About Operation on Integers
- Multiplication Operation on Integers
- For example, Finding the Product of \( 2\times (-2) \) using a Number Line.
- Multiplication of Integers
- Multiplying Integers with the Same Sign
- Multiplying Integers with the Different Signs
- Examples on Multiplying Integers
- Multiplying the Expressions
- Examples on Multiplying the Expressional Number
- Examples of Multiplication of Integers in Real Life Situations
- Frequently Asked Questions
Multiplication Operation on Integers
Integers
Integers are all negative and positive numbers, which include -1, -2, -3… and also 1, 2, 3,… and so on. Integers do not have a decimal or a fractional part, hence, we can say that integers are whole numbers which can be positive, negative or zero.
Integer multiplication using a number line
Multiplication on a number line refers to the use of a number line to perform multiplication operations on a set of numbers. A number line is a horizontally extending infinite line with digits placed at regular spacing on both sides. The application of a multiplication operation to a given list of numbers on a number line is known as multiplication on a number line. Repeated addition is another name for multiplication.
Case 1: Multiplication of positive integers
There is a simple rule to follow when multiplying two or more positive integers. For positive numbers we’ll move to the right side of the number line.
Case 2: Multiplication of negative integers
1. Use the even-odd rule to multiply more than two negative numbers.
2. Count the number of negative signs in the multiplication.
3. The product is positive when there are an even number of negative numbers multiplied. We’ll move to the right side of the number line because this is a multiplication of positive numbers.
4. The product is negative when there are an odd number of negative numbers multiplied. We’ll move to the left side of the number line because this is a multiplication of negative numbers.
For example, Finding the Product of \( 2\times (-2) \) using a Number Line.
Here there is one negative number, -2, and it is odd. Hence, we’ll move to the left side of the number line.
Step 1: Start at the beginning from 0.
Step 2: On the number line, make two groups of two equal intervals. The first group corresponds to the first 0 to -2 jump.
Step 3: The second group is equivalent to the next jump from -2 to -4.
We will reach -4 on the number line, forming 2 distinct groups.
Therefore, \( 2\times (-2)=-4 \).
Multiplication of Integers
Integer multiplication is the process of repeatedly adding positive and negative numbers or both, or simply integers. Integer multiplication is carried out in the same way as regular multiplication. However, because integers include both negative and positive numbers, we must remember certain rules or conditions when multiplying them, as shown in the example below.
Let’s evaluate 2 (- 3 + 7) using two methods:
Method 1: In parenthesis, evaluate:
2 (- 3 + 7) = 2(4)
= 8
Method 2: Using the distributive property to solve a problem:
2 (- 3 + 7) = 2( -3) + 2(7)
= -6 + 14
= 8
2(-3) must equal to ‘-6’ for the distributive property to be true. This helps in forming the rules for multiplying integers.
Multiplying Integers with the Same Sign
The product of two integers with the same sign will give a positive result.
For example, \( 2\times 3 = 6 \)
\( -2\times (-3) = 6 \)
Multiplying Integers with the Different Signs
The product of two integers with opposite signs will give a negative result.
For example, \( 2\times (-3) = -6 \)
\( -2\times 3 = -6 \)
The following steps can be applied when multiplying integers:
Step 1: Determine the number’s absolute value.
Step 2: Calculate the absolute value of the product.
Step 3: Once you have the product, apply the rules or conditions to determine the product’s sign.
Examples on Multiplying Integers
a) Find the product of \( -5\times (-5)=? \).
Solution:
Both the integers have the same (-) sign. So the product will be positive.
\( -5\times (-5)=25 \)
The product is 25.
b) Find the product of \( -4\times (-9)=? \).
Solution:
Both the integers have the same (-) sign. So the product will be positive.
\( -4\times (-9)=36 \)
The product is 36.
c) Find the product of \( 12\times (-2)=? \).
Solution:
Both the integers have different signs. So the product will be negative.
\( 12\times (-2)=-24 \)
The product is -24.
d) Find the product of \( 4\times (-6)=? \).
Solution:
Both the integers have different signs. So the product will be negative.
\( 12\times (-2)=-24 \)
The product is -24.
Multiplying the Expressions
A number, a variable, or a combination of numbers, variables, and operation symbols make up an expression.
We know that,
- Three positive integers add up to a positive number.
- Three negative integers add up to a negative number.
- When two positive integers are multiplied, the result is positive.
- When two negative integers are multiplied together, then the result is a positive number.
These facts can be applied when multiplying expressions.
The expression \( (-3)^2 \) indicates the multiplication of the number in parenthesis, -3, by itself, that is \( (-3\times -3) \). The expression \( -3^2 \),
is 9 as the square of negative numbers is positive.
