Mathematical Constants

Intriguing, ubiquitous, and at times mysterious, numerical constants set the allowable limits for all universal phenomena. Whether your questions involves π, Avogadro's number, Planck's constant, the atomic mass unit, or any of the other multitudes of immutable numbers used in science, this is the category where they should be asked.

3,237 Questions
Numbers
Mathematical Constants
Complex Numbers

How do you enter an imaginary number in a scientific calculator?

Some scientific calculators can't handle complex or imaginary numbers. If you happen to have a special calculator that does, probably the manual will tell you how to enter them.

The HP 48 and up series does. It depends on if your calculator is in Polar Coordinate mode or X-Y coordinate mode, but a quick way to get the imaginary number i (regardless of which mode the calculator is currently in), is to press -1, then 'square root' button.

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Math and Arithmetic
Mathematical Constants

What does 24x5 equals?

120

410411412
Mathematical Constants

What is double of 19?

Well if 10 add 10 is 20 and 9 add 9 is 18 and u add them together its

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Science
Gravity
Mathematical Constants

Does Gravitational constant equals 6.67300 x 10-11 m3 kg 1 s2?

No.

G = 6.674 * 10-11 m3kg-1s-2

The powers for mass and time were incorrect.

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Numbers
Mathematical Constants
Complex Numbers

How do you multiply imaginary numbers?

First, let's make sure we are not confusing imaginary numbers with complex numbers.

Imaginary (sometimes called "pure imaginary" for clarity) numbers are numbers of the form ai, where a is a real number and i is the principal square root of -1.

To multiply two imaginary numbers ai and bi, start by pretending that i is a variable (like x).

So ai x bi = abi2. But since i is the square root of -1, i2=-1. So abi2=-ab.

For example, 6i x 7i =-42.

5i x 2i =-10.

(-5i) x 2i =-(-10)= 10.

Complex numbers are numbers of the form a+bi, where a and b are real numbers. a is the real part, bi is the imaginary part.

To multiply two complex numbers, again, just treat i as if it were a variable and then in the final answer, substitute -1 wherever you see i2.

Hence (a+bi)(c+di) = ac + adi + bci + dbi2 which simplifies to ac-db + (ad+bc)i.

For example:

(2+3i)(4+5i) = 8 + 10i +12i + 15i2= 8 + 10i + 12i - 15 = -7 + 22i

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Physics
Numbers
Mathematical Constants

What is the dimensional formula of gravitational constant?

Well, let's see:

Force of gravity = G M1 M2 / R2

So G = (force) x (distance)2 / (mass)2 = (M L / T2) x (L2) / (M2) = (M L3) / (M2 T2) =

(Length)3 (Mass)-1(Time)-2

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Math and Arithmetic
Numbers
Mathematical Constants

What is a googol plus a googol?

2 googol

305306307
Chemistry
Elements and Compounds
Mathematical Constants

What is the molar mass of SiC?

Molar Mass of Carbon + Molar Mass of Silicon = Molar Mass of SiC.

12.0107 + 28.0855 = 40.0962 g / mol.

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Math and Arithmetic
Numbers
Mathematical Constants

What are the first 100 numbers of pi 3.14..?

3.1415926535 8979323846 2643383279 5028841971 6939937510

5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960

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Math and Arithmetic
Numbers
Mathematical Constants

How many zeroes are in 1 billion?

There are 9 in the US, but possibly 12 in other countries, because there are two scales that use the same number names. The short scale advances names by thousands (thousand, million, billion, trillion, quadrillion) while the long scale advances numbers by millions, with intermediate names (e.g. milliard) for the thousands of those units.

--- Short Scale ---

In the 'short scale', a million is a thousand thousand, and a billion is a thousand million (1 x 109)

million 1,000,000 (6 zeros)

billion 1,000,000,000 (9 zeros)

trillion 1,000,000,000,000 (12 zeros)

*All English speaking countries now use the short scale.

(The UK changed to the short scale in 1974.)

(Some non-English speaking countries [Brazil, Bulgaria, Estonia, Indonesia, Iran, Israel, Latvia, Lithuania, Myanmar, Romania, Russia, Turkey, Ukraine and Wales] call the 9 zero number a billion (or sometimes a milliard) and the 12 zero number a trillion. Greece also uses the short scale, but with different names.)

