y$' divides ‘$x$’ without leaving any leftover, then '$x$' is divisible by '$y$' evenly.
Example:
When 21 is divided by 3, it is divided into three equal parts, each with a value of 7.
Property 2:
When an integer is divided by another integer, the division algorithm is the sum of the quotient's product and the divisor's remainder, where the remainder equals the dividend.
More specifically,
\[\text{Dividend} =\text{ Quotient} \times \text{Divisor} + \text{Remainder}\]
The dividend is the number by which we divide.
The divisor is the integer by which we divide.
The obtained result is known as the quotient.
The remainder is the remaining integer.
Property 3:
When you divide an integer by one, the quotient is the number itself.
Example:
\[\Rightarrow \dfrac{7}{1}=7\]
Property 4:
The quotient of an integer divided by itself is 1.
Example:
\[\Rightarrow \dfrac{5}{5}=1\]
Property 5:
The quotient of any positive or negative integer divided by zero is unknown. As a result, dividing any positive or negative number by zero has no value.
Example:
\[\Rightarrow \dfrac{3}{0}=\text{undefined}\]
Property 6:
The quotient is zero when zero is divided by any positive or negative integer.
Example:
\[\Rightarrow \dfrac{0}{5}=0\]
Property 7:
When one integer is divided by another integer that is a multiple of ten, such as 10, 100, 1000, and so on, the decimal point must be moved to the left.
Examples:
\[\Rightarrow \dfrac{123}{10}=12.3\]
\[\Rightarrow \dfrac{123}{100}=1.23\]
Property 8:
\[\dfrac{\text{Positive}}{\text{Positive}}=\text{Positive}\]
Example: \[\dfrac{12}{3}=4\]
\[\dfrac{\text{Positive}}{\text{Negative}}=\text{Negative}\]
Example: \[\dfrac{12}{-3}=-4\]
\[\dfrac{\text{Negative}}{\text{Positive}}=\text{Negative}\]
Example: \[\dfrac{-12}{3}=-4\]
\[\dfrac{\text{Negative}}{\text{Negative}}=\text{Positive}\]
Example: \[\dfrac{-12}{-3} = 4\]
Properties of Division of Integers with Examples
Integer Division has the following properties:
Closure property
Commutative property
Associative property
Division by zero
Division by 1
Closure Property under Division of Integers:
The division's closure property states that the result of dividing two whole numbers is not always a whole number. Division does not close whole numbers, so \[\dfrac{a}{b}\] is not necessarily a whole number. According to the property, \[\dfrac{35}{7}=5\] (whole number), yet \[\dfrac{5}{15}=\dfrac{1}{3}\] (not a whole number).
Commutative Property under Division of Integers:
Whole number division is not commutative. If a and b are both whole numbers, then \[\dfrac{a}{b} \ne \dfrac{b}{a}\].
Consider the following example: a = 14, b = 7.
\[\dfrac{14}{7} \ne \dfrac{7}{14}\].
Associative Property under Division of Integers:
The Associative feature does not apply to whole number divisions. If a, b and c are the three whole numbers, then \[\dfrac{a}{\dfrac{b}{c}} \ne \dfrac{\dfrac{a}{b}}{c}\].
Example:
\[\dfrac{24}{\dfrac{12}{2}}\ne \dfrac{\dfrac{24}{12}}{2}\].
Division by Zero:
Any entire number that is divided by zero (apart from zero) has no definition or meaning.
Example:
\[\Rightarrow \dfrac{3}{0}=\text{undefined}\]
The result is zero when zero is divided by a full number that is not zero.
Example:
\[\Rightarrow \dfrac{0}{5}=0\]
Division by 1:
Any whole number that is not zero and is divided by one will result in the whole number itself as the quotient.
Example:
\[\Rightarrow \dfrac{7}{1}=7\]
Solved Problems
1. Apply the division of integers principles to the following statement to find the solution: \[(-20)\div (-5)\div (-2)\].
