infinity divided by any number

No - any number we pick will be infinitely far away from $\infty$. As we cannot guess the exact number, we consider it as a length of a number or infinity. Infinity divided by any finite number is infinity. Its still infinity. It all depends on the defintions of $\infty$ , "number" and "divides". As we cannot guess the exact number, we consider it as a length of a number or infinity. infinity, the concept of something that is unlimited, endless, without bound. Enjoy and manage TV, high-speed Internet, phone, and home security services that work seamlessly together anytime, anywhere, on any device. Which Teeth Are Normally Considered Anodontia. You will find people all around you who will say infinity is not a number and they will stop there, but there are a set of math classes in your future (Calculus) that are all about dealing with numbers that are virtually infinite--they call them limits. As such, the expression 1/infinity is actually undefined. Thinking and playing with infinity is a popular thing to do, especially for people your age. To do so correctly however, you need to understand wh You can divide that by any number you want, other than zero, and the result will still be zero. Any number divided by infinity is equal to 0. Sometimes on Family Guy when there about to take someones heart out they say, calimar or maybe its spelled different. _____ 1. Any finite number divided by infinity is a number infinitesimally larger than, but never equal It is really great to be thinking of stuff like this. (An infinite set can be put in one-to-one correspondence with a proper subset of itself - we can match the integers with the even integers by pairing $n$ with $2n$). Beginning in the 6th century BC with the Pythagoreans, with Greek mathematics the Ancient Greeks began a systematic study of mathematics as a subject in its own right. __CONFIG_colors_palette__{"active_palette":0,"config":{"colors":{"62a54":{"name":"Main Accent","parent":-1}},"gradients":[]},"palettes":[{"name":"Default Palette","value":{"colors":{"62a54":{"val":"var(--tcb-skin-color-0)"}},"gradients":[]},"original":{"colors":{"62a54":{"val":"rgb(19, 114, 211)","hsl":{"h":210,"s":0.83,"l":0.45,"a":1}}},"gradients":[]}}]}__CONFIG_colors_palette__, {"email":"Email address invalid","url":"Website address invalid","required":"Required field missing"}, __CONFIG_colors_palette__{"active_palette":0,"config":{"colors":{"f3080":{"name":"Main Accent","parent":-1},"f2bba":{"name":"Main Light 10","parent":"f3080"},"trewq":{"name":"Main Light 30","parent":"f3080"},"poiuy":{"name":"Main Light 80","parent":"f3080"},"f83d7":{"name":"Main Light 80","parent":"f3080"},"frty6":{"name":"Main Light 45","parent":"f3080"},"flktr":{"name":"Main Light 80","parent":"f3080"}},"gradients":[]},"palettes":[{"name":"Default","value":{"colors":{"f3080":{"val":"rgba(23, 23, 22, 0.7)"},"f2bba":{"val":"rgba(23, 23, 22, 0.5)","hsl_parent_dependency":{"h":60,"l":0.09,"s":0.02}},"trewq":{"val":"rgba(23, 23, 22, 0.7)","hsl_parent_dependency":{"h":60,"l":0.09,"s":0.02}},"poiuy":{"val":"rgba(23, 23, 22, 0.35)","hsl_parent_dependency":{"h":60,"l":0.09,"s":0.02}},"f83d7":{"val":"rgba(23, 23, 22, 0.4)","hsl_parent_dependency":{"h":60,"l":0.09,"s":0.02}},"frty6":{"val":"rgba(23, 23, 22, 0.2)","hsl_parent_dependency":{"h":60,"l":0.09,"s":0.02}},"flktr":{"val":"rgba(23, 23, 22, 0.8)","hsl_parent_dependency":{"h":60,"l":0.09,"s":0.02}}},"gradients":[]},"original":{"colors":{"f3080":{"val":"rgb(23, 23, 22)","hsl":{"h":60,"s":0.02,"l":0.09}},"f2bba":{"val":"rgba(23, 23, 22, 0.5)","hsl_parent_dependency":{"h":60,"s":0.02,"l":0.09,"a":0.5}},"trewq":{"val":"rgba(23, 23, 22, 0.7)","hsl_parent_dependency":{"h":60,"s":0.02,"l":0.09,"a":0.7}},"poiuy":{"val":"rgba(23, 23, 22, 0.35)","hsl_parent_dependency":{"h":60,"s":0.02,"l":0.09,"a":0.35}},"f83d7":{"val":"rgba(23, 23, 22, 0.4)","hsl_parent_dependency":{"h":60,"s":0.02,"l":0.09,"a":0.4}},"frty6":{"val":"rgba(23, 23, 22, 0.2)","hsl_parent_dependency":{"h":60,"s":0.02,"l":0.09,"a":0.2}},"flktr":{"val":"rgba(23, 23, 22, 0.8)","hsl_parent_dependency":{"h":60,"s":0.02,"l":0.09,"a":0.8}}},"gradients":[]}}]}__CONFIG_colors_palette__. Similarly, since:\begin{align*}\dfrac{0}{\infty}&=0\cdot\left(\dfrac{1}{\infty}\right)\\-\dfrac{10}{\infty}&=-10\cdot\left(\dfrac{1}{\infty}\right),\end{align*}then we get that both $0/\infty$ and $-10/\infty$ are equal also to zero. There is no largest "real number" and so there is no "highest (real) number" that $11$ can be divided by. How could magic slowly be destroying the