--- Long Scale ---

In the 'long scale' a billion is a million million, with 12 zeros (1 x 1012):

million 1,000,000 (6 zeros)

billion 1,000,000,000,000 (12 zeros)

trillion 1,000,000,000,000,000,000 (18 zeros)

milliard (thousand million) = 1,000,000,000 (9 zeros)

billiard (thousand billion) = 1,000,000,000,000,000 (15 zeros)

*Most non-English speaking countries use the long scale.

(Some countries like Canada, Puerto Rico and South Africa use both, depending on whether English is being spoken or not.

Some countries (Bangladesh, India, Nepal, Pakistan, China, Japan, North and South Korea) have their own names and systems of numbering.) Researchers need to be careful with numbers greater than one million in English historical documents, and in numbers from countries other than their own.
one it is in the question

244245246
Roman Numerals
Mathematical Constants

What is XL in Roman Numerals?

40

212223
Math and Arithmetic
Algebra
Mathematical Constants

What is the biggest number other than infinity?

Goggle plex

110111112
Units of Measure
Mathematical Constants

What is the si unit of planck's constant?

h=6.62606896×10−34

J·s

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Math and Arithmetic
Mathematical Analysis
Numbers
Mathematical Constants

What is zero divided by zero?

In ordinary mathematics, you may not divide by zero. It is considered undefined.

Consider the two situations:

For the inverse of multiplication 0/0 = a, a could be any number to satisfy a x 0 = 0. At the same time, division of any nonzero number, a/0 = b, there is no number a such that b x 0 = a.

---

In nearly every known algebraic structure, 0/0 is an undefinable term. This means that, based on the rules that govern most of the mathematical systems we use, there isn't just one, single, definable value for the term 0/0, and believe it or not, the reason for this isn't because we're dividing by zero, it's because the division relation is defined by another relation, multiplication. You see, when we talk about "divide," what we really mean is "multiply by the inverse." For example, x/y actually means, x*y-1 where y-1 is the inverse of y. The inverse of a number is defined to be the number which, when multiplied by the original number, equals one; e.g. x*x-1 = 1. Now, in the algebraic structures we're all familiar with, any number multiplied by zero is defined to be equal to zero; e.g. 0*x = 0. So, using these definitions, what does dividing by zero, which actually means multiplying by the inverse of zero, equal? In other words, x*0-1 = ? Well, to isolate x, you would need to cancel out 0-1, but how? As anyone who's taken any sort of algebra knows, the method of isolation in these cases would be to multiply 0-1 by 0 because, as stated above, x*x-1 = 1, therefore 0*0-1 = 1. But wait, didn't I also just say that 0*x = 0? That would mean that 0*0-1 = 0, which would mean that 0 = 1. That, my friends, is called a contradiction. Zero does not equal one; therefore the term 0-1 can't be defined.

This answer may seem unsatisfactory to some people. There's got to be a way to work around this pesky contradiction, right? Actually, there is! In the branch of mathematics called abstract algebra, there exists an algebraic structure called a wheelwhich is required to have division defined everywhere within it. Therefore, in this particular algebraic structure, 0/0 must exist or else the structure isn't a wheel.

But wait, 0/0 is undefined, right? How could you ever satisfy this requirement for a wheel then?

That's easy; all you have to do is define it! Specifically, you give this quantity, 0/0, some specific algebraic properties, and then, if it ever comes up in an equation, you manipulate it within the equation using the properties you've given it. Isn't that convenient?!

"Preposterous!" you may say. "You can't simply make something up which has no tangible or rational analogue, that's cheating!" Well my dear skeptic, may I direct your attention to the following little marvel, √(-1), otherwise known as "the imaginary number," or i. That's right, I said imaginary, as in, "doesn't exist." You see, nothing multiplied by itself in our nice little world of mathematical rationality can possibly be a negative number. Unless, of course, you define something to be as such. Then...Presto! The absurd is now reality!

