# I Product with an undefined term

#### mathman

Summary
product with undefined term
Let c=ab. Let b be undefined (but finite). If a=0, is c undefined or does c=0?

#### jedishrfu

Mentor
Since B is finite and we assume B is element of the Reals and A is 0 then C must be 0.

"0 times any real number = 0" is a property of 0.

#### Math_QED

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What does $b$ finite but undefined even mean??

#### fresh_42

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2018 Award
Or what does c=ab mean? For the conclusion 0b=0 we need the distributive law and a loop for a.

#### WWGD

Gold Member
Summary: product with undefined term

Let c=ab. Let b be undefined (but finite). If a=0, is c undefined or does c=0?
It is 0, afaik. Using limits, let $\epsilon \rightarrow 0$ assume a is not really zero but very small and you can make the product very small thanks to the bound on c: $ab < Max(b) \frac {\epsilon}{ Max(b)}$ for all $\epsilon$.

#### fresh_42

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2018 Award
It is 0, afaik.
Not necessarily. Even if we assume $0$ as neutral element of addition, and the unwritten operation as a multiplication, it is not clear if we have the distributive law, or another combination of the two operations. Furthermore, we need an additive inverse and an element $a\neq 0$. If we lack any of these, then I don't know a valid proof for $0\cdot b=0$.

#### Stephen Tashi

What does $b$ finite but undefined even mean??
Perhaps it means that the vlerth level of b is less than 16, but we don't know what a vlerth level is.

• Math_QED

#### WWGD

Gold Member
Not necessarily. Even if we assume $0$ as neutral element of addition, and the unwritten operation as a multiplication, it is not clear if we have the distributive law, or another combination of the two operations. Furthermore, we need an additive inverse and an element $a\neq 0$. If we lack any of these, then I don't know a valid proof for $0\cdot b=0$.
See my previous. This is used e.g. to conclude that lim h-->0 hSin(1/h)=0 . The expression is bounded above by $Max(a)\frac{\epsilon}{Max(a)}$ for all $\epsilon$.

#### fresh_42

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2018 Award
See my previous. This is used e.g. to conclude that lim h-->0 hSin(1/h)=0 . The expression is bounded above by $Max(a)\frac{\epsilon}{Max(a)}$ for all $\epsilon$.
Yes, but you already make a hidden assumption, namely that we are talking about ordinary numbers.

#### WWGD

Gold Member
Yes, but you already make a hidden assumption, namely that we are talking about ordinary numbers.
How are they not ordinary numbers? Arent they all assumed to be Real numbers? Whatever value the bounded undefined takes, the product will be bounded above by epsilon.

#### fresh_42

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2018 Award
How are they not ordinary numbers? Arent they all assumed to be Real numbers?
Who knows? I always think in terms of algebras, in which multiplication is not a natural given extension of school counting. But I admit that distribution is one of the very basic and here crucial properties, which is usually assumed.

• jedishrfu

#### WWGD

Gold Member
Who knows? I always think in terms of algebras, in which multiplication is not a natural given extension of school counting.
Fair enough, I was assuming numbers were Real. If not, it is a whole different situation. When we talk about bounded I guess we assume a modulus or valuation.

#### WWGD

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But in what setting outside of the Reals does a question like this make sense? Maybe @mathman can clarify the setting?

#### fresh_42

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2018 Award
But in what setting outside of the Reals does a question like this make sense? Maybe @mathman can clarify the setting?
I have found another proof, which only uses the distributive law, and characteristic not two. Again, everything depends on the distributive law. Given that we can conclude with $a-a=0$, either for an $a\neq 0$ or for $a=0$ with $\operatorname{char} \neq 2$.

However, if distribution is not given, then everything is possible.

#### WWGD

Gold Member
But there was a notion of boundedness used, so there is a norm or valuation too.

