Understanding Property 9 of Negative Numbers in Calculus

In summary, the steps are as follows: -(-a) x b = -(a x b)-([-a] + a) \cdot b = [(-a) + a] \cdot b-(-a) \cdot = -(a \cdot b)-But are these steps truly valid; how I can know that, when factoring the b, the negative symbol isn't appended to the b; isn't this what I am trying to prove, that the negative symbol is appended to the a?
  • #1
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Hello,

I am embarking to read Spivak's book on Calculus, and have come across some difficulty with something that is perhaps rather trivial. In the third edition, there is a section entitled Basic Properties of Numbers. Near the end of page 7, the author begins discussing how he will use property 9 to derive the fact that, when negative numbers are multiplied, the result is a positive number. Before this, he must show that (-a) x b = -(a x b) is true. To prove this, we must show that
(-a) x b + a x b = [(-a) + a] x b, which I can follow. Next, he says that, because this is true, then we can add -(a x b) to both sides of the equation:

[itex](-a) \cdot b + a \cdot b + [-(a \cdot b)] = [(-a) + a] \cdot b + [-(a \cdot b)][/itex]

Clearly, the first term on the RHS of the equation will yield zero.

[itex](-a) \cdot b + a \cdot b + [-(a \cdot b)] = -(a \cdot b)[/itex]

[itex][-a + a - a] \cdot b = -(a \cdot b)[/itex]

[itex](-a) \cdot = -(a \cdot b)[/itex]

But are these steps truly valid; how I can know that, when factoring the b, the negative symbol isn't appended to the b; isn't this what I am trying to prove, that the negative symbol is appended to the a?
 
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  • #2
Bashyboy said:
Hello,

I am embarking to read Spivak's book on Calculus, and have come across some difficulty with something that is perhaps rather trivial. In the third edition, there is a section entitled Basic Properties of Numbers. Near the end of page 7, the author begins discussing how he will use property 9 to derive the fact that, when negative numbers are multiplied, the result is a positive number. Before this, he must show that (-a) x b = -(a x b) is true. To prove this, we must show that
(-a) x b + a x b = [(-a) + a] x b, which I can follow. Next, he says that, because this is true, then we can add -(a x b) to both sides of the equation:

[itex](-a) \cdot b + a \cdot b + [-(a \cdot b)] = [(-a) + a] \cdot b + [-(a \cdot b)][/itex]

Clearly, the first term on the RHS of the equation will yield zero.

[itex](-a) \cdot b + a \cdot b + [-(a \cdot b)] = -(a \cdot b)[/itex]

[itex][-a + a + a] \cdot b = -(a \cdot b)[/itex]

[itex](-a) \cdot = -(a \cdot b)[/itex]

But are these steps truly valid; how I can know that, when factoring the b, the negative symbol isn't appended to the b; isn't this what I am trying to prove, that the negative symbol is appended to the a?

Are those the actual steps listed in the text? The next-to-last line looks a bit suspect. It seems like it should be

##(-a)\cdot b+a\cdot b=(-a+a)\cdot b## by distributivity of multiplication over addition
##(-a)\cdot b+a\cdot b=0\cdot b## by definition of additive inverses
##(-a)\cdot b+a\cdot b=0## from results related to the definition of the additive identity and distributivity of multiplication over addition
##[(-a)\cdot b+a\cdot b]+(-(a\cdot b))=0+(-(a\cdot b))## because addition is well-defined as a binary operation (?)
##[(-a)\cdot b+a\cdot b]+(-(a\cdot b))=-(a\cdot b)## by definition of additive identity
##(-a)\cdot b+[a\cdot b+(-(a\cdot b))]=-(a\cdot b)## by associativity of addition
##(-a)\cdot b+0=-(a\cdot b)## by definition of additive inverse
##(-a)\cdot b=-(a\cdot b)## by definition of additive identity

It's possible that some of those steps were left out, but that's what is going on as far as I can tell.
 
  • #3
Oh, yes, I see now. Thank you very much.
 

1. What is Property 9 of Negative Numbers in Calculus?

Property 9 of Negative Numbers in Calculus states that when subtracting a positive number from a negative number, the result is equal to adding the absolute value of the positive number to the negative number.

2. Why is Property 9 important to understand in Calculus?

Property 9 is important because it allows us to easily solve equations and inequalities involving negative numbers. It also helps us understand the behavior of negative numbers in relation to positive numbers in mathematical operations.

3. How does Property 9 relate to the concept of absolute value?

Property 9 is closely related to the concept of absolute value because it involves finding the distance between a negative number and a positive number on a number line. By adding the absolute value of a positive number to a negative number, we are essentially finding the distance between them.

4. Can you provide an example of Property 9 in action?

Sure, if we have the equation -5 - 3, we can rewrite it as -5 + (-3), which according to Property 9, is equal to -5 + 3 = -2. This means that subtracting a positive number (3) from a negative number (-5) is the same as adding the absolute value of the positive number (3) to the negative number (-5).

5. How can understanding Property 9 help in real-life situations?

Property 9 can help in real-life situations such as calculating change when making purchases or understanding temperature changes. For example, if the temperature drops by 5 degrees and then increases by 3 degrees, we can use Property 9 to find the overall change as -5 + 3 = -2 degrees.

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