'e' in javascript numbers - javascript

I need to understand the following:
when I type 4e4 in Google Chrome's console it returns 40000.
Can anyone help me to understand what is e in javascript numbers and what is the algorithm working for this?
Thanks in advance

4e4 is a floating-point number representation.
It consists of:
Sign - S(+ or -)
Mantissa - M(some number, normalized: 1.x where x is some sequence of digits)
Exponent - E(represents a power of 10 that is Mantissa(M) multiplied with)
It is also a way of how floating-point numbers are stored on the system. For instance, for single-precision we get:
single-precision floating-point number representation
Together, it gives us:
-1^S * M * p^E where p is the basis of the numerical system
So, in common sense, p can be anything so that 4e4 could be also 4 * 5^4 if p == 5
As we usually work with decimal values p is equal to 10
And as was answered before, 4e4 == 4 * 10^4 (as 4 is a decimal value in this case)

'e' in a number like that is the same as 'times 10 to the power of'
3.2e6 is 3.2x10^6

4*10^4
If the number was 4e5 this would be equal to 4*10^5
etc.

Related

How to get Power of big number in decimal?

How to get big power of 2 in decimal or
how to convert big exponential value into decimal value.
I want 2 to the power of 128 in decimal not exponential
what I did till now
tofixed(+exponent)
which again given me the same value.
var num = Math.pow(2, 128);
Actual result = 3.402823669209385e+38
expected some decimal value not exponential value.
You could use BigInt, if implemented.
var num = BigInt(2) ** BigInt(128);
console.log(num.toString());
console.log(BigInt(2 ** 128).toString());
3.402823669209385e+38 is a decimal number (in string form, because it's been output as a string). It's in scientific notation, specifically E-notation. It's the number 3.402823669209385 times 100000000000000000000000000000000000000.
If you want a string that isn't in scientific notation, you can use Intl.NumberFormat for that:
console.log(new Intl.NumberFormat().format(Math.pow(2, 128)));
Note: Although that number is well outside the range that JavaScript's number type can represent with precision in general (any integer above Number.MAX_SAFE_INTEGER [9,007,199,254,740,991] may be the result of rounding), it's one of the values that is held precisely, even at that magnitude, because it's a power of 2. But operations on it that would have a true mathematical result that wasn't a power of 2 would almost certainly get rounded.
I think the default power function won't be able to the results you want.
You can refer to the article below to understand how to create an Power function with big number by yourself.
Demo code is not JS but still quite understandable.
↓
Writing power function for large numbers

Math.pow gives wrong result

I was trying to repeat a character N times, and came across the Math.pow function.
But when I use it in the console, the results don't make any sense to me:
Math.pow(10,15) - 1 provides the correct result 999999999999999
But why does Math.pow(10,16) - 1 provide 10000000000000000?
You are producing results which exceed the Number.MAX_SAFE_INTEGER value, and so they are not accurate any more up to the unit.
This is related to the fact that JavaScript uses 64-bit floating point representation for numbers, and so in practice you only have about 16 (decimal) digits of precision.
Since the introduction of BigInt in EcmaScript, you can get an accurate result with that data type, although it cannot be used in combination with Math.pow. Instead you can use the ** operator.
See how the use of number and bigint (with the n suffix) differ:
10 ** 16 - 1 // == 10000000000000000
10n ** 16n - 1n // == 9999999999999999n
Unlike many other programming languages, JavaScript does not define different types of numbers, like integers, short, long, floating-point etc.
JavaScript numbers are always stored as double-precision floating-point numbers, following the international IEEE 754 standard.
This format stores numbers in 64 bits, where the number (the fraction) is stored in bits 0 to 51, the exponent in bits 52 to 62, and the sign-in bit 63:
Integers (numbers without a period or exponent notation) are accurate up to 15 digits.

Javascript scientific notation floats vs integers explained

I am working with js numbers and have lack of experience in that. So, I would like to ask few questions:
2.2932600144518896
e+160
is this float or integer number? If it's float how can I round it to two decimals (to get 2.29)? and if it's integer, I suppose it's very large number, and I have another problem than.
Thanks
Technically, as said in comments, this is a Number.
What you can do if you want the number (not its string representation):
var x = 2.2932600144518896e+160;
var magnitude = Math.floor(Math.log10(x)) + 1;
console.log(Math.round(x / Math.pow(10, magnitude - 3)) * Math.pow(10, magnitude - 3));
What's the problem with that? Floating point operation may not be precise, so some "number" different than 0 should appear.
To have this number really "rounded", you can only achieve it through string (than you can't make any operation).
JavaScript only has one Number type so is technically neither a float or an integer.
However this isn't really relevant as the value (or rather representation of it) is not specific to JavaScript and uses E-Notation which is a standard way to write very large/small numbers.
Taking this in to account 2.2932600144518896e+160 is equivalent to 2.2932600144518896 * Math.pow(10,160) and approximately 229 followed by 158 zeroes i.e. very flippin' big.

