probability- a number between 0 and 1 that reflects the chances that a random event will occur, where 0 is the complete absence of probability of the event occurring, and 1 means that the event in question will definitely occur.
The probability of event E is a number from to 1.
The sum of the probabilities of mutually exclusive events is equal to 1.
empirical probability- probability, which is calculated as the relative frequency of an event in the past, extracted from the analysis of historical data.
The probability of very rare events cannot be calculated empirically.
subjective probability- probability based on a personal subjective assessment of an event without regard to historical data. Investors who make decisions to buy and sell shares often act based on considerations of subjective probability.
prior probability -
The chance is 1 in... (odds) that an event will occur through the concept of probability. The chance of an event occurring is expressed through probability as follows: P/(1-P).
For example, if the probability of an event is 0.5, then the chance of the event is 1 out of 2 because 0.5/(1-0.5).
The chance that an event will not occur is calculated using the formula (1-P)/P
Inconsistent probability- for example, the price of shares of company A takes into account possible event E by 85%, and the price of shares of company B only takes into account 50%. This is called inconsistent probability. According to the Dutch Betting Theorem, inconsistent probability creates profit opportunities.
Unconditional probability is the answer to the question “What is the probability that the event will occur?”
Conditional probability- this is the answer to the question: “What is the probability of event A if event B occurs.” Conditional probability is denoted as P(A|B).
Joint probability- the probability that events A and B will occur simultaneously. Denoted as P(AB).
P(A|B) = P(AB)/P(B) (1)
P(AB) = P(A|B)*P(B)
Rule for summing up probabilities:
The probability that either event A or event B will happen is
P (A or B) = P(A) + P(B) - P(AB) (2)
If events A and B are mutually exclusive, then
P (A or B) = P(A) + P(B)
Independent events- events A and B are independent if
P(A|B) = P(A), P(B|A) = P(B)
That is, it is a sequence of results where the probability value is constant from one event to the next.
A coin toss is an example of such an event - the result of each subsequent toss does not depend on the result of the previous one.
Dependent Events- these are events where the probability of the occurrence of one depends on the probability of the occurrence of another.
The rule for multiplying the probabilities of independent events:
If events A and B are independent, then
P(AB) = P(A) * P(B) (3)
Total probability rule:
P(A) = P(AS) + P(AS") = P(A|S")P(S) + P (A|S")P(S") (4)
S and S" are mutually exclusive events
expected value a random variable is the average of the possible outcomes of a random variable. For event X, the expectation is denoted as E(X).
Let’s say we have 5 values of mutually exclusive events with a certain probability (for example, a company’s income was such and such an amount with such a probability). The expected value is the sum of all outcomes multiplied by their probability:
Dispersion of a random variable is the expectation of square deviations of a random variable from its expectation:
s 2 = E( 2 ) (6)
Conditional expected value is the expected value of a random variable X, provided that the event S has already occurred.
(listed in order of increasing accuracy)
Determination methods
Number e can be defined in several ways.
- Over the limit: e = lim x → ∞ (1 + 1 x) x (\displaystyle e=\lim _(x\to \infty )\left(1+(\frac (1)(x))\right)^(x) )(second remarkable limit). e = lim n → ∞ n n ! n (\displaystyle e=\lim _(n\to \infty )(\frac (n)(\sqrt[(n)](n}}} !}(this follows from the Moivre-Stirling formula).
- As the sum of the series: e = ∑ n = 0 ∞ 1 n ! (\displaystyle e=\sum _(n=0)^(\infty )(\frac (1)(n}} !} or 1 e = ∑ n = 2 ∞ (− 1) n n ! (\displaystyle (\frac (1)(e))=\sum _(n=2)^(\infty )(\frac ((-1)^(n))(n}} !}.
- As singular a (\displaystyle a), for which ∫ 1 a d x x = 1. (\displaystyle \int \limits _(1)^(a)(\frac (dx)(x))=1.)
- As the only positive number a (\displaystyle a), for which it is true d d x a x = a x . (\displaystyle (\frac (d)(dx))a^(x)=a^(x).)
