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(She's now doing a maths PhD.). ( Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Why does a rope attached to a block move when pulled? {\displaystyle a^{2n}+b^{2n}} Since an empty product, like an empty sum should be neutral and not affect the product of the numbers coming in after it, the multiplying machine should be set at 1 before it starts work. 2 48 = 281474976710656. Any exponent of (1/2)is actually the square root of that number. de Moivre's formula to show that , so adding multiples Can you have more than 1 panache point at a time? Therefore, the numbers 1, 2, 4, 8, 16, 31, 62, 124 and 248 add up to 496 and further these are all the numbers that divide 496. 1. 2 * x does not mean adding 2 to itself x times. rev2023.6.2.43474. Some more examples: it's 64! Colour composition of Bromine during diffusion? A number to the power of negative one is equal to one over that number. As long as the base is greater than one, the same thing happens. properties of exponents, so we can find e to the power of any complex number the easy-to-remember-rules and the mathematical proofs) and reveals the underlying "sense" to it. answer to another question, where it is shown that &= 2\cdot \frac{1}{2}\\ Even & odd numbers of negatives. Negative exponents in the denominator of a fraction get moved up to the numerator and become positive exponents. For example, The exponent of any number represents how many times to use the number in a multiplication. ALERT! Incidentally, the best way to teach fractions is by having pizza for dinner. I up-voted your answer, it is a valid approach, and more clearly stated than the comment, although I might think about reordering it, depending which part of those steps they intuitively find agreement with. Another thing to note about exponents is that the number 1 can be used, as well: {eq}3^{1} = 3 {/eq} Since only one value of three is represented by the "power of one," the result is simply 3 . Why is Bb8 better than Bc7 in this position? When you decrease the exponent, you divide by 2. Hello and welcome to MSE. What is the one-to-one property and why does it work for logarithms? Technically What's the term for the coefficient that change any floating point number to its next or previous value? 2^0=&2^{7-7}\\ If a number is raised to a negative power, it can be rewritten as 1 divided by that number raised to the power, e.g., 2^-2 = 1/2^2. 2 46 = 70368744177664. 1 and -1 to different powers. In a connection with nimbers, these numbers are often called Fermat 2-powers. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. @QiaochuYuan "Intuitively this is because if you add 2 to itself zero times, you get zero." This works for any negative power, not only the power of one. What does "Welcome to SeaWorld, kid!" m And so on. Can a judge force/require laywers to sign declarations/pledges? and if $x$ approaches minus infinity then what happens? a^(b+ic) as having only one value (in much the same way as we think ( Here we observe that 2 0 is in a 0 format so by using the exponent rule we can say that 2 0 = 1. +1, This is the one which made sense to me when I was learning it. How common is it to take off from a taxiway? Is there a mathematical term for raising a a number to the power of its own value. That this can be added/multiplied to anything without resulting in a change should be accepted. {\displaystyle a^{n}+b^{n}=(a^{p})^{m}+(b^{p})^{m}} Should the Beast Barbarian Call the Hunt feature just give CON x 5 temporary hit points, Use of Stein's maximal principle in Bourgain's paper on Besicovitch sets. I need help to find a 'which way' style book featuring an item named 'little gaia', How to determine whether symbols are meaningful. Likewise, suppose we are multiplying numbers in a multiplying machine. Go backward to What is the Square Root of i? Actually, let's back up a little and use our calculator to get the answer to our example; 2.14 ^ 2.14 = 5.09431. How could a person make a concoction smooth enough to drink and inject without access to a blender? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Note: Applying the same argument to any number different from $0$ gives the same result. Thinking of $2^n$ as how much we multiply by if we double $n$ times makes it clear that it should be $1$ if we don't double at all. It is more interesting to consider $2^0$ to be $1$ than giving up. @Yishai Well, if you go slowly through the proof showing them step-by-step, then there is a huge chance that they get it. It is a good idea to start each sentence with a capital letter. 