Let's talk about imaginary numbers for a moment. Our newly defined yet still rather imaginary friend, i, was apparently not content on simply having a nice, comfy little existence within the realm of obscure mathematics, oh no no no. It decided to defy logic and become a fairly common number; popping up all over the place, even in (you're going to love this) actual, real-life applications. For example, anyone who's ever done some form of electromagnetic wave analysis, through the fields of engineering, physics, etc., LOVES i and will gladly bow down and kiss its feet upon command (God bless ei(ωt-kr)). Why? Because of the very thoughtful relation that it's given to us known as "Euler's formula:" eiθ = cos(θ) + isin(θ). Step back a minute and look at that. The irrational, real number, e (2.71828...) exponentiated to the product of a real number, θ, and the imaginary number, i, is equal to a simple trigonometric expression involving two basic functions. In fact (you may want to sit down for this), if the value for θ happens to be π (3.14159...), another irrational, real number mind you, the trigonometric expression on the right hand side of Euler's formula reduces to exactly -1. Let's write that out: eiπ = -1. We call that "Euler's identity," although it should really be called, "THE MOST INCREDIBLE MATHEMATICAL EXPRESSION, EVER!"

But enough about i, let's get back to our newest friend, 0/0. As stated earlier, the problem with 0/0 isn't the fact that we're dividing by zero, it's the fact that the division relation is defined by multiplication. Well, how do we fix that? Simple! Change the definition of divide! Instead of x/y = x*y-1, it's now going to equal x*/y, where "/" is defined as a unary operation analogous to the reciprocal operation.

OK, another quick aside. A unary operator is an operator that only needs one input to work. For example, you only need one number to perform the operation of negation. For instance, negating the number 1 is simply -1. This is opposed to a binaryoperator. Binary operators include many of the guys we're all familiar with; like addition, multiplication, subtraction, etc. To make this clearer, consider the addition operation. It would make no sense to write 1 +. You need another number after the "+" to satisfy the operation; 1 + 2, for example, hence the term binary.

So, with our trusty new unary operator "/" in hand, we're going to look at the number 0/0 again. 0/0 is no longer defined as 0*0-1 like it was before. Now, it's defined as 0*/0, and in our world, not only does /0 ≠ 0-1, but 0*x doesn't have to equal 0 either. Isn't abstraction fun?! Ok, so 0/0 is officially defined, now let's give it some properties!

How about, x + 0/0 = 0/0 and x*0/0 = 0/0. Awesome! Why not go ahead and make a more general rule as well: (x + 0y)z = xz + 0y. OK! Well, we're certainly off to a good start, I'd say. I'll leave the complete derivation of the algebraic structure known as the wheel to the experts, please see the corresponding link below.

I'll end this answer with a final note for those who think that this entire concept of "defining the undefined" is ridiculous. Consider the following sets of numbers: The prime numbers, P; the set of all real numbers with exactly two natural number factors.

The natural numbers, N; the set of all integers greater than or equal to 0.

The integers, Z; the set of all real numbers without remainders or decimals.

The rational numbers, Q; the set of all real numbers that can be expressed as an integer divided by a non-zero integer.

The irrational numbers, I; the set of all real numbers that aren't rational.

Now consider this:

The imaginary number, i, is undefined in I.

The ratio pi, or π (3.14159...), is undefined in Q.

The common fraction 1/2 is undefined in Z.

All of the negative numbers, including -1, are undefined in N.

The number 4 is undefined in P.

Yet, these "undefined" numbers are hardly mysterious to us. We just broadened our definition of definable to include the "undefined" ones, and life became good again. 0/0 is not quite, but nearly, the same idea.

--------------------------------------------------------

I once asked one of my professor lecturers at University this and his answer was any value you want (or need).

0/0 is used as a limit in Calculus.

Consider any curve y = f(x)

Take a point (x, f(x)) on that curve.

The slope of that point is the slope of the tangent at that point.

The slope of the tangent is close to the slope of a small chord between the point (x, y) = (x, f(x)) and a point a small distance h away (x+h, f(x+h)), which can be found by: m = (f(x+h) - f(x))/((x+h) - x) = (f(x+h) - f(x))/h

The smaller the value of h, the closer the chord is to the tangent and the closer the slope of the chord is to the slope of the tangent and thus the slope of the curve at that point.

As h tends towards 0, f(x+h) tends towards f(x) and the expression m = (f(x+h) - f(x))/h tends towards 0/0.