world? Ans: Dividing any number by zero does not make sense, because in maths, dividing by zero can be interpreted as multiplying by zero. He was in the service of King Hiero II of Syracuse. Raju told Shyam that any number divided by zero is undefined. Explanation: Zero divided by any number is always 0. I); 3. However, if we will use the definition of a limit of a function and evaluate the function $\frac{1}{x}$, where $x$ becomes larger and larger, we will see that the function $\frac{1}{x}$ approaches a particular number. Home | About | Contact | Copyright | Report Content | Privacy | Cookie Policy | Terms & Conditions | Sitemap. Infinity is a never ending quantity - and Answer: 5 to the power of 5 can be expressed as 55 = 5 5 5 5 5 = 3,125. Theres no solution, so any non-zero number divided by 0 is undefined. Kill the fear of Maths. Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. Regardless of what large number were dividing by our answer is 0 and by letting this large Remember that It is really great to be thinking of stuff like this. You will find people all around you who will say infinity is not a number and they will sto However, we can find a way to target this problem that is valid and acceptable. Mathematicians (except in very, informal discussions) never speak of this number equals infinity. They say this number approaches infinity, meaniung lets look at what happens when this number gets bigger than any number we can possibly think of., So when you speak of infinity divided by a real number you are still talking of a number thats bigger than any number you could name. The cookie is used to store the user consent for the cookies in the category "Other. In mathematics, a limit of a function occurs when x gets larger and larger as it approaches infinity, and 1/x gets smaller and smaller as it approaches zero. Whereas a number represents a specific quantity, infinity does not define given quantity. Any number divided by infinity is equal to 0. This cookie is set by GDPR Cookie Consent plugin. In ordinary arithmetic, this does not have a well-defined meaning, since infinity is a mathematical Rahul says to Shyam that any number divided by zero is undefined. But how can we be closer to $\infty$? In particular, /4=. Solving $1/\infty$ is the same as solving for the limit of $1/x$ as $x$ approaches infinity, so using the definition of limit, 1 divided by infinity is equal to $0$. The first modern equivalent of numeral zero comes from a Hindu astronomer and mathematician Brahmagupta in 628. I); 2. By clicking Accept All, you consent to the use of ALL the cookies. , , 0 0, 0, and 1 that a mathematical function may assume by formal substitution. In normal cases, the value of Since we can never locate the largest real number, we use infinity instead to represent these very large numbers. Let's assume (falsely) that at some point, our $x_n$ get close to a limit $L=\infty$. any other positive number (I / I = and so on); 4. [I am always amazed when several people give very long answers to a question that they do not "like." As a result, we reached a dead end. When the Degree of the function is: greater than 0, the limit is infinity (or infinity) less than 0, the limit is 0. The expression $1^{\infty}$ is considered one of the indeterminate forms, meaning that it will have different answers depending on which situation it was used. If nothing is shared, then no one will get any toffees. Infinity CAN be divided or multiplied. This is a contradiction because 1 will never be equal to 2. Our online coding, design, chess and math courses are designed to suit kids' learning pace. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. If you divide any number by infinity then answer will be zero. In the case where the numerator is infinity and the denominator is a real number, say $c$, then\begin{align*}\dfrac{\infty}{c}=\infty.