Properties
| Proof of irrationality |
|---|
| Let's pretend that e (\displaystyle e) rational. Then e = p / q (\displaystyle e=p/q), Where p (\displaystyle p)- whole, and q (\displaystyle q)- natural. Hence p = e q (\displaystyle p=eq)Multiplying both sides of the equation by (q − 1) ! (\displaystyle (q-1) !}, we get p (q − 1) ! = eq! = q! ∑ n = 0 ∞ 1 n ! = ∑ n = 0 ∞ q ! n! = ∑ n = 0 q q ! n! + (\displaystyle p(q-1)!=eq!=q!\sum _(n=0)^(\infty )(1 \over n=\sum _{n=0}^{\infty }{q! \over n!}=\sum _{n=0}^{q}{q! \over n!}+\sum _{n=q+1}^{\infty }{q! \over n!}} !}We transfer ∑ n = 0 q q ! n! (\displaystyle \sum _(n=0)^(q)(q! \over n} !} to the left: ∑ n = q + 1 ∞ q ! n! = p (q − 1) ! − ∑ n = 0 q q ! n! (\displaystyle \sum _(n=q+1)^(\infty )(q! \over n=p(q-1)!-\sum _{n=0}^{q}{q! \over n!}} !}All terms on the right side are integers, therefore the sum on the left side is integer. But this sum is also positive, which means it is not less than 1. On the other side, ∑ n = q + 1 ∞ q ! n! = ∑ m = 1 ∞ q ! (q + m) ! = ∑ m = 1 ∞ 1 (q + 1) . . . (q + m)< ∑ m = 1 ∞ 1 (q + 1) m {\displaystyle \sum _{n=q+1}^{\infty }{q! \over n!}=\sum _{m=1}^{\infty }{q! \over (q+m)!}=\sum _{m=1}^{\infty }{1 \over (q+1)...(q+m)}<\sum _{m=1}^{\infty }{1 \over (q+1)^{m}}}Summing up the geometric progression on the right side, we get: ∑ n = q + 1 ∞ q ! n!< 1 q {\displaystyle \sum _{n=q+1}^{\infty }{q! \over n!}<{1 \over q}}Because the q ≥ 1 (\displaystyle q\geq 1), ∑ n = q + 1 ∞ q ! n!< 1 {\displaystyle \sum _{n=q+1}^{\infty }{q! \over n!}<1}We get a contradiction. |
- Number e (\displaystyle e) transcendentally. This was first proven in 1873 by Charles Hermite. Transcendence of number e (\displaystyle e) follows from Lindemann's theorem.
- It is assumed that e (\displaystyle e)- normal number, that is, the frequency of appearance of different digits in its notation is the same. Currently (2017) this hypothesis has not been proven.
- Number e is a computable (and therefore arithmetic) number.
- e i x = cos (x) + i ⋅ sin (x) (\displaystyle e^(ix)=\cos(x)+i\cdot \sin(x)), see Euler's formula, in particular
- Formula connecting numbers e (\displaystyle e) And π (\displaystyle \pi ), so-called Poisson integral or Gauss integral ∫ − ∞ ∞ e − x 2 d x = π (\displaystyle \int \limits _(-\infty )^(\infty )\ e^(-x^(2))(dx)=(\sqrt (\pi )))
- For any complex number z the following equalities are true: e z = ∑ n = 0 ∞ 1 n ! z n = lim n → ∞ (1 + z n) n . (\displaystyle e^(z)=\sum _(n=0)^(\infty )(\frac (1)(n}z^{n}=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{n}.} !}