7 x 7 x 7 x 7 = ? Connect and share knowledge within a single location that is structured and easy to search. Playing a game as it's downloading, how do they do it? For any number a, except 0, a0 = 1. VS "I don't like it raining.". Nearly all processor registers have sizes that are powers of two, 32 or 64 being very common. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Different exponents affect the value of a number: when raised to the power of zero, any number equals one; when raised to an even power, negative numbers yield positive results; and when raised to an odd power, negative numbers yield negative results. So after 1 day, number of $ = 3 = 1x3 = 3^1 When I was a kid, I had to teach this concept (and fractions) to some kids in a lower grade. Try thinking in terms other than math. 1. Starting with2 the last digit is periodic with period4, with the cycle 2486, and starting with4 the last two digits are periodic with period20. In this example: 82 = 8 8 = 64. +1 A bit long, yet the core "you haven't started doubling them yet" is brilliant :). so for there always exit a unit value. What $2^n$ really represents above is an endomorphism of the free commutative monoid on an apple, which is much less concrete than an apple. In words: 8 2 could be called "8 to the power 2" or "8 to the second power", or simply "8 squared". b Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Using QGIS Geometry Generator to create labels between associated features in different layers. Why is a raised to the power of Zero is 1? Why does the bool tool remove entire object? The limit of $e^x$ as $x$ approaches infinity is $\infty$. Tell her to look down and tell you how many balls do you see? Hence $2^0 = 1$. It makes perfectly good sense to add and multiply complex numbers, and speech to text on iOS continually makes same mistake. Why is Bb8 better than Bc7 in this position? Four There is no single value to "ln a ": there are lots of . the expression i^i has an infinite set of possible values. Go up to Question Corner Index Go forward to Raising a Number to a Complex Power Switch to text-only version (no graphics) Access printed version in PostScript format (requires PostScript printer) Go to University of Toronto Mathematics Network No number is no number. how to explain that Prob[heads, tails] = 2 * Prob[heads, heads] to a student? For a < 0, a x is undefined in the reals for irrational x. if number < base: # If number is equal to 1, it's a power (base**0). It only takes a minute to sign up. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Determining on what domain a complex Logarithm is analytic. Indeed, the number of endofunctions on an n-element set, Is there a name for raising a number to its own value as a power, CEO Update: Paving the road forward with AI and community at the center, Building a safer community: Announcing our new Code of Conduct, AI/ML Tool examples part 3 - Title-Drafting Assistant, We are graduating the updated button styling for vote arrows. The negative exponent rule says that when doing calculations with negative exponents, any negative exponents in the top half of a fraction get moved to the bottom half and become positive exponents. Applying this to the geometric progression 31, 62, 124, 248, 496 (which results from 1, 2, 4, 8, 16 by multiplying all terms by 31), we see that 62 minus 31 is to 31 as 496 minus 31 is to the sum of 31, 62, 124, 248. real numbers. (except when b and c are both rational numbers), because instead Where the number xis called the base, whereas (1/2)is the power or exponent of the expression. three. Answer: As already explained, the answer to (-1)0 is 1 since we are raising the number -1 (negative 1) to the power zero. It's intuitive what it means to add different amounts of apples, and it's intuitive what it means to have zero apples. This is what the formula up above gives you. 2 42 = 4398046511104. "Zero." Both addition, multiplication, exponentiation and the operation you are describing are particular cases of the following recursive function $H: \mathbb{N}^3 \to \mathbb{N}$ (called "hyperoperation"): $$H(n, a, b)= \begin{cases}b+1 & \quad n=0 \\ a & \quad n = 1 \\ 0 & \quad n = 2, b = 0 \\ 1 & \quad n \geq 3, b = 0 \\ H(n-1,a, H(n,a,b-1)) \end{cases}$$. Every power of 2 (excluding 1) can be written as the sum of four square numbers in 24 ways. Explanation: According to the zero exponent rule or zero property of exponents, any number raised to the power of zero always equals 1. Speed up strlen using SWAR in x86-64 assembly. I think that the only things that differentiate