In other words, 0/0 is the limit of (f(x+h) - f(x))/h as h tends towards 0.

But as this chord tends towards the tangent at the point (x, f(x)) on the curve y = f(x), 0/0 must be the slope of the tangent.

Clearly not every point of a non-linear curve has the same slope, thus 0/0 is any value you want (or need).

As the chord tends towards having zero length (when h = 0), (f(x+h) - f(x))/h will tend towards a constant value, a limit, which is the slope of the tangent.

The "trick" that calculus uses is that as h never reaches 0 but tends towards 0 it is possible to divide by h, and then see what happens when h becomes 0, ie when the original expression became 0/0, since (f(x+h) - f(x))/h = (f(x+0) - f(x))/0 = (f(x) - f(x))/0 = 0/0 when h = 0.

For example, take the curve y = x³ - 2x² + 5x + 3; what is the value of the slope of that line?

slope = lim{h→0} (f(x+h) = f(x))/h

= lim{h→0} ((x+h)³ - 2(x+h)² + 5(x+h) + 3 - (x³ - 2x² + 5x + 3))/h

Expanding the brackets:

= lim{h→0} (x³ + 3x²h +3xh² + h³ - 2x² - 4xh - 2h² + 5x + 5h + 3 - x³ + 2x² - 5x - 3)/h

Simplifying:

= lim{h→0} (3x²h +3xh² + h³ - 4xh - 2h² + 5)/h

Since h ≠ 0, it is possible to divide by h:

= lim{h→0} 3x² +3xh + h² - 4x - 2h + 5

Now the limit can be found by letting h = 0:

= 3x² - 4x + 5

Thus the slope of y = x³ - 2x² + 5x + 3 is given by m = 3x² - 4x + 5 at any value for x.

This value m, which is normally written as f'(x) is the first derivative of f(x), also written as dy/dx.

The slope of any line y = f(x) is given by y = f'(x).

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Mathematical Constants
Atomic Mass

How do calculate atomic mass?

Number of protons plus the number of neutrons (electrons don't matter as they have such a small mass).

163164165
Algebra
Numbers
Mathematical Constants

How do you use decimal numbers in real life?

One situation is when shopping for grocery items ... things will be priced using the decimal system: Tomato's, $1.59 per pound. Knowing that 1.59 is less than 1.75 could be beneficial.

155156157
Math and Arithmetic
Numbers
Mathematical Constants

How many zeros are in 1 trillion?

The "short scale" used in the US, the UK, and elsewhere assigns number names (thousand, million, billion, trillion, quadrillion) by thousands.

On the short scale, there are 12 zeros in a trillion(1012).

It is 1,000,000,000,000.

The traditional long scale, still used in some areas (Europe, Latin America), advances number names by millions, with intermediate names for the thousands (milliard, billiard, trilliard). On the long scale, a billion is a million million, and a trillion is a million billion.

So on the long scale, there are 18 zeros in a trillion (1018).

It is 1,000,000,000,000,000,000.

131132133
Brain Teasers and Logic Puzzles
Mathematical Constants

What number comes after 999?

1000.

8910
Math and Arithmetic
Business Accounting and Bookkeeping
Definitions
Mathematical Constants

What is the definition of labor ratios?

cost relationship between direct and indirect.

151152153
Algebra
Geometry
Numbers
Mathematical Constants

What are the first 10 numbers in pi?

π (pi) = 3.141592653

596061
Math and Arithmetic
Numbers
Mathematical Constants

How much is a googol?

10^100 = The number 1 followed by 100 Zeros

135136137
Math and Arithmetic
Numbers
Mathematical Constants

Who many zero does a googol have?

It is called gogool, and it has 100 zeroes.

131132133
Mathematical Constants
Atomic Mass

How do you calculate effective atomic mass?

Aeff=(Zeff)/((Z/A)eff)

Zeff=wi*Zi

wi is weight fraction in ith element in compound. for example water has 0.112 H and 0.888 O.

(Z/A)eff=wi*Zi/Ai

123124125
Math and Arithmetic
Irrational Numbers
Mathematical Constants

What is the square root of pi?

.3333333

828384
Mathematical Constants

What is 0.00004 in standard form?

0.00004

001

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