- Number e expands into an infinite continued fraction as follows (a simple proof of this expansion involving Padé approximants is given in): e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , 1 , 8 , 1 , 1 , 10 , 1 , … ] (\displaystyle e=), that is e = 2 + 1 1 + 1 2 + 1 1 + 1 1 + 1 4 + 1 1 + 1 1 + 1 6 + 1 1 + 1 1 + 1 8 + 1 1 + 1 1 + 1 10 + 1 1 + … (\displaystyle e=2+(\cfrac (1)(1+(\cfrac (1)(2+(\cfrac (1)(1+(\cfrac (1)(1+(\cfrac (1)() 4+(\cfrac (1)(1+(\cfrac (1)(1+(\cfrac (1)(6+(\cfrac (1)(1+(\cfrac (1)(1+(\cfrac (1)(8+(\cfrac (1)(1+(\cfrac (1)(1+(\cfrac (1)(10+(\cfrac (1)(1+\ldots )))))) )))))))))))))))))))))))))
- Or equivalent to it: e = 2 + 1 1 + 1 2 + 2 3 + 3 4 + 4 … (\displaystyle e=2+(\cfrac (1)(1+(\cfrac (1)(2+(\cfrac (2)() 3+(\cfrac (3)(4+(\cfrac (4)(\ldots )))))))))))
- To quickly calculate a large number of signs, it is more convenient to use another expansion: e + 1 e − 1 = 2 + 1 6 + 1 10 + 1 14 + 1 … (\displaystyle (\frac (e+1)(e-1))=2+(\cfrac (1)(6+( \cfrac (1)(10+(\cfrac (1)(14+(\cfrac (1)(\ldots )))))))))
- e = lim n → ∞ n n ! n. (\displaystyle e=\lim _(n\to \infty )(\frac (n)(\sqrt[(n)](n}}.} !}
- Representation of Catalan: e = 2 ⋅ 4 3 ⋅ 6 ⋅ 8 5 ⋅ 7 4 ⋅ 10 ⋅ 12 ⋅ 14 ⋅ 16 9 ⋅ 11 ⋅ 13 ⋅ 15 8 ⋅ 18 ⋅ 20 ⋅ 22 ⋅ 24 ⋅ 26 ⋅ 28 ⋅ 30 ⋅ 32 17 ⋅ 19 ⋅ 21 ⋅ 23 ⋅ 25 ⋅ 27 ⋅ 29 ⋅ 31 16 ⋯ (\displaystyle e=2\cdot (\sqrt (\frac (4)(3)))\cdot (\sqrt[(4)](\frac (6 \cdot 8)(5\cdot 7)))\cdot (\sqrt[(8)](\frac (10\cdot 12\cdot 14\cdot 16)(9\cdot 11\cdot 13\cdot 15)) )\cdot (\sqrt[(16)](\frac (18\cdot 20\cdot 22\cdot 24\cdot 26\cdot 28\cdot 30\cdot 32)(17\cdot 19\cdot 21\cdot 23\ cdot 25\cdot 27\cdot 29\cdot 31)))\cdots )
- Representation through the work: e = 3 ⋅ ∏ k = 1 ∞ (2 k + 3) k + 1 2 (2 k − 1) k − 1 2 (2 k + 1) 2 k (\displaystyle e=(\sqrt (3))\ cdot \prod \limits _(k=1)^(\infty )(\frac (\left(2k+3\right)^(k+(\frac (1)(2)))\left(2k-1\ right)^(k-(\frac (1)(2))))(\left(2k+1\right)^(2k))))
- Through Bell numbers
E = 1 B n ∑ k = 0 ∞ k n k ! (\displaystyle e=(\frac (1)(B_(n)))\sum _(k=0)^(\infty )(\frac (k^(n))(k}} !}
Story
This number is sometimes called non-feathered in honor of the Scottish scientist Napier, author of the work “Description of the Amazing Table of Logarithms” (1614). However, this name is not entirely correct, since it has a logarithm of the number x (\displaystyle x) was equal 10 7 ⋅ log 1 / e (x 10 7) (\displaystyle 10^(7)\cdot \,\log _(1/e)\left((\frac (x)(10^(7))) \right)).
The constant first appears tacitly in an appendix to the English translation of Napier's above-mentioned work, published in 1618. Behind the scenes, because it contains only a table of natural logarithms determined from kinematic considerations, but the constant itself is not present.