these two cases are conventions and experiences. Can Bitshift Variations in C Minor be compressed down to less than 185 characters? Why is two to the power of zero equal to binary one? Each of these is in turn equal to the binomial coefficient indexed by n and the number of 1s being considered (for example, there are 10-choose-3 binary numbers with ten digits that include exactly three 1s). x to do the calculation gives rise to infinitely many different possible \begin{align*}2^0 &= 2^{1-1}\\ more. 3. But in the domain of complex numbers, the polynomial so this formula can be used as a definition of what e^x means when If you go into a grocery store with an order, you expect the cash register to read 0 before any items are rung up. each of this 1 dollar triples by itself to become 9 after 1day(as stated above), By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. or simply "8 squared". 2 I need help to find a 'which way' style book featuring an item named 'little gaia'. Connect and share knowledge within a single location that is structured and easy to search. The exponent $1$ 'gives the number $1$ the power to transform into $3$. [ a 0 . Basically, therules for exponentiation states thatIf n is a positive integer and x is anyreal number, then ancan be represented as: Hence 2 to the power of 1/2 can be written as 2(1/2)which is a fractional exponent. How would you explain the concept to a child, other than the teachers "that is just the rule" approach? Exponents with negative fractional bases. Unfortunately, I understood it too early, and never got any pizza. for some integer n, so the possible values of I think that it is worth pointing out that, strictly speaking, explaining $2^0$ equals $1$ is not really what people do in this situation. But you can multiply apples by numbers; that is, you can start with $1$ apple, then double the number of apples you have to get $2$ apples, then double the number of apples you have to get $4$ apples, and so forth. ,n). Well, that means you haven't started doubling them yet, so you still have $1$ apple. (where n! This expression equals i i. ? Now p cannot divide 16 or it would be amongst the numbers 1, 2, 4, 8 or 16. n 2 rev2023.6.2.43474. That's the button to make a power of 10. 1.4 Doing all this on your calculator. I want to extend the answer by @Qiaochu Yuan. Hence,xto the power of 1/2 can be written as x. so for example in every exponetation/multiplication there is a hidden 1, x1 is always there but invisible its still part of the value hidden in every single calculation, its just not written, adding x1 makes no difference, Let's consider, their is 1bacteria in a dish. Let q be 4, then p must be 124, which is impossible since by hypothesis p is not amongst the numbers 1, 2, 4, 8, 16, 31, 62, 124 or 248. as the infinite sum expression for cos c, and the imaginary part is Learn more about Stack Overflow the company, and our products. When you do this and split the sum into To find 2 to the power of 0, we can write it in exponent form as 2 0 , where 2 is base and 0 is power. between 1 and 10 and a power of 10. However, when a is not real there is no one natural choice of logarithm &= 2^1\cdot 2^{-1}\\ Remember, if their is no bacteria or no money,that means it cant multiply. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. a not combining at all) has the same effect as combining with the identity. Just to answer the second part of your question, yes you can write 9 3 / 2 = ( 9 1 / 2) 3 = ( 9) 3. for , just as 2 and -2 are both square roots of 4. the number in a multiplication. i Interpreting the logarithm as a sum of simple poles along the negative real axis. Your answer contains a common mistake that even very smart mathematicians seem to make quite often. And since 31 does not divide q and q measures 496, the fundamental theorem of arithmetic implies that q must divide 16 and be amongst the numbers 1, 2, 4, 8 or 16. to irrational and then to complex values of x, you need to rewrite 'The Multiplicative Identity'. Any exponent of (1/2) is actually the square root of that number. It's also interesting to note that all these values of are lim x a x = , lim x a x = 0. for any a > 1. lim x + a x = 0, lim x a x = . So, 2 * 0 indicates you are adding up 0 copies of 2, not that you are adding 2 to itself 0 times. The pattern continues where each pattern has starting point 2k, and the period is the multiplicative order of 2modulo5k, which is (5k) = 4 5k1 (see Multiplicative group of integers modulo n). 