Math.pow() not giving precise answer

I've used Math.pow() to calculate the exponential value in my project.
Now, For specific values like Math.pow(3,40), it returns 12157665459056929000.
But when i tried the same value using a scientific Calculator, it returns 12157665459056928801.
Then i tried to traverse the loop till the exponential value :
function calculateExpo(base,power){
base = parseInt(base);
power = parseInt(power);
var output = 1;
gameObj.OutPutString = ''; //base + '^' + power + ' = ';
for(var i=0;i<power;i++){
output *= base;
gameObj.OutPutString += base + ' x ';
}
// to remove the last comma
gameObj.OutPutString = gameObj.OutPutString.substring(0,gameObj.OutPutString.lastIndexOf('x'));
gameObj.OutPutString += ' = ' + output;
return output;
}
This also returns 12157665459056929000.
Is there any restriction to Int type in JS ?
This behavior is highly dependent on the platform you are running this code at. Interestingly even the browser matters even on the same very machine.
<script>
document.write(Math.pow(3,40));
</script>
On my 64-bit machine Here are the results:
IE11: 12157665459056928000
FF25: 12157665459056929000
CH31: 12157665459056929000
SAFARI: 12157665459056929000
52 bits of JavaScript's 64-bit double-precision number values are used to store the "fraction" part of a number (the main part of the calculations performed), while 11 bits are used to store the "exponent" (basically, the position of the decimal point), and the 64th bit is used for the sign. (Update: see this illustration: http://en.wikipedia.org/wiki/File:IEEE_754_Double_Floating_Point_Format.svg)
There are slightly more than 63 bits worth of significant figures in the base-two expansion of 3^40 (63.3985... in a continuous sense, and 64 in a discrete sense), so hence it cannot be accurately computed using Math.pow(3, 40) in JavaScript. Only numbers with 52 or fewer significant figures in their base-two expansion (and a similar restriction on their order of magnitude fitting within 11 bits) have a chance to be represented accurately by a double-precision floating point value.
Take note that how large the number is does not matter as much as how many significant figures are used to represent it in base two. There are many numbers as large or larger than 3^40 which can be represented accurately by JavaScript's 64-bit double-precision number values.
Note:
3^40 = 1010100010111000101101000101001000101001000111111110100000100001 (base two)
(The length of the largest substring beginning and ending with a 1 is the number of base-two significant figures, which in this case is the entire string of 64 digits.)
Haskell (ghci) gives
Prelude> 3^40
12157665459056928801
Erlang gives
1> io:format("~f~n", [math:pow(3,40)]).
12157665459056929000.000000
2> io:format("~p~n", [crypto:mod_exp(3,40,trunc(math:pow(10,21)))]).
12157665459056928801
JavaScript
> Math.pow(3,40)
12157665459056929000
You get 12157665459056929000 because it uses IEEE floating point for computation. You get 12157665459056928801 because it uses arbitrary precision (bignum) for computation.
JavaScript can only represent distinct integers to 253 (or ~16 significant digits). This is because all JavaScript numbers have an internal representation of IEEE-754 base-2 doubles.
As a consequence, the result from Math.pow (even if was accurate internally) is brutally "rounded" such that the result is still a JavaScript integer (as it is defined to return an integer per the specification) - and the resulting number is thus not the correct value, but the closest integer approximation of it JavaScript can handle.
I have put underscores above the digits that don't [entirely] make the "significant digit" cutoff so it can be see how this would affect the results.
................____
12157665459056928801 - correct value
12157665459056929000 - closest JavaScript integer
Another way to see this is to run the following (which results in true):
12157665459056928801 == 12157665459056929000
From the The Number Type section in the specification:
Note that all the positive and negative integers whose magnitude is no greater than 253 are representable in the Number type ..
.. but not all integers with large magnitudes are representable.
The only way to handle this situation in JavaScript (such that information is not lost) is to use an external number encoding and pow function. There are a few different options mentioned in https://stackoverflow.com/questions/287744/good-open-source-javascript-math-library-for-floating-point-operations and Is there a decimal math library for JavaScript?
For instance, with big.js, the code might look like this fiddle:
var z = new Big(3)
var r = z.pow(40)
var str = r.toString()
// str === "12157665459056928801"
Can't say I know for sure, but this does look like a range problem.
I believe it is common for mathematics libraries to implement exponentiation using logarithms. This requires that both values are turned into floats and thus the result is also technically a float. This is most telling when I ask MySQL to do the same calculation:
> select pow(3, 40);
+-----------------------+
| pow(3, 40) |
+-----------------------+
| 1.2157665459056929e19 |
+-----------------------+
It might be a courtesy that you are actually getting back a large integer.

Math.random and arithmetic shift

If I have the following code in JavaScript:
var index1 = (Math.random() * 6) >> 0;
var index2 = Math.floor(Math.random() * 6);
The results for index1 or index2 are anywhere between 0 and 6.
I must be confused with my understanding of the >> operator. I thought that by using arithmetic shift that the results for index1 would be anywhere between 1 and 6.
I am noticing, however that I don't need to use Math.floor() or Math.round() for index1 if I use the >> operator.
I know I can achieve this by adding 1 to both indexes, but I was hoping there was a better way of ensuring results are from 1 to 6 instead of adding 1.
I'm aware that bitwise operators treat their operands as a sequence of 32 bits (zeros and ones), rather than as decimal, hexadecimal, or octal numbers. For example, the decimal number nine has a binary representation of 1001. Bitwise operators perform their operations on such binary representations, but they return standard JavaScript numerical values.
UPDATE:
I saw the original usage in this CAAT tutorial on line 26 and was wondering whether that would actually return a random number between 1 and 6 and it seems it would only ever return a random number between 0 and 6. So you would never actually see the anim1.png fish image!
Thank you in advance for any enlightenment.
Wikipedia says '(Arithmetic right shifts for negative numbers would be equivalent to division using rounding towards 0 in one's complement representation of signed numbers as was used by some historic computers.)'
Not exactly an answer, but the idea is that >> 0 is really specific and shouldn't be used in general for getting a range between 1 and 6.
Most people would tell you to do
Math.floor((Math.random()*10)+1);

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