The constant itself was first calculated by the Swiss mathematician Jacob Bernoulli while solving the problem of the limiting value of interest income. He discovered that if the original amount $ 1 (\displaystyle \$1) and is calculated annually once at the end of the year, then the total amount will be $ 2 (\displaystyle \$2). But if the same interest is calculated twice a year, then $ 1 (\displaystyle \$1) multiplied by 1 , 5 (\displaystyle 1(,)5) twice, getting $ 1 , 00 ⋅ 1 , 5 2 = $ 2 , 25 (\displaystyle \$1(,)00\cdot 1(,)5^(2)=\$2(,)25). Interest accrual quarterly results in $ 1, 00 ⋅ 1, 25 4 = $ 2,441 40625 (\displaystyle \$1(,)00\cdot 1(,)25^(4)=\$2(,)44140625), and so on. Bernoulli showed that if the frequency of interest calculations is increased indefinitely, then the interest income in the case of compound interest has a limit: lim n → ∞ (1 + 1 n) n . (\displaystyle \lim _(n\to \infty )\left(1+(\frac (1)(n))\right)^(n).) and this limit is equal to the number e (≈ 2.718 28) (\displaystyle e~(\approx 2(,)71828)).
$ 1.00 ⋅ (1 + 1 12) 12 = $ 2.613 035... (\displaystyle \$1(,)00\cdot \left(1+(\frac (1)(12))\right)^( 12)=\$2(,)613035...)
$ 1, 00 ⋅ (1 + 1,365) 365 = $ 2,714,568... (\displaystyle \$1(,)00\cdot \left(1+(\frac (1)(365))\right)^( 365)=\$2(,)714568...)
So the constant e (\displaystyle e) means the maximum possible annual profit at 100% (\displaystyle 100\%) per annum and maximum frequency of interest capitalization.
The first known use of this constant, where it was denoted by the letter b (\displaystyle b), found in Leibniz's letters to Huygens, -1691.
Letter e (\displaystyle e) Euler began using it in 1727, it is first found in a letter from Euler to the German mathematician Goldbach dated November 25, 1731, and the first publication with this letter was his work “Mechanics, or the Science of Motion, Explained Analytically,” 1736. Respectively, e (\displaystyle e) usually called Euler number. Although some scientists subsequently used the letter c (\displaystyle c), letter e (\displaystyle e) was used more often and is now the standard designation.
Although this connection at first glance seems completely unobvious (scientific mathematics, it would seem, is one thing, and economics and finance are quite another), but once you study the history of the “discovery” of this number, everything becomes obvious. In fact, no matter how the sciences are divided into different seemingly unrelated branches, the general paradigm will still be the same (in particular, for the consumer society - “consumer” mathematics).
Let's start with a definition. e is the base of the natural logarithm, a mathematical constant, an irrational and transcendental number. Sometimes the number e is called the Euler number or the Napier number. Denoted by the lowercase Latin letter “e”.
Since the exponential function e^x is integrated and differentiated “into itself,” logarithms based on the base e are accepted as natural (although the very name of “naturalness” should be in great doubt, because all mathematics is essentially based on artificially invented ones, divorced from nature fictitious principles, and not at all on natural ones).
This number is sometimes called Nepier in honor of the Scottish scientist Napier, author of the work “Description of the Amazing Table of Logarithms” (1614). However, this name is not entirely correct, since Napier did not directly use the number itself.
The constant first appears tacitly in an appendix to the English translation of Napier's above-mentioned work, published in 1618. Behind the scenes, because it contains only a table of natural logarithms determined from KINEMATIC considerations, but the constant itself is not present.
The constant itself was first calculated by the Swiss mathematician Bernoulli (according to the official version in 1690) while solving the problem of the limiting value of INTEREST INCOME. He found that if the original amount was $1 (the currency is completely unimportant) and compounded 100% per annum once at the end of the year, the final amount would be $2. But if the same interest is compounded twice a year, then $1 is multiplied by 1.5 twice, resulting in $1.00 x 1.5² = $2.25. Compounding interest quarterly results in $1.00 x 1.254 = $2.44140625, and so on. Bernoulli showed that if the frequency of interest calculation INCREASES INFINITELY, then the interest income in the case of compound interest has a limit - and this limit is equal to 2.71828...
