Multiplication division integers

An integer is a number from the set of negative and positive numbers, including zero, that has no decimal or fractional element. Integers include the numbers -5, 0, 1, 5, 8, 97, and 343. Z represents a collection of numbers that includes:

• Positive Integers: A positive integer is one that is greater than zero. For instance: 1, 2, 3…
• Negative Integers: A negative integer is one that is less than zero. For example: -1, -2, -3…
• The number zero is neither negative nor positive. It’s a whole number.

Z = {…-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6…}

Multiplication and Division of Integers: 

Integer multiplication and division are two of the most fundamental operations on numbers. Integer multiplication is the same as repeated addition, which involves adding an integer a certain number of times. For example, 5 × 3 implies adding 5 three times, i.e.  5 + 5 + 5 = 15.

Integer division denotes the equal grouping or division of an integer into a certain number of groups. For example, -6 ÷ 2 denotes splitting -6 into two equal halves, yielding -3. In this topic, we will study more about integer multiplication and division. 

Multiplication of integers:

Integer multiplication is the process of adding positive and negative numbers over and over again, or simply integers. When it comes to integer multiplication, the following situations must be considered: 

• 2 positive numbers multiplied
• 2 negative numbers multiplied
• 1 positive and 1 negative number multiplied

When two positive signs are multiplied together, the result is Positive x Positive = Positive = 6 × 5= 30.

When two negative signs are multiplied together, the result is Negative x Negative = Positive = –6 × –3 = 18.

When you multiply numbers with one negative and one positive sign, Negative x Positive = Negative = –3 × 5 = –15.

Integer multiplication is quite similar to standard multiplication. However, because integers can contain both negative and positive values, we must remember certain criteria or conditions while multiplying integers, as we saw in the previous section. Let’s see how to multiply integers.

• Step 1: Determine the numbers’ absolute values.
• Step 2: Calculate the absolute value product.
• Step 3: After obtaining the product, identify the sign of the number in accordance with the rules or criteria.

Let’s look at an example to better comprehend the stages. Multiply  -8 and 9.

Step 1: Find the absolute value of -8 and 9.

|-8| equals 8 and |9| equals 9.

Step 2: Calculate the product of absolute value values 8 and 9.

8 × 9 = 72

Step 3: Using the multiplication of integers principles, determine the sign of the product. If a negative number is multiplied by a positive number, the product is a negative number, according to the multiplication of integers rule.

As a result, – 8 × 9 = – 72.

Division of integers: 

Integer division entails grouping of objects. It contains both positive and negative numbers. Division of numbers, like multiplication, involves the same instances.

• 2 positive numbers divided
• 2 negative numbers divided
• dividing two numbers that are both positive and negative

Positive ÷ Positive Equals Positive, 18 ÷ 3 = 6 when you divide numbers with two positive signs

Negative ÷ Negative = Positive, –18 ÷ –3 = 6 when you divide numbers with two negative signs.

Negative ÷ Positive = Negative, –18 ÷ 3 = –6 when you divide numbers with one negative sign and one positive sign.

To summarise and make things simple, the two most important elements to keep in mind while multiplying or dividing integers are:

The result is always negative when the signs are different. 

When both the signs are same, the result is always positive.

Properties of Multiplication and Division of Integers:

When two or more numbers are connected by a multiplication or division operation, the characteristics of multiplication and division of integers let us discover the relationship between them. When it comes to multiplication and division of integers, there are a few properties to consider.

The following are properties associated with integer multiplication and division:

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Identity Property

1) For each mathematical action, the set is closed according to the closure property. When adding, subtracting, or multiplying integers, they become closed. Under division, however, they are not closed.

2) Interchanging the locations of arithmetic operations in an operation does not alter the outcome, according to the commutative property. The commutative property applies to the addition and multiplication of integers, but not to the division of integers.

3) Changing the grouping of numbers has no effect on the operation’s result, according to the associative property. The associative property is true for the addition and multiplication of two numbers, but not for the division of two integers.

4) The distributive property asserts that operand a can be divided among operands b and c as (a × b + a × c), i.e., a (b + c) = (a ×b + a × c) for any expression of the type a (b + c), which means a × (b + c). Over addition and subtraction, multiplication of integers is distributive. When it comes to integer division, the distributive property does not apply.

5) When it comes to integer multiplication, 1 is the multiplicative identity. In the case of integer division, there is no identity element.