That ought to read something like: Does this polynomial have 7 roots, counting multiplicity?(x-3)^4( x-4)(x-5)(x-8)^2
does multiplicity = 7?
Is that a typo? I've always thought that simple roots have multiplicity 1.x = 4 with multiplicity 0
I'm trying to understand what multiplicity means in this context above. In other words lets say hypothetically I added another value (.7) to this multi-set
{0, 0, 0, 0.3 0.5, 0.5, 0.6, 0.7, 1, 1, 1} .. would this still have a multiplicity of 9?
thanks
Yes, that is one way to interpret it.
English Language Argument: "Multiplicity" necessarily refers to a plurality, not to a unity. "multiplicity of 1" is not a
meaningful expression. A single occurrence has "no multiplicity". If you would like to count that as "1", feel free.
Just be very clear what you want your students to put on an exam.
Like I said, opinions vary.
It is true that some texts and discussions (using a polynomial as an example) define "multiplicity" as "How many times does the factor or zero appear?"
About those factors and zeros, it I clear, but you're going with a falsehood by claiming "multiplicity"to mean something else.
This is what lookagain defined, nicely. However, I always have found this definition confusing, for the two reasons I stated above. I prefer to refer to
"degree" or "order". These are much more clearly and specifically defined.
One more time to emphasize ambiguity.
In lookagain's polynomial example, these two things can be said.
1) There are 8 zeros, counting multiplicities.
2) Ignoring multiplicities, we see 4 distinct zeros.
In #1, we count the linear factors as having multiplicity = 1. This gets us to 8 zeros. It does not get us to multiplicity = 8 for the entire polynomial. That still doesn't mean anything.
In #2, we are ignoring multiplicity, so we cannot be counting multiplicity = 1. There is no multiplicity. We're ignoring it! Yet, we still get to 4 distinct zeros. We must be counting something other than multiplicity.
Is that a typo? I've always thought that simple roots have multiplicity 1.
There are truths (read: rigid facts) in portions of mathematics. There are variations and inconsistencies in the definitions of many mathematical terms.There is no "truth" in mathematics.
mmm4444bot said:Math has a lot of variability, both in the way it's taught and used:
"zero is either positive or negative"
versus
"zero is neither positive nor negative"
e^x = y implies that ln(y) = x
versus
e^x = y implies that log(y) = x
0^0 is undefined
versus
0^0 = 1
There are truths (read: rigid facts) in portions of mathematics.
There are variations and inconsistencies in the definitions of many mathematical terms.
Second example: Those logarithm words are different, but they are still are used for the same base.
Third example: 0^0 is undefined.