\end{align*}. Infinity +1 is still an infinity. This kind of thing is calculated by slicing up curves into infinitely (Approaching infinite anyway) thin slices and adding the slices together to get an exact result. If you divide a number by an undefined value, there is no way to find the result since you divide by an undefined value. The answer will still be infinity You see when we say infinite we actually mean that the number is indefiniteor in other words; we have no idea Since infinity divided by infinity is definitely not equal to zero, we can answer right INFINITI | Luxury SUVs, Crossovers, Sedans, and Coupes All-New QX60 Take on Life in Style Proudly introducing our completely reimagined three-row SUV. But the limit of the expression 1/x as x tends to zero is infinity. Infinity divided by any number is still infinity. Therefore, 5 divided by infinity equal to zero. Thinking and playing with infinity is a popular thing to do, especially for people your age. undefined plus infinity is infinity (I + I = I); 9. Science Advisor. Archimedes is considered the father of mathematics because of his notable inventions in mathematics and science. So, 0/n = 0 and n/0= infinite. It's related to a question that came to my mind, namely which is the highest number you can divide 11 by. _The point where the two axes intersect is called the origin. Its a shorthand expression for the limiting, case of a number that is bigger than any number you can name. minus a finite number is infinity (I - f = I); but 12. We say all of these things are just undefined.. But opting out of some of these cookies may affect your browsing experience. These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc. In mathematics, expressions like 1/0 are undefined. Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. The online classes for kids at CodingHero help your child develop skills, not only in math and science but also in critical life skills like problem-solving, critical thinking, communication, organization, and planning. Most people have some conception of things that have no bound, no boundary, no limit, no end. $$11\equiv 0\mod{x}$$ Advertisement cookies are used to provide visitors with relevant ads and marketing campaigns. Theres no number that you can multiply by zero to get a non-zero number. Why is IVF not recommended for women over 42? Is $10$ closer to $\infty$ than $100$ is? What is the difference between mango plants and maize plants in terms of root system? 4 Answers aaja Come. Also, in some calculators such as the TI-Nspire, 1 divided by infinity can be evaluated as 0. Now, let us divide 5 by infinity. This website uses cookies to improve your experience while you navigate through the website. Thus, we have:\begin{align*}\dfrac{2}{\infty}=2\cdot0=0.\end{align*}Therefore, $2/\infty$ is also zero. However, if we will use the definition of a limit of a function and evaluate the function $1/x$, where $x$ becomes larger and larger, we will see that the function $1/x$ approaches a particular number. It does not store any personal data. Julia got by = (that is, the product of all prime numbers). How can I calculate this limit at infinity? But the answer would The cookies is used to store the user consent for the cookies in the category "Necessary". However, when they have dealt with it, it was just a symbol used to represent a really, really large positive or really, really large negative number and that $10^{100000000000000000}$ is a very large number, still, we can find another larger number than this, we just need to add 1 to it, and we already have one. (If you started counting really fast for billions of years, you would never get to infinity.) If you add one to infinity, you still have infinity; you dont have a bigger number. At first, you may think that infinity divided by infinity equals one. The online classes offered by CodingHero helping the kids learn: Copyright 2023 GoalPath Education Private Ltd, all rights reserved. @Bill_K Fwiw, my previous comment and downvote weren't intended as a violent response, simply an opinion of the pedagogical value of your answer OP's question. You can see this with limits and some calculus rules. In fundamental operations, we know that any number, except 0, when divided by itself, equals one. Answer: The answer is as follows Step-by-step explanation: Whatever 'infinity divided by 2' should mean, it should be some which, when multiplied by , gives 'infinity'. Spanish Help If you were working within the rational numbers (fractions), you could divide by any rational number (including any integer) except for zero. However, when any number be it infinity when divided by zero will give answer as infinity. EMMY NOMINATIONS 2022: Outstanding Limited Or Anthology Series, EMMY NOMINATIONS 2022: Outstanding Lead Actress In A Comedy Series, EMMY NOMINATIONS 2022: Outstanding Supporting Actor In A Comedy Series, EMMY NOMINATIONS 2022: Outstanding Lead Actress In A Limited Or Anthology Series Or Movie, EMMY NOMINATIONS 2022: Outstanding Lead Actor In A Limited Or Anthology Series Or Movie. This website uses cookies to improve your experience while you navigate through the website. So, in this case, there are no items to be shared, hence, no