#### Mark44

Mentor
Let c=ab. Let b be undefined (but finite). If a=0, is c undefined or does c=0?
Instead of making assumptions about what is being asked here, why don't we hold off until @mathman returns to clarify what sorts of things a, b, and c are?
Clearly if a, b, and c are members of the real or complex field, the question is trivial and undeserving of an 'A' tag.

• jedishrfu

#### mathman

Clarification: I was thinking about expressions like xsin(1/x) for x=0. Real numbers and ordinary multiplication.

#### WWGD

Gold Member
Clarification: I was thinking about expressions like xsin(1/x) for x=0. Real numbers and ordinary multiplication.
Then just let x approach 0 as $\epsilon/(Max(Sin(1/x))= \epsilon/1$ so that $xsin(1/x) < \epsilon \cdot 1 = \epsilon$ for all $\epsilon$

#### mathman

I am well aware that the limitng process gives 0. However there seems to be a school of thought which asserts that sin(1/0) being undefined, multiplying by 0 results in an undefined product.

An interest variation is about $f(x)=x^2sin(1/x)$, with $f'(x)=2xsin(1/x)-cos(1/x)$ for $x\ne 0$, which is undefined as $x\to 0$, but $f'(0)=0$

#### jedishrfu

Mentor
Computer math follows that trend when some factor isn't computable then a NaN is returned and this taints the rest of the calculation resulting in a NaN as the final answer.

#### WWGD

Gold Member
Then just let x approach 0 as $\epsilon/(Max(Sin(1/x))= \epsilon/1$ so that $xsin(1/x) < \epsilon \cdot 1 = \epsilon$ for all $\epsilon$
Then the operation of multiplication is undefined. Axioms/rules only stipulate multiplication within the Real numbers.

#### Stephen Tashi

An interest variation is about $f(x)=x^2sin(1/x)$, with $f'(x)=2xsin(1/x)-cos(1/x)$ for $x\ne 0$, which is undefined as $x\to 0$, but $f'(0)=0$
f'(x) is defined as $\lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$, so $f'(x)$ doesn't exist at a value $x$ where $f(x)$ doesn't exist. So I assume you are defining $f(0)$ to be $\lim_{x \rightarrow 0}x^2 sin(1/x) = 0$.

I don't understand how this example illustrates the question in your original post.

#### Mark44

Mentor
However there seems to be a school of thought which asserts that sin(1/0) being undefined, multiplying by 0 results in an undefined product.
Yes, since 1/0 is undefined, then so is sin(1/0), so the product 0 * sin(1/0) is undefined.
Let c=ab. Let b be undefined (but finite). If a=0, is c undefined or does c=0?
You can't even say that b (= sin(1/0) here) is finite.

However, you can work with limits to establish that $\lim_{x \to 0} x \cdot \sin(\frac 1 x)$ exists and is 0, which is easy to show without resorting to the limit definition. I.e., $\forall x \gt 0, -x \le x \cdot \sin(\frac 1 x) \le x$. Since both bounds approach 0 as x approaches 0, so must the expression in the middle.

#### Mark44

Mentor
Computer math follows that trend when some factor isn't computable then a NaN is returned and this taints the rest of the calculation resulting in a NaN as the final answer.
But computer floating point math isn't relevant here because of the inherent limitations of representing very small or very large numbers in a finite number of bits, particularly floating point numbers as described by IEEE-754, which describes different situations in which NANs (both signalling and nonsignalling) are produced. I'm aware that there are libraries that support larger numbers and higher precision than the 2-, 4-, 8, or 10-byte floating point numbers described in IEEE-754, but I'm not familiar with the limitations of those libraries. In any case, if these libraries store numbers have less than infinite precision, and signal events such as underflow and the like, then they too have limitations.

• jedishrfu

#### jedishrfu

Mentor
At the time, I felt the computer math rule of returning a NaN was useful in explaining how an inner expression that’s undefined can taint the final result making it undefined not that computer math was needed to answer the question.

My apologies for sidetracking the thread.