3. Is there a shortcut for raising 2 to the power of a number (e.g. The sum of no number is $0$. You'll discover a short way of writing very long numbers. the theory about infinite sums can also be extended to complex numbers, n How about this: There's always an implicit 1 in the expansion: $$2^{3} = 2 \cdot 2 \cdot 2 \cdot 1 = 8$$. The prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024 (210). Given $2^{x}=129$, why is it that I can use the natural logarithm to find $x$? Therefore, if z is any complex number for . If the ratio of frequencies of two pitches is a power of two, then the interval between those pitches is full octaves. The sum of the reciprocals of the powers of two is 1. For $a \lt 0$, $a^x$ is undefined in the reals for irrational $x$, Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. I would try to explain it in terms of exponents, $2=2^{1}=2^{1+0}=2^{1}\times 2^{0}=2\times 2^{0}$. For example, five to the negative one power equals one over five, or 1/5. We know that i^3=i^2\cdot i i3 = i2 i. Learn more about Stack Overflow the company, and our products. Is there a term for the process of "taking a logarithm"? 2 * x means you are adding up x copies of 2. So in "no" days, number of dollar = (3^0 + 3^0 + 3^0) = 3, which is the initial value. Home Page, University of Toronto Mathematics Network Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. are some simple rules to use with exponents. The first few powers of 210 are slightly larger than those same powers of 1000 (103). Why not just take out a calculator, let her punch in any number she likes, and then let her hit square root over and over. Why does bunched up aluminum foil become so extremely hard to compress? Is there a way to tap Brokers Hideout for mana? Why is Bb8 better than Bc7 in this position? and $H(4, a, b)$ is the exact binary operation on $a$ and $b$ of iterated exponentiation. n The best method is to work backward with division. 2. i It would be more interesting to consider $f(0)$ to be 1 than giving up. Also, we can say that if the exponent is zero then the result is always 1. notation, or E notation on a calculator ("E" stands for "Exponent"). of positive integers, the series, converges to an irrational number. Any exponent of (1/2) is actually the square root of that number. The best answers are voted up and rise to the top, Not the answer you're looking for? But for repeatedly multiplying by $2$, it isn't necessarily, since you can't multiply apples and apples (at least, not in a way that makes sense to a child). of doing the calculation writing a = e^(ln a), you could also when you have Vim mapped to always print two? ) 2 Number raised to power of irrational number. This is also the only way to wrap your head around negative powers. The best we can do is to convince a child of the following facts: 1. Now that we have 'the answer' and the portion attributable to the integer component of our exponent, let's determine the increase contributed by our decimal component; (5.09431/4.5796) = 1.112392. ( Home Page, University of Toronto Mathematics Network {\displaystyle 2^{x}{\tbinom {n}{x}}.}. Its cardinality is 2n. I like this a lot, and I've never thought of it precisely this way. to de Moivre's formula: Now we know what e raised to an imaginary power is. What does "Welcome to SeaWorld, kid!" (I'm really curious what her response to this argument will be, actually. Limit as x approaches infinity for weird function, The Limits at Infinity of e to the power of sinx - x. When a is real it is more "natural" to use the ordinary Then, this 3dollar is actually 3 x 1 dollars (i.e 1+1+1) 2 43 = 8796093022208. &= \frac{2}{2}\\ When a number is divided by zero, it results in undifined. 2 47 = 140737488355328. Differentiating the process by which we calculate something as to that which a process or calculation actually means might be important. different magnitudes). z are . A prime number that is one less than a power of two is called a Mersenne prime. So it might be better to first get her used to variable exponent (or variables of any kind, really). How to prevent amsmath's \dots from adding extra space to a custom \set macro? [citation needed]. But I was an odd kid who enjoyed formal logic and whose idea of fun was teaching my little sister simultaneous equations. No need to do negatives just yet; it suffices to only show that $\frac{2^m}{2^n}=2^{m-n}$, and then consider what happens when you divide a number by itself @Yishai Why that argument is not convincing? So in general we can say that the physical meaning of m^0 is some unit value at 'no' time :). For example, 640 = 32 20, and 480 = 32 15. x What happens if you double your apples zero times? What is the one-to-one property and why does it work for logarithms? The numbers that can be represented as sums of consecutive positive integers are called polite numbers; they are exactly the numbers that are not powers of two. