Proving Even Number Additive Inverse is Even

In summary, the proof that an even number's additive inverse is also even involves using the definition of even numbers and the property of additive inverses. An example of this proof is shown using the even number 8 and its additive inverse -8. This proof also relates to the properties of addition and has a similar proof for odd numbers. This concept is important in mathematics as it helps understand the behavior of even and odd numbers under addition and is a fundamental concept used in higher level math.
  • #1
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Homework Statement


How do I show that the additive inverse, or negative, of an even number is an even number ( direct proof) ?


Homework Equations





The Attempt at a Solution



I have no idea. Anybody?
 
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  • #2
Really? Let [tex]a=2n[/tex] be an even number then the additive inverse [tex]b[/tex] is defined by [tex]a+b=0[/tex], so [tex]b=-a=-2n[/tex], and so...
 
  • #3
alternatively, start "an even number is a + a, so …" :wink:
 
  • #4
Thanks! Makes sense now.
 

1. How do you prove that an even number's additive inverse is also even?

The proof involves using the definition of even numbers, which states that an even number can be divided by 2 without a remainder. We can show that the additive inverse of an even number can also be divided by 2 without a remainder, making it even.

2. Can you provide an example to illustrate this proof?

Sure, let's take the even number 8. Its additive inverse is -8. When we divide -8 by 2, we get -4, which is an integer. This proves that the additive inverse of 8 is also even.

3. How does this proof relate to the properties of addition?

This proof is based on the property of additive inverses, which states that the sum of a number and its additive inverse is always equal to 0. In other words, for an even number n, n + (-n) = 0. Since 0 is an even number, this shows that the additive inverse of an even number is also even.

4. Is there a similar proof for odd numbers?

Yes, there is. The proof for odd numbers is based on the same concept, but using the definition of odd numbers. An odd number cannot be divided by 2 without a remainder, but when we add its additive inverse, the result can be divided by 2 without a remainder, making it an even number.

5. Why is this proof important in mathematics?

This proof is important because it helps us understand the relationship between even and odd numbers and how they behave under addition. It also shows that the properties of addition hold true for all types of numbers, including even and odd numbers. This proof is also a fundamental concept in higher level math and is used to prove more complex theories and theorems.

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