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PrathameshR
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What's the difference between something (eg a function or a matrix) becoming zero and it becoming identically zero?
Illustrations will be helpful. Thanks
Illustrations will be helpful. Thanks
This really helped. ThanksI like Serena said:Hi PrathameshR ;)
There is no real mathematical distinction.
Identical zero is merely an emphasis to indicate it's 'more' zero than might otherwise be thought.
When we say that a function is identical to zero, we want to emphasize that we really mean the zero-function, which is zero everywhere in its domain.
Saying that a function is zero should mean the same thing (that it's the zero-function), but some authors are a bit sloppy, and they might mean that the function just becomes zero for a certain value in its domain.
Same thing for a matrix.
A matrix that is zero means the same thing as a matrix that is identical to zero - it's the matrix with only zeroes.
That is as opposed to a matrix that has a zero.
I'm going to disagree slightly with this.I like Serena said:There is no real mathematical distinction.
Identical zero is merely an emphasis to indicate it's 'more' zero than might otherwise be thought.
Mark44 said:I'm going to disagree slightly with this.
For example the equation ##x^2 - 2x + 1 = 0## is a true statement only when x = 1. OTOH, the equation ##x^2 - 2x + 1 - (x - 1)^2 = 0## is true for all real values of x. The expression ##x^2 - 2x + 1 - (x - 1)^2## is identically zero.
I didn't say anything about functions, and explicitly wrote "the expression ##x^2 - 2x + 1 - (x - 1)^2##.I like Serena said:This is indeed exactly where the ambiguity occurs.
When we write ##x^2 - 2x + 1 - (x - 1)^2##, it's somewhat ambiguous if we're talking about a specific function value, or about the function in general.
What I wrote was not in the context of functions, but if you treat it as such, with ##f(x) = x^2 - 2x + 1 - (x - 1)^2##, then this is a function that is identically zero. I.e., ##f(x) \equiv 0##. That was not my intent, though. Instead, I was distinguishing between an expression that is zero for a particular value of the variable (conditional equality) versus another one that was identically zero.I like Serena said:Literally speaking, the expression ##x^2 - 2x + 1 - (x - 1)^2## is not a function - it's a specific function value.
I like Serena said:It's just that it's not uncommon that an author intends the corresponding function.
So saying ##x^2 - 2x + 1 - (x - 1)^2## is identical to zero is intended to mean that ##x\mapsto x^2 - 2x + 1 - (x - 1)^2## is zero.
Or alternatively that the function given by ##f(x)=x^2 - 2x + 1 - (x - 1)^2## is zero.
The word identical here is used to disambiguate, although it's not really what the word identical means (mathematically it just means the same thing as equal to).
Note that 'zero' here is ambiguous as well, since it's not clear whether it's a function value that is zero, or the function itself that is the zero-function.
Mark44 said:I didn't say anything about functions, and explicitly wrote "the expression ##x^2 - 2x + 1 - (x - 1)^2##.
What I wrote was not in the context of functions, but if you treat it as such, with ##f(x) = x^2 - 2x + 1 - (x - 1)^2##, then this is a function that is identically zero. I.e., ##f(x) \equiv 0##. That was not my intent, though. Instead, I was distinguishing between an expression that is zero for a particular value of the variable (conditional equality) versus another one that was identically zero.
I don't agree that the concept of functions necessarily needs to be part of such a discussion. One can write ##\forall x \in \mathbb R, \sin^2(x) + \cos^2(x) - 1 = 0## without either implicitly or explicitly stating that the left side is a function. I'm not saying it's wrong to do so, just that it's not necessary.I like Serena said:Whether we use the word 'function' or not, when we talk about an expression to be identical to zero, we mean the function that assigns a value according to the given expression to each value that is in its domain instead of the actual expression.
I like Serena said:(And without specification of the domain, it is assumed to be ##\mathbb R## minus any points for which the expression is not defined.)
Mark44 said:I don't agree that the concept of functions necessarily needs to be part of such a discussion. One can write ##\forall x \in \mathbb R, \sin^2(x) + \cos^2(x) - 1 = 0## without either implicitly or explicitly stating that the left side is a function. I'm not saying it's wrong to do so, just that it's not necessary.
I agree completely. There is no need to bring in functions, fields, equivalence operators, or other concepts from advanced algebra.FactChecker said:"identically zero" means that for all legitimate values of variables, the result is zero. That is quite different from saying that something equals zero for some particular values of variables.
The number zero is a numerical value that represents the absence of quantity or amount. It is neither positive nor negative and is placed at the center of the number line.
Identically zero refers to a mathematical expression or function that always results in a value of zero, regardless of the input value. It is a special case of zero where there is no variation or difference.
The main difference between zero and identically zero is that zero is a specific numerical value, while identically zero is a concept that describes a function or expression. Zero can have different representations, such as 0, 0.00, or 0/1, while identically zero remains the same regardless of the form it is written in.
For example, the function f(x) = 0 is always equal to zero, regardless of the value of x. This is identically zero. However, the function g(x) = 1/x becomes zero when x approaches infinity, but it is not identically zero as it can have different values for different inputs.
Understanding the difference between zero and identically zero is crucial in mathematics because it can lead to different interpretations and solutions. It is especially important in calculus and analysis, where small differences can have significant implications. It also helps in identifying and solving mathematical equations and functions accurately.