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. its confusing, look you said if a>1 then the result will be zero, and you did not told about if a is beteen 0 and 1. i am assuming from your answer that if a>1 then limit x--> + infinity or - infinity of a to the power x is = 0. A byte is now considered eight bits (an octet), resulting in the possibility of 256 values (28). Are the imaginary zero, the complex zero, and the real zero distinct numbers? Powers of two are often used to measure computer memory. It is the fact that zero is the neutral element of addition that makes it a valid start for the register; starting there will not cheat you on your order or give you any unfair advantage. What is neutral there? Weisstein, Eric W. What are the possible values for z? Binary prefixes have been standardized, such as kibi(Ki) meaning 1,024. Learn more about Stack Overflow the company, and our products. p Anyway, my daughter is now 14 and I think gets it. The sum 31 multiplied by 16 (the 5th term in the series) equals 496, which is a perfect number. Colour composition of Bromine during diffusion? Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. a1 = a. . rev2023.6.2.43474. like a^(b+ic) has many different values. Take each one and stack them vertically on a table. It doubles every hour (i.e x2) In this case, the corresponding notes have the same name. This is illustrated in the "I don't like it when it is rainy." 1) I saw in a book that "the limit as $x$ approaches positive infinity of $e^x$ equals $0$" I want to ask about this? If $a>1$ then $\lim_{x\to\infty}a^x=\infty$ and $\lim_{x\to -\infty}a^x=0$. The Negative Exponents. For example, in the original Legend of Zelda the main character was limited to carrying 255rupees (the currency of the game) at any given time, and the video game Pac-Man famously has a kill screen at level256. Many Thanks This is the exact same argument as @user93957 uses above.. @MSalters: not really. The pattern continues where each pattern has starting point 2k, and the period is the multiplicative order of 2modulo5k, which is (5k) = 4 5k1 (see Multiplicative group of integers modulo n). (The term byte once meant (and in some cases, still means) a collection of bits, typically of 5 to 32 bits, rather than only an 8-bit unit.) Then ask her to tell you how many balls do you see now? Dotted or otherwise modified notes have other durations. Put another way, they have fairly regular bit patterns. This occurs when In a context where only integers are considered, n is restricted to non-negative values, [1] so there are 1, 2, and 2 multiplied by itself a certain number of times. Let $f(x) = \frac {\sin x} x$. From there, you can segue into negative exponents, if you'd like. 1. Therefore, 31 cannot divide q. Is there liablility if Alice scares Bob and Bob damages something? This is very clear and breaks down into steps comparable with what the child already understands. Therefore, x (1/2) = x. mean? Thus, we need to recall some basic rules: Any number raised to the power of 0 equals 1. Some of the other answers provide good ways to convince a child of these facts. Or, to be concrete, if someone gives you two apples zero times, you have zero apples. I am interested in knowing what i to the power of i is. do it by writing , or by writing Why does a rope attached to a block move when pulled? affect the truth of the equality). squared times two cubed is not the same as 8 raised to the power two plus Basically, therules for exponentiation states that, if n is a positive integer and x is any real number, then ancan be represented as: Hence xto the power of 1/2 can be written as x(1/2)which is a fractional exponent. the definition in a way that makes sense even when r is complex. This can all be done using positive integers, it doesn't even need fractions or negative numbers. In particular, a $0$-cube, of any side length, is a point, and so exactly one $0$-cube of side length $1$ fits into a $0$-cube of side length $2$. That would indicate there are x + 1 copies of 2 total. b + ic as follows: This answers the question you asked. numbers in scientific notation. , As a consequence, numbers of this form show up frequently in computer software. fit in the window? $5^5 , 1692^{1692}$. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. VS "I don't like it raining.". +1 As a classroom teacher I have found this argument to be effective with low- and average-skilled middle and high school students. then the number of bacterial present is "1", You have 1 dollar. Learn how and when to remove these template messages, Learn how and when to remove this template message, Multiplicative group of integers modulo n, quadruple-precision floating-point format, sum of the reciprocals of the powers of two, sum of the reciprocals of the squared powers of two, "Powers of 2 Table - - - - - - Vaughn's Summaries", "Mersenne Prime Discovery - 2^82589933-1 is Prime! Please also tell me what would happen if $a$ is positive number. p :(. to write 127,680,000 in scientific notation, change the number to a number 565,000 74 @Djaian, but the multiplication rule can be verified by a kid: just add sufficiently many times: you shoul dget the same result as in the table. So, when you go from 21 to 20, of course you divide by 2, which gives you 1. So 3^0 means the money has not yet started multiplying. Let us first look at what an "exponent" is: The exponent of a number says how many times to use. I have also included the code for my attempt at that. b I think you can improve the quality of your posts by being a bit more formal. Is it possible? As long as the base is greater than one, the same thing happens. Is linked content still subject to the CC-BY-SA license? The powers of 210 values that have less than 25% deviation are listed below: It takes approximately 17 powers of 1024 to reach 50% deviation and approximately 29 powers of 1024 to reach 100% deviation of the same powers of 1000. CEO Update: Paving the road forward with AI and community at the center, Building a safer community: Announcing our new Code of Conduct, AI/ML Tool examples part 3 - Title-Drafting Assistant, We are graduating the updated button styling for vote arrows. n This multiplication rule tells us that we can simply add the exponents when multiplying two powers with the same base. The best answers are voted up and rise to the top, Not the answer you're looking for? Learn more about Stack Overflow the company, and our products. Now if zero is raised to a negative power, it would be like: 0^-1 what simplifies to 1/0^1 what simplifies to 1/0. Corresponding signed integer values can be positive, negative and zero; see signed number representations. e ^ ( b + ic) = ( e ^ b ) ( e ^ ( ic )) = ( e ^ b ) ( (cos c) + i (sin c )) Finally, for a real number a, you can define a ^ ( b + ic) by writing a = e ^ (ln a ): This answers the question you asked. What is the first science fiction work to use the determination of sapience as a plot point? principal value of would be --the case where The reverse is also true. Why isn't the logarithm defined as its absolute value? Therefore, 2 (1/2) = 2 or 1.4142. (If n is odd, then an + bn is divisible by a+n, and if n is even but not a power of 2, then n can be written as n=mp, where m is odd, and thus In fact, they. values for a^(b+ci). Is there a name for expressions of the form $n^n$? gives no power of transformation), so $3^0$ gives no power of transformation to the number $1$, so $3^0=1$. The first ten powers of 2 for non-negative values of n are: Because two is the base of the binary numeral system, powers of two are common in computer science. rev2023.6.2.43474. Is the natural logarithm actually unique as a multiplier? Each of these equalities is true (you can check them using 1,2,. . Also, if $a<0$ the limit $\lim_{x\to\infty}a^x$ is not defined since $a^{\frac{b}{2}}$ does not exist for any $b>0$. The exponent of a number says how many times to use the number in a multiplication.. Now, if a is a complex number Can a rule be formulated to explain this to 7 year old? Scientific For example, the sum of the first 5 terms of the series 1 + 2 + 4 + 8 + 16 = 31, which is a prime number. Assume it triples every day (i.e x3). Go up to Question Corner Index Go forward to The Origin of Complex Numbers Switch to text-only version (no graphics) Access printed version in PostScript format (requires PostScript printer) Go to University of Toronto Mathematics Network @MSalters: Sorry for asking this, but did you read my post before commenting on it? speaking, the expression a^(b+ic) has infinitely many possible values For more about representing signed numbers see two's complement. The exponent $0$ provides $0$ power (i.e. i^3=-i, i^4=1, i^5=i, etc. rather than "Gaudeamus igitur, *dum iuvenes* sumus!"? For any numbers a, b, and c, ab x ac = ab+c. Answer: Any number to the power of 0 is 1. If $0

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