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What is the Multiplicative Identity for Integers?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The Multiplicative Identity for Integers is a special integer that, when multiplied by any other integer, leaves that integer unchanged. It acts like a 'mirror' for multiplication. This special integer is 1.
Simple Example
Quick Example
Imagine you have 5 cricket balls. If you multiply this by 1, you still have 5 cricket balls (5 x 1 = 5). The number of balls doesn't change. Similarly, if you have -300 rupees due to your friend, multiplying by 1 still means you owe -300 rupees (-300 x 1 = -300).
Worked Example
Step-by-Step
Let's find the result when we multiply the integer -15 by the Multiplicative Identity.
STEP 1: Identify the given integer. Here, it is -15.
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STEP 2: Recall the Multiplicative Identity for Integers. It is 1.
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STEP 3: Perform the multiplication: -15 x 1.
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STEP 4: According to the property of the Multiplicative Identity, any integer multiplied by 1 remains the same.
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STEP 5: So, -15 x 1 = -15.
ANSWER: The result is -15.
Why It Matters
Understanding the Multiplicative Identity is fundamental in all of mathematics and science. In computer science, it's crucial for programming calculations, and in data science, it helps in scaling data without changing its core value. Engineers use it to simplify complex equations, making design and analysis easier.
Common Mistakes
MISTAKE: Thinking the Multiplicative Identity is 0. | CORRECTION: The Multiplicative Identity is 1. Multiplying by 0 always gives 0, which changes the original number.
MISTAKE: Confusing Multiplicative Identity with Additive Identity. | CORRECTION: The Multiplicative Identity is 1 (for multiplication). The Additive Identity is 0 (for addition, where adding 0 doesn't change the number).
MISTAKE: Applying the concept only to positive integers. | CORRECTION: The Multiplicative Identity (1) works for ALL integers – positive, negative, and zero.
Practice Questions
Try It Yourself
QUESTION: What is 17 multiplied by the Multiplicative Identity? | ANSWER: 17
QUESTION: If a number 'x' multiplied by 1 equals -45, what is the value of 'x'? | ANSWER: -45
QUESTION: A factory produces 250 cars per day. If they multiply this production rate by the Multiplicative Identity for 3 days, what is the total number of cars produced in those 3 days? | ANSWER: 750 cars (250 x 1 = 250 cars/day; 250 x 3 = 750 cars in 3 days)
MCQ
Quick Quiz
Which of the following statements about the Multiplicative Identity for Integers is true?
It is 0, and multiplying by it makes any integer 0.
It is 1, and multiplying by it leaves any integer unchanged.
It is -1, and multiplying by it changes the sign of any integer.
It is 1, and multiplying by it always increases the integer's value.
The Correct Answer Is:
B
The Multiplicative Identity is indeed 1, and its unique property is that multiplying any integer by 1 results in the same integer, leaving it unchanged. Options A, C, and D describe incorrect properties or values.
Real World Connection
In the Real World
Imagine you are using a calculator app on your phone. If you type in '85' and then press 'x' followed by '1' and then '=', the display will still show '85'. This simple function relies on the Multiplicative Identity. In computer programming, when scaling data or performing calculations, multiplying by 1 is often implicitly used to maintain original values or confirm data integrity.
Key Vocabulary
Key Terms
INTEGER: A whole number (positive, negative, or zero) | MULTIPLICATION: The process of finding the product of two or more numbers | IDENTITY: A number that leaves another number unchanged when a specific operation is performed | PROPERTY: A characteristic or attribute of a mathematical operation or number
What's Next
What to Learn Next
Now that you understand the Multiplicative Identity, you're ready to explore the Multiplicative Inverse for Integers. This concept builds on identity and will show you how to 'undo' multiplication, which is very useful in solving equations!
