If you take the number 7 out of the real number line, then

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In summary, the conversation discusses the concept of 7 and its relation to infinitesimals and the set [7,7]. It is mentioned that if 7 is removed from the number line, it leaves the real numbers with a point removed, which has zero length. The set [7,7] is identified as the best choice for representing 7, but the concept of "infinitesimal" needs to be defined before any further questions can be answered. The conversation ends with a comparison between [7,7] and the number 7, and how they may or may not share similar properties.
  • #1
student34
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what can be said about 7? Is it like an infinitesimal? Is it smaller or bigger? What is it? Is it just [7,7]?
 
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  • #2
Your question is too vague. Explain what you really have in mind.
 
  • #3
student34 said:
what can be said about 7? Is it like an infinitesimal? Is it smaller or bigger? What is it? Is it just [7,7]?
If you remove 7 from the number line, then you get the real numbers with a point removed. A point has zero length.
 
  • #4
Of the choices given, "[7, 7]", which is exactly the same as the set notation "{7}" is best. It is a single point. You would have to define exactly what you mean by "infinitesimal" in this context before anyone could answer your other questions.
 
  • #5
mathman said:
Your question is too vague. Explain what you really have in mind.

If I take out the number 7, then it seems as though I have something very small but perhaps larger than an infinitesimal like 5/n→∞. Because, we know that 5/n→∞ = 0, but 7 = 7, not 0.
 
  • #6
HallsofIvy said:
Of the choices given, "[7, 7]", which is exactly the same as the set notation "{7}" is best. It is a single point. You would have to define exactly what you mean by "infinitesimal" in this context before anyone could answer your other questions.

The "infinitesimal" that I was thinking of is something like 3/n→∞.
 
  • #7
student34 said:
3/n→∞.

What you've written doesn't make any sense. Did you mean something like
[tex] \lim_{n\to \infty} \frac{3}{n} = 0 [/tex]?

It's not clear to me why you think this is related to the number 7. The length of the interval [7,7] is completely unrelated to the number 7 itself
 
  • #8
Office_Shredder said:
What you've written doesn't make any sense. Did you mean something like
[tex] \lim_{n\to \infty} \frac{3}{n} = 0 [/tex]?

Yes, I just don't know how to use the proper notation on this forum.

It's not clear to me why you think this is related to the number 7. The length of the interval [7,7] is completely unrelated to the number 7 itself

Doesn't [7,7] and 7 have similar properties?
 
  • #9
student34 said:
Doesn't [7,7] and 7 have similar properties?

Does [tex] \mathbb{R} [/tex] and [tex] \{ \mathbb{R} \} [/tex] have similar properties? The first being the set of real numbers, and the second being a set which contains a single element, namely the set of real numbers. The answer is no, not in general. Another good example is ∅, the empty set, and {∅}, the set containing only the empty set (in particular it is not empty!)

Any claims that [7,7] (which is the set containing only the number 7, i.e. {7} as has been mentioned above) and 7 the number share similar properties should be clarified in detail: what properties might these share?
 

1. What happens to the rest of the numbers on the real number line if you take out 7?

If you take out 7 from the real number line, it creates a gap or hole in the line. This means that there is now a discontinuity in the number line at that point. The numbers on either side of the gap are still considered real numbers, but the number 7 itself is no longer included in the set of real numbers.

2. Can you still perform mathematical operations on the numbers on either side of the gap?

Yes, you can still perform mathematical operations on the numbers on either side of the gap. The gap created by removing 7 does not affect the properties of the real numbers, so basic operations such as addition, subtraction, multiplication, and division can still be performed.

3. How does removing 7 affect the order of the numbers on the real number line?

Removing 7 does not affect the order of the numbers on the real number line. The numbers on either side of the gap are still ordered from least to greatest, and the gap itself does not change the relative positions of the numbers.

4. Is the gap created by removing 7 considered a real number?

No, the gap created by removing 7 is not considered a real number. It is simply an empty point on the number line where the number 7 used to be. Real numbers are defined as all rational and irrational numbers, and the gap does not fit this definition.

5. How does removing 7 affect the properties of the real numbers?

Removing 7 does not affect the properties of the real numbers. The real numbers still maintain their fundamental properties, such as closure under addition and multiplication, commutativity, associativity, and distributivity. The gap created by removing 7 does not change these properties.

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