Does 1 have a degree of 1 or 0?

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Discussion Overview

The discussion revolves around the concept of the degree of the number 1 and the degree of the zero polynomial, exploring whether 1 has a degree of 1 or 0, and the implications of these definitions in polynomial terms.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that 1 is a degree zero polynomial, arguing that a degree one polynomial would require a variable term.
  • Others question the relevance of the degree of 1, suggesting that it may not matter in certain contexts.
  • A participant mentions that for every real r, 1 can be expressed as 1^r, implying a broader interpretation of its degree.
  • There is a discussion about the degree of the zero polynomial, with some stating it is considered -∞ to maintain certain mathematical rules.
  • One participant expresses confusion about the concept of a variable, indicating a limited understanding beyond high school mathematics.
  • Another participant clarifies that constants are polynomials of degree 0, challenging a previous claim about the degree of 0 being -∞.
  • Some participants note that while constants like 1 have a degree of 0, the zero polynomial is treated differently in terms of degree.

Areas of Agreement / Disagreement

Participants express differing views on the degree of 1 and the zero polynomial, with no consensus reached on whether the degree of 0 should be considered -∞ or if constants are strictly degree 0 polynomials.

Contextual Notes

There are unresolved assumptions regarding the definitions of polynomial degrees and the treatment of the zero polynomial, which may affect the interpretations presented.

student34
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Or is it both?
 
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Degree as what? A polynomial?
 
Does it matter?

For every real r, we have 1=1^r
 
Office_Shredder said:
Degree as what? A polynomial?

yes, as a polynomial
 
arildno said:
Does it matter?

For every real r, we have 1=1^r

oh yeah
 
It's a degree zero polynomial - if it was degree one it would have a variable term.
 
a*z^0 is a zero'th degree monomial in z, a first degree monomial in "a".
 
Office_Shredder said:
It's a degree zero polynomial - if it was degree one it would have a variable term.

Oh, so even though 5 has a power of 1, is it still considered a degree of 0?
 
arildno said:
a*z^0 is a zero'th degree monomial in z, a first degree monomial in "a".

Ok, but what degree polynomial is 0 then?
 
  • #10
Do you understand the concept of a variable?
 
  • #11
arildno said:
Do you understand the concept of a variable?

I have grade 12 algebra and grade 12 calculus, but any meaning of a variable beyond those courses, I am not sure.

I found an answer to the degree of 0; apparently it's -∞, !?
 
  • #12
"I found an answer to the degree of 0; apparently it's -∞, !?"

No, you haven't. Constants are polynomials of degree 0.

What do you mean ``12 grade algebra and 12 grade calculus''?
 
  • #13
statdad said:
"I found an answer to the degree of 0; apparently it's -∞, !?"

No, you haven't. Constants are polynomials of degree 0.

I found it in my notes from my first year math course in university.

What do you mean ``12 grade algebra and 12 grade calculus''?

You have "12" and "grade" switched around.
 
  • #14
statdad said:
"I found an answer to the degree of 0; apparently it's -∞, !?"

No, you haven't. Constants are polynomials of degree 0.
Some people do consider the degree of the zero polynomial to be -∞, so as to preserve rules like deg fg = deg f + deg g.
 
  • #15
The degree of a polynomial, in variable x, is the highest power of x. We can write "1" as "[itex]1x^0[/itex]" so "degree 0". The reason for the distinction between the '0' polynomial (degree [itex]-\infty[/itex]) and the '1' (or any non-zero number) polynomial (degree 0) is that we could, theoretically, write 0 as "[itex]0x^n[/itex]" for any n.
 

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