one will get any item. The addition of $2$ will not change the result of this equation. Combined with geometric insights we can extend familiar numbers/systems to obtain projective spaces or the Riemann Sphere, both of which accommodate notions of infinity. Infinity divided by a finite number is infinite (I / f = As x approaches 0 from the left, y approaches negative infinity. Infinity over infinity is Undefined. divided by infinity is zero. We also use third-party cookies that help us analyze and understand how you use this website. What is Probability Definition, Terminologies, Uses & Examples, Altitude of a Triangle(Definition & Properties), SouthGeorgia&SouthSandwichIslands 500. Required fields are marked *. Such a division can be We assume that $\infty/\infty=1$. If the question was asked "does there exist a largest number that $11$ can be divided by" without any additional clarification about context then the answer would be no. Infinity divided by infinity is undefined, meaning it is not equal to any real number. It depends on the problem. Given a finite sequence, can we always find a relation that generates that sequence? Let $a_n$ be a sequence, let $b_n=(a_1+a_2++a_n)*1/n$. In other words, $11$ divided by $x$ leaves no remainder or fractional part, then the solution would be $x=11$. Let us Julia thinks is evenly divisible by everything. You need infinite numbers to talk about and compare amounts that are unending, but some unending amounts some infinities are literally bigger than others. Infinity is more a concept than a number. Therefore, 5 divided by infinity equal to zero. Can you drive a forklift if you have been banned from driving? If infinity divided by any number is still infinity, then perhaps infinity is a prime number after all. Is it true that zero divided by any number is zero? The Latest Innovations That Are Driving The Vehicle Industry Forward. In mathematics, division by infinity is division where the divisor (denominator) is infinity. First published Thu Apr 29, 2021. In mathematics, they use infinity to represent the largest number among the set of real numbers, which we know does not exist. Zero divided by any number is always 0. The extended real numbers however is not the usual context to be working in. Thanks Advertisement Advertisement ananyabadjena ananyabadjena Answer: the answer is not known yet Therefore 0 is an even number. 2022 - 2023 Times Mojo - All Rights Reserved Lets again start with an assumption that infinity divided by infinity is equal to one, i.e., $ \frac {\infty}{\infty}= 1$. Infinity is defined to be greater than any number, so there can not be two numbers, both infinity, that are different.However, when dealing with limits, one can approach a non-infinite value for a function involving infinity. As far as I know, $\infty$ is always interpreted distinct from the usual numbers (by usual I mean real numbers) (including integers), that is, you have an axiom (or a theorem), $$\forall x \in \mathbb{Z} , x \neq \infty.$$ Hence $\infty$ could not possibly be the greatest integer number dividing 11 since it then would be an integer contradicting this axiom (or a theorem, whatever you wanna call it). We can make the terms $x_n$ as close as we want to $L$, as long as we go beyond a sufficiently high $n$. It is the only integer (and, in fact, the only real number) that is neither negative nor positive. 0 is your answer (not a number close to zero). What is zero divided infinity? In other words, $1$ divided by $\infty$ does not equal a number or is undefined. The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". Let us take 1 divided by 0. So, 99 = . on the infinity. What do the parents perceive as their role to the Day Care worker? That would give you a valid fraction as In this sense, infinity minus 1 is still infinity. However, $1$ divided by $\infty$ does equal a limit approaching $0$. In the limit x→∞ of the function f(x)/g(x), we havelimx→∞ f(x)/g(x) = limx→∞ x2/x = limx→∞ x = ∞;so, both f(x) and g(x), in that limit, equal infinity, but f(x)/g(x) ≠ 1. B)Enter the the Ksp expression forC2D3 in terms of the molar solubility x. Jimmy aaja -M.I.A. operations of arithmetic. Just say that it equals undefined. In summary with all of this, we can say that zero over 1 equals zero. Hence, infinity is not a real number since we will never find the largest real number. However, any real number divided by infinity is equal to undefined, because you can never finish dividing something into infinite number of parts. This 'boundless sequence' infinity is the same one that is represented by $\int^\infty$, $\sum^\infty$ and in derivatives. Therefore its consistent to say 30 = 1. If you believe that, then infinity is not a number. 