$1.00×(1+1/12)12 = $2.613035…
$1.00×(1+1/365)365 = $2.714568… - in the limit the number e
Thus, the number e actually historically means the maximum possible ANNUAL PROFIT at 100% per annum and the maximum frequency of interest capitalization. And what do the laws of the Universe have to do with it? The number e is one of the important building blocks in the foundation of the monetary economy of loan interest in a consumer society, under which from the very beginning, even at the mental philosophical level, all the mathematics used today was adjusted and sharpened several centuries ago.
The first known use of this constant, where it was denoted by the letter b, appears in Leibniz's letters to Huygens, 1690-1691.
Euler began to use the letter e in 1727, it first appears in a letter from Euler to the German mathematician Goldbach dated November 25, 1731, and the first publication with this letter was his work “Mechanics, or the Science of Motion, Explained Analytically,” 1736. Accordingly, e is usually called the Euler number. Although some scholars subsequently used the letter c, the letter e was used more often and is the standard designation today.
It is not known exactly why the letter e was chosen. Perhaps this is due to the fact that the word exponential (“indicative”, “exponential”) begins with it. Another suggestion is that the letters a, b, c and d were already in fairly common use for other purposes, and e was the first "free" letter. It is also noteworthy that the letter e is the first letter in the surname Euler.
But in any case, to say that the number e somehow relates to the universal laws of the Universe and nature is simply absurd. This number by the concept itself was initially tied to the credit and financial monetary system, and in particular through this number (but not only) the ideology of the credit and financial system indirectly influenced the formation and development of all other mathematics, and through it all other sciences (after all, without exception, science calculates something using the rules and approaches of mathematics). The number e plays an important role in differential and integral calculus, which through it is actually also connected with the ideology and philosophy of maximizing interest income (one might even say it is connected subconsciously). How is the natural logarithm related? Establishing e as a constant (along with everything else) led to the formation of implicit connections in thinking, according to which all existing mathematics simply cannot exist in isolation from the monetary system! And in this light, it is not at all surprising that the ancient Slavs (and not only them) managed perfectly well without constants, irrational and transcendental numbers, and even without numbers and numbers in general (letters acted as numbers in ancient times), different logic, different thinking in the system in the absence of money (and therefore everything connected with it) makes all of the above simply unnecessary.
Describing e as “a constant approximately equal to 2.71828...” is like calling pi “an irrational number approximately equal to 3.1415...”. This is undoubtedly true, but the point still eludes us.
Pi is the ratio of the circumference to the diameter, the same for all circles. It is a fundamental proportion common to all circles and hence is involved in calculating circumference, area, volume and surface area for circles, spheres, cylinders, etc. Pi shows that all circles are related, not to mention the trigonometric functions derived from circles (sine, cosine, tangent).
The number e is the basic growth ratio for all continuously growing processes. The e number allows you to take a simple growth rate (where the difference is only visible at the end of the year) and calculate the components of this indicator, normal growth, in which with every nanosecond (or even faster) everything grows a little more.
The number e is involved in both exponential and constant growth systems: population, radioactive decay, percentage calculation, and many, many others. Even step systems that do not grow uniformly can be approximated using the number e.
Just as any number can be thought of as a "scaled" version of 1 (the base unit), any circle can be thought of as a "scaled" version of the unit circle (with radius 1). And any growth factor can be viewed as a "scaled" version of e (the "unit" growth factor).
So the number e is not a random number taken at random. The number e embodies the idea that all continually growing systems are scaled versions of the same metric.
Concept of exponential growth
Let's start by looking at the basic system that doubles for a certain period of time. For example:
- Bacteria divide and “double” in number every 24 hours
- We get twice as many noodles if we break them in half
- Your money doubles every year if you make 100% profit (lucky!)
And it looks something like this:
Dividing by two or doubling is a very simple progression. Of course, we can triple or quadruple, but doubling is more convenient for explanation.
Mathematically, if we have x divisions, we end up with 2^x times more good than we started with. If only 1 partition is made, we get 2^1 times more. If there are 4 partitions, we get 2^4=16 parts. The general formula looks like this:
height= 2 x
In other words, a doubling is a 100% increase. We can rewrite this formula like this:
height= (1+100%) x
This is the same equality, we just divided “2” into its component parts, which in essence is this number: the initial value (1) plus 100%. Smart, right?