Ans: Given that this phrase contains many operations and that we must divide three numbers, we will apply the BODMAS rule in this situation. The Initial Step is \[(-20)\div (-5)\]. Now, we may find the answer by dividing -20 by -5, which is 4. Since \[\text{Negative} \div \text{Negative} =\text{Positive}\], the number 4 is a positive integer. The new expression is therefore \[{4} \div \left({-2}\right)\]. The result of dividing 4 by -2 is -2 since \[\text{Positive}\div \text{Negative} = \text{Negative}\].
As a result, \[(-20)\div (-5)\div (-2) = -2\].
2. What is 91 divided by 7?
Ans: \[91 \div 7\]
\[\Rightarrow \dfrac {91}{7}=13\]
3. Find the value of \[\left[32 + 2 \times 17+(-6)\right]\div 15\].
Ans: \[\left[32 + 2 \times 17+(-6)\right]\div 15\]
\[= \left[32 + 34 +(-6)\right]\div 15\]
\[= \left[66-6\right]\div 15\]
\[= \left[60\right]\div 15\]
= 4
Key Features
1 is the smallest positive integer, and -1 is the largest negative number.
Operations on integers are governed by the BODMAS rule. Any of the subsequent constitute "operations": Brackets, Squares, Powers, Square Roots, Division, Multiplication, Addition and Subtraction.
Integer Division is not associative in nature.
Practice Questions
1. Evaluate \[\left[36 \div (-9)\right] \div \left[(-24)\div 6\right]\].
Ans: 1
2. Divide 20 by 10.
Ans: 2
FAQs on Properties of Division of Integers Explained
1. What are the main properties related to the division of integers?
The primary properties of integers discussed in mathematics are the Closure, Commutative, Associative, and Identity properties. However, unlike addition and multiplication, the division of integers does not satisfy any of these fundamental properties. Key rules specifically apply to the division of integers, such as division by 1, division by itself, and division involving zero.
2. Why is the division of integers not considered closed?
The division of integers is not closed because the result of dividing one integer by another is not always an integer. For an operation to be closed, the result must belong to the same set of numbers. For example, if we divide the integer 7 by the integer 2 (i.e., 7 ÷ 2), the result is 3.5. Since 3.5 is a decimal and not an integer, the closure property fails for the division of integers.
3. Does the commutative property apply to the division of integers? Explain with an example.
No, the division of integers is not commutative. The commutative property states that changing the order of the numbers does not affect the result (a ÷ b = b ÷ a). This is not true for integer division. For instance, (-8) ÷ 4 = -2, but 4 ÷ (-8) = -0.5. As the results are different, the commutative property does not hold.
4. What are the rules for dividing integers with different signs (positive and negative)?
The sign of the quotient when dividing integers depends on the signs of the dividend and divisor. The rules are as follows:
- If you divide two integers with the same sign (both positive or both negative), the result is always positive. Example: (-12) ÷ (-3) = 4.
- If you divide two integers with different signs (one positive and one negative), the result is always negative. Example: 15 ÷ (-5) = -3.
5. Why is there no identity element for the division of integers?
An identity element must work commutatively. For division, while any integer 'a' divided by 1 results in 'a' (a ÷ 1 = a), the reverse operation, 1 ÷ a, does not equal 'a' (unless a = 1 or -1). For 1 to be a true identity element for division, it would need to satisfy the property regardless of its position. Since it fails this condition, division is said to have no identity element.
6. What is the difference between dividing an integer by 0 and dividing 0 by an integer?
This is a crucial concept with very different outcomes:
- Dividing an integer by 0: Division of any integer by zero is undefined. It is not possible to determine a value for an expression like a ÷ 0.
- Dividing 0 by an integer: Zero divided by any non-zero integer is always zero. For example, 0 ÷ 10 = 0, because 0 × 10 = 0.
7. How does the associative property fail for the division of integers? Provide an example.
The associative property, which states (a ÷ b) ÷ c = a ÷ (b ÷ c), does not hold for the division of integers because grouping the numbers differently changes the result. Consider the integers -24, 4, and -2:
- Grouped one way: ((-24) ÷ 4) ÷ (-2) = (-6) ÷ (-2) = 3
- Grouped another way: (-24) ÷ (4 ÷ (-2)) = (-24) ÷ (-2) = 12
Since 3 is not equal to 12, the division of integers is proven to be non-associative.