3 Answers C5H5N in water > C5H5NH+ & OH- Kb = [C5H5NH+] [OH-] / [C5H5N] 1.5e-9 = [x] 1. The cookie is used to store the user consent for the cookies in the category "Performance". We can now start to use the term extremely large number when referring to infinity so that we can understand better how to perform these operations with infinities. We still get an extremely large number. negative infinity equals zero because if we divide 1 by a negative The non-negative integers and the rational numbers are systems of numbers which are carefully constructed to obey clearly stated rules - the need for careful definition arises because of the kind of question you are asking. \end{align*} Lets find whether $ \frac {\infty}{\infty}$ is equal to $1$ or not. El subjuntivo In fact infinity multiplied by infinity is also equal to infinity. Who controlled the House of Representatives in 1982? Then, we use the fact that $\infty+\infty=\infty$. The following table, Table 1, shows the value of $1/x$ as $x$ gets larger and larger. In normal cases, the value of something divided by 0 has not been set yet, so its undefined. There is no number that you can multiply by 0 to get a non-zero number. Calculate the pH of a solution of 0.157 M pyridine. In mathematics, a limit of a function occurs when x gets larger and larger as it approaches infinity, and 1/x gets smaller and smaller as it approaches zero. 4. In that context, yes you are correct. Infinity divided by any finite number is infinity. For example, if zero is to be divided by any number, this means 0 items are to be shared or distributed among the given number of people. Numbers that are only divisible by themselves and one are called prime numbers. The smallest prime number is 2 and it goes on. You may believe that 1 also has the qualities to become a prime number. However, though it is divisible by one and itself, 1 Is 1 so it is only divisible by 1 number. Why is number 1 not a prime number? When the numerator is greater than 1 or less than 1, will the expression be still equal to zero? When 0 is divided by any number the quotient is? We use the information we obtained in the previous section. Infinity is not a real number and is only used as a representation of an extremely large real number. In this article, we defined infinity in a clearer manner and used it to perform operations and evaluate expressions with infinities. That's what confuses me. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Take, for example, 2x divided by x, when x is infinity. So, 0/n = 0 and n/0= infinite. zero This means that $e^{\infty}$ we multiply $e$ by itself infinitely many times. It depends on what you mean by 'infinity' as a mathematical object. Now, we want to know the answer when we divide $1$ by infinity, denoted as $\frac{1}{\infty}$, which we know does not exist since there exists no number that is largest among all. It indicates a state of endlessness or having no boundaries in terms of space, time, or other quantities. Infinity Your answer has the flavor of an encouragement. So we find another way to show that infinity divided by infinity is indeed undefined. A number which is not zero is said to be nonzero. We use cookies to ensure that we give you the best experience on our website. Answers and Replies Nov 4, 2016 #2 FactChecker. Or Infinity is not a number so you cannot add or subtract it. Who, Table of Contents What is Probability?Examples of ProbabilityProbability Definition MathBasic Concepts of, Table of Contents What is the Altitude of a Triangle?Properties of Altitude, Web & mobile App Development Course For Kids, Artificial Intelligence Foundation Course For Kids, Varthur Main Road, Marathahalli, Bangalore, India, 560037. The expression $1/\infty^2$ is equal to zero. However, there are number systems that do include infinity. Something that never ends minus one, would still never end. You have infinity raised to a negative power which means that you have one divided by infinity raised to some (positive) power and that will be zero. How we determine type of filter with pole(s), zero(s)? Advertisement cookies are used to provide visitors with relevant ads and marketing campaigns. Since infinity is just a concept and is not a fixed number, hence the operations like addition, subtraction, multiplication, or division do not give similar results as obtained by other numbers when operated by these operations. Whatever we say 0/0 equals to, we contradict one crucial property of numbers or another. unlimited extent of time, space, or quantity : boundlessness. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. When a number is divided by infinity it is always equal to zero. We know that, any number divided by 0(zero) is infinity, then Let us suppose the n Is infinity divided by infinity equal to 1? Work Plz. We can even say that 1 divided by An Insight into Coupons and a Secret Bonus, Organic Hacks to Tweak Audio Recording for Videos Production, Bring Back Life to Your Graphic Images- Used Best Graphic Design Software, New Google Update and Future of Interstitial Ads. infinitesimally larger than, but never equal to, zero (f / I = 1 / is not a number. larger than any number. So, we have:\begin{align*}\dfrac{\infty}{\infty}&=\dfrac{\left(\infty+\infty\right)}{\infty}\\&=\dfrac{\infty}{\infty}+\dfrac{\infty}{\infty}\\\end{align*}, Since $\infty/\infty=1$, then this should be true:\begin{align*}\dfrac{\infty}{\infty}&=\dfrac{\infty}{\infty}+\dfrac{\infty}{\infty}\\1&=1+1\\1&=2.\end{align*}. How to calculate the limit $\lim_{n\to\infty} \frac{\sqrt{n+1} (1+n^2)}{\sqrt{n} (2 +n^2+2n)}$? We can say that zero over zero equals undefined. And of course, last but not least, that were a lot of times faced with, is 1 divided by zero, which is still undefined. @user647486 Care to justify your statement? A root of a function is also sometimes known as a zero of .. Thus,$\infty/\infty$ is undefined. To do so correctly however, you need to understand what we actually mean by infinity and what we mean by a "number.". We use the earlier information we gathered that $\lim_{x\to\infty}\dfrac{1}{x}$ is equal to zero. If you are working with non-negative integers, then there is no infinite integer, and you can't divide by something which doesn't exist. How to rename a file based on a directory name? You must apply the L'Hospital theorem by deriving the numerator and the denominator of the equation that gave you infinity over infinity.-----------------Why ∞/∞ is not 1One could think that ∞/∞ = 1, but this is wrong.The answer depends on the kind of infinity: in fact, there are different kinds of infinity.For example, consider f(x) = x2 and g(x) = x. Is 1% of infinity still infinity? Let us take 1 divided by 0. Save my name, email, and website in this browser for the next time I comment. If you continue to use this site we will assume that you are happy with it. Unless we are talking about limits and orders of magnitude. Let's consider the second type of infinity, ostensibly represented by your definition $\lim_{n\infty}x_n=L$. Solving $\frac{1}{\infty}$ is the same as solving for the limit of $\frac{1}{x}$ as $x$ approaches infinity, so using the definition of limit, $1$ divided by infinity is equal to 0. Is infinity infinity defined? Greeking Out3. There are, however, different "sizes of infinity." Also if you divide any number by infinity you get zero. It just flat out doesn't make sense in this context. rev2023.1.18.43173. Infinity is a concept, an idea not a number. Thus the answer to your question is 0. Since $\frac{1}{\infty}$ is equal to zero, therefore, something divided by infinity becomes zero. In general, any number divided by 0 equals to infinity. First of all we must know what is infinity, For a normal high school student infinity is a rational/irration number or quantity but if we go in the 6 Is the value of Infinity divided by 0 indeterminate? (PS, I haven't done much calculus in decades, if this is factually wrong please feel free to fix it rather than just blindly downvoting. The online training program in coding imbibes in your kid creativity and problem-solving skills apart from improving kids academic performance. infinity is not a number but is is treated as if it is something (If It Is At All Possible). If we have f(x) = 2x and g(x) = x, both f(x) and g(x) equal infinity (for x→∞), butlimx→∞ f(x)/g(x) = limx→∞ 2x/x = limx→∞ 2 = 2 ≠ 1.So you see that infinity is something to check everytime!--------------Addition: Since infinity is not a set number, you cannot assume that infinity divided by infinity would equal one. These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc. The origin is also identified as the point (0, 0). Whom life had made ugly in the story of dodong and teang? How is a limit as it goes to infinity unable to simulate what would happen if you actually were able to act on infinity? Calculate the pH of a solution of 0.157 M pyridine.?