Examples on Multiplying the Expressional Number
a) Find \( (-4)^2=? \)
Solution: \( (-4)^2=(-4)\times (-4) \) (Write \( (-4)^2 \) as the repeated multiplication)
\( =16\) (Multiply)
b) Find \( -4^2=? \)
Solution: \( -4^2=-(4\times 4) \) (Write \( 4^2 \) as the repeated multiplication)
\( = -16\) (Multiply)
c) Find \( (-7)^2=? \)
Solution: \( (-7)^2=(-7)\times (-7) \) (Write \( (-7)^2 \) as the repeated multiplication)
\( =49\) (Multiply)
d) Find \( -7^2=? \)
Solution: \( -7^2=-(7\times 7) \) (Write \( 7^2 \) as the repeated multiplication)
\( =-49\) (Multiply)
Examples of Multiplication of Integers in Real Life Situations
Example 1:
a. Find the product of \( -7\times (-8)=? \).
Solution: Both the integers have the same (-) sign, so the product will be positive.
\( -7\times (-8)=56 \)
The product is 56.
b. Find the product of \( 4\times (-5)=? \).
Solution: Both the integers have different signs, so the product will be negative.
\( 4\times (-5)=-20 \)
The product is -20.
Example 2:
a. Find \( -3\times 10\times (-6)=? \).
Solution:
\( -3\times 10\times (-6)=-3\times (-6)\times 10 \) (Commutative property of multiplication)
\( =18\times 10 \) (Multiply -3 and -6)
\( =180 \) (Multiply 18 and 10)
b. Find \( -6(-3+5)+7=? \).
Solution:
\( -6(-3+5)+7=-6(2)+7 \) (Performing operation in parentheses)
\( =-12+7 \) (Multiply -6 and 2)
\( =-5 \) (Multiply -12 and 7)
Example 3:
You use your computer to solve a number puzzle. You begin with a score of 300 points. You complete the puzzle in 9 minutes and 50 seconds and make four errors. Every error is worth -30 points, and every second under 10 minutes is worth 1 bonus point. So, how did you perform in solving the number puzzle on the computer?
Solution: When solving a puzzle, we are given options that either help in gaining points or make us lose points. After completing the puzzle, we must calculate our score. To solve this problem, we’ll use a verbal model. Calculate the total of the starting points, penalty points, and time bonuses.
First, we have to calculate the time bonus.
Time bonus = 10 min – 9 min 50 sec (Subtracting the time difference)
= 10 sec
Score = Starting points + number of errors x penalty per error + time bonus
= 300 + 4(-30) + 10 (Substituting the values from the given question)
= 300 + (-120) +10 (Multiplying 4 and – 30)
= 180 + 10 (using common properties of addition)
= 190 (Simplified)
So, the score is 190 points.
Example 4:
Jogging burns 20 calories per minute. How many calories will be burned after 30 minutes of jogging?
Solution: Given that, 20 calories are burnt per minute of jogging,
Calories burned after jogging for 30 minutes = ?
Calories burned after jogging for 30 minutes = 30 20
= 600 calories [By multiplication]
Hence, 600 calories are burned after 30 minutes.
Example 5:
Find the product of \( 4\times (-2) \) using a number line?
Solution:
Here there is one negative number, -2, that is odd. Hence, we’ll move to the left side of the number line.
Step 1: Start at the beginning from 0.
Step 2: On the number line, make four groups of two equal intervals. The first group corresponds to the first 0 to -2 jump.
Step 3: The second group is equivalent to the next jump from -2 to -4. The third group jumps from -4 to -6, and the fourth group jumps from -6 to -8.
We will reach -8 in this manner forming four distinct groups.
Hence, \( 4\times (-2) =-8 \).
According to the associative property, the multiplication of \( (-a)\times b \) or the multiplication of \( a\times (-b) \) results in \( -ab \). So, both are the same. As per the property, it states that when three or more numbers are multiplied, regardless of the order of multiplicands, the product is the same.
From the rules of multiplication of integers, The product of three positive integers yields a positive result.
For example, \( 2\times 1\times 3 \)
\( =(2\times 3)\times 1 \) (Using Associative property)
\( =6\times 1 \) (Multiply 2 and 3)
\( =6 \) (Multiplication property of 1)
From the rules of multiplication of integers, the product of two positive integers and one negative integer is a negative result.
For example, \( 3\times 5\times -1 \)
\( =(3\times 5)\times -1 \)
\( =15\times -1 \) (Multiply 3 and 5)
\( =-15 \) (Multiply 15 and -1)