F = 0.00000000006674 N·m²/kg² × 5,972,000,000,000,000,000,000,000 kg × 73,480,000,000,000,000,000,000 kg / (384,400,000 m)². Suppose that you've taken up astronomy recently and would like to know the gravitational force acting between the Earth and the Moon. For the calculations, we need the masses of the two objects (denote the Earth's by M₁ and the Moon's by M₂) and the distance between them (denoted by R). We have:

and the circumference is... actually, the 40,075 km doesn't look that bad, does it? Well, we could use a length converter and change it to 4.0075 × 10⁴ km, but is it better that way? If we needed to change it to millimeters, then maybe it'd be a better idea, but the kilometer form seems perfectly usable. As its name suggests, it is the most frequently used form of logarithm. It is used, for example, in our decibel calculator. Logarithm tables that aimed at easing computation in the olden times usually presented common logarithms, too. You may notice that even though the frequency of compounding reaches an unusually high number, the value of (1 + r/m)ᵐ (which is the multiplier of your initial deposit) doesn't increase very much. Instead, it becomes somewhat stable: it's approaching a unique value already mentioned above, e ≈ 2.718281. The new computational procedure was instrumental in the field of astronomy. Napier's scientific activities coincided with the era of new developments in astrophysics. As a result, many astronomers were struggling with endless calculations to detect the position of the planets using Copernicus's theory of the solar system. Johannes Kepler, at the time working on his famous laws of planetary motions, was among them.the absolute value of n tells us how many places we have to move the point, and the sign of n indicates if it should be to the right (for n positive) or the left (for n negative). Therefore, converting to standard form is all about choosing the power of 10 in such a way that the b in the formula is between 1 and 10. We said that the number b should be between 1 and 10. This means that, for example, 1.36 × 10⁷ or 9.81 × 10⁻²³ are in standard form, but 13.1 × 10¹² isn't because 13.1 is bigger than 10. We could, however, convert it to standard form by saying that: This time, we indeed see the digits as the first factors in each multiplication. Moreover, the second factors have a lot in common - they consist of a single 1 with some zeros (possibly none). Don't ask us how they found the mass of the Earth, as there isn't any scale big enough to weigh the entire planet. As for the circumference, talk to Eratosthenes. It might seem artificial to write a sum of the products, like 1×100 or 4×1, but that's just what the expanded form is.

There is a valuable lesson here: writing numbers in standard form is not always the way to go. It's all about simplicity of notation, but, at the end of the day, it pretty much boils down to a matter of personal preference (or your teacher's if you're writing a test).Welcome to the standard form calculator, where we'll learn how to write a number in standard form. "What is the standard form?" Well, we'll get to the standard form definition soon enough. But let's just say that standard form in math and physics (quite often called scientific notation) is a neat way of dealing with very large or very small values. It's quite troublesome to write all the zeros of a number in every line of our calculations. Preferably, we can use standard form exponents and write the same thing with just a few symbols. That's why we made this standard form converter – to help you with just that. Now that we've seen how to write a number in standard form, it's time to convince you that it's a useful thing to do. Of course, we know that you're most probably learning all of this for the pure pleasure of grasping yet another part of theoretical mathematics, but it doesn't hurt to take a look at physics or chemistry from time to time. You know, those two minor branches of mathematics. Still, we might wish to decompose it even further. After all, we wanted to see the digits themselves (i.e., as one-digit numbers) and not some " complicated" expression like 0.07. Therefore, we can also write: Anyway, if scientists had to write all of those zeros every time they calculated something about our planet, they'd waste ages! It's much easier to recall how to write a number in standard form and say that the mass of Earth is, in fact, times 4.73 ≅ 10

lg ( 5.89 × 4.73 ) = lg ( 5.89 ) + lg ( 4.73 ) ≅ 0.770115 + 0.6748611 \text{lg}(5.89 \times 4.73) =\text{lg}(5.89) + \text{lg}(4.73) ≅ 0.770115 + 0.6748611 lg ( 5.89 × 4.73 ) = lg ( 5.89 ) + lg ( 4.73 ) ≅ 0.770115 + 0.6748611 patterns with an unfortunate bit of repetition and an outer let: let (count_str, item) = match (it.next(), it.next()) { A foot (symbol: ft) is a unit of length. It is equal to 0.3048 m, and used in the imperial system of units and United States customary units. The unit of foot derived from the human foot. It is subdivided into 12 inches. Frequently asked questions to convert 1.65 Meters into Feet If you want to compute a number's natural logarithm, you need to choose a base that is approximately equal to 2.718281. Conventionally this number is symbolized by e, named after Leonard Euler, who defined its value in 1731. Accordingly, the logarithm can be represented as logₑx, but traditionally it is denoted with the symbol ln(x). You might also see log(x), which also refers to the same function, especially in finance and economics. Therefore, y = logₑx = ln(x) which is equivalent to x = eʸ = exp(y).Now, this looks even worse than the previous example; it doesn't have commas in between! Thankfully, there are tools - like our standard form calculator - to make our lives easier. So, what is the standard form of the above numbers?