Of course, we can substitute any other number (50%, 25%, 200%) instead of 100% and get the growth formula for this new coefficient. The general formula for x periods of the time series will be:
height = (1+growth) x
This simply means that we use the return rate, (1 + gain), "x" times in a row.
Let's take a closer look
Our formula assumes that growth occurs in discrete steps. Our bacteria wait and wait, and then bam!, and at the last minute they double in number. Our profit on interest on the deposit magically appears exactly after 1 year. Based on the formula written above, profits grow in steps. Green dots appear suddenly.
But the world is not always like that. If we zoom in, we can see that our bacterial friends are constantly dividing:
The green fellow does not arise out of nothing: he slowly grows out of the blue parent. After 1 period of time (24 hours in our case), the green friend is already fully ripe. Having matured, he becomes a full-fledged blue member of the herd and can create new green cells himself.
Will this information change our equation in any way?
Nope. In the case of bacteria, half-formed green cells still can't do anything until they grow up and separate completely from their blue parents. So the equation is correct.
Each of the functions E tests the specified value and returns TRUE or FALSE depending on the result. For example, the function EMPTY returns the Boolean value TRUE if the value being tested is a reference to an empty cell; otherwise, the boolean value FALSE is returned.
Functions E are used to obtain information about a value before performing a calculation or other action on it. For example, to perform a different action when an error occurs, you can use the function ERROR in combination with the function IF:
= IF( ERROR(A1); "An error has occurred."; A1*2)
This formula checks for an error in cell A1. When an error occurs, the function IF returns the message "An error occurred." If there are no errors, the function IF calculates the product A1*2.
Syntax
EMPTY(value)
EOS(value)
ERROR(value)
ELOGIC(value)
UNM(value)
NETTEXT(value)
ETEXT(value)
function argument E are described below.
meaning Required argument. The value being checked. The value of this argument can be an empty cell, an error value, a Boolean value, text, a number, a reference to any of the listed objects, or the name of such an object.
Function | Returns TRUE if |
|---|---|
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The value argument refers to an empty cell |
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The value argument refers to any error value other than #N/A |
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The value argument refers to any error value (#N/A, #VALUE!, #REF!, #DIV/0!, #NUM!, #NAME?, or #EMPTY!) |
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The value argument refers to a boolean value |
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The value argument refers to the #N/A error value (value not available) |
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ENETEXT |
The value argument refers to any element that is not text. (Note that the function returns TRUE if the argument refers to an empty cell.) |
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The value argument refers to a number |
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The value argument refers to the text |
Notes
Arguments in functions E are not converted. Any numbers enclosed in quotation marks are treated as text. For example, in most other functions that require a numeric argument, the text value "19" is converted to the number 19. However, in the formula ISNUMBER("19") this value is not converted from text to number, and the function ISNUMBER returns FALSE.
Using functions E It is convenient to check the results of calculations in formulas. Combining these features with the function IF, you can find errors in formulas (see examples below).
Examples
Example 1
Copy the sample data from the following table and paste it into cell A1 of a new Excel worksheet. To display the results of formulas, select them and press F2, then press Enter. If necessary, change the width of the columns to see all the data.
Copy the sample data from the table below and paste it into cell A1 of a new Excel worksheet. To display the results of formulas, select them and press F2, then press Enter. If necessary, change the width of the columns to see all the data.
Data |
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|---|---|---|
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Formula |
Description |
Result |
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EMPTY(A2) |
Checks if cell C2 is empty |
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ERROR(A4) |
Checks if the value in cell A4 (#REF!) is an error value |
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Checks whether the value in cell A4 (#REF!) is the error value #N/A |
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Checks whether the value in cell A6 (#N/A) is the error value #N/A |
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Checks whether the value in cell A6 (#N/A) is an error value |
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ISNUMBER(A5) |
Tests whether the value in cell A5 (330.92) is a number |
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ETEXT(A3) |
Checks whether the value in cell A3 ("Region1") is text |
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