What is the highest number this javascript expression can evaluate to? What is the lowest number? Why?

+(''+Math.random()).substring(2)

Extra credit: How many different values can the expression evaluate to? Can it be every value from the minimum to the maximum, or are some intermediate values not obtainable due to rounding issues?

Response to Daniel's answer (deleted, was 10000000000000000 max, 0 min):

I was playing around in Chrome's console and got this:

    Math.random();

>> 0.00012365682050585747

    '12365682050585747'.length

>> 17

    12365682050585747 > 10000000000000000

>> true

... so 10000000000000000 can't be the max!

What is the highest number this javascript expression can evaluate to? What is the lowest number? Why?

+(''+Math.random()).substring(2)

Extra credit: How many different values can the expression evaluate to? Can it be every value from the minimum to the maximum, or are some intermediate values not obtainable due to rounding issues?

Response to Daniel's answer (deleted, was 10000000000000000 max, 0 min):

I was playing around in Chrome's console and got this:

    Math.random();

>> 0.00012365682050585747

    '12365682050585747'.length

>> 17

    12365682050585747 > 10000000000000000

>> true

... so 10000000000000000 can't be the max!

• javascript
• floating-point

• @no: I guess it depends on what the precision of numbers returned by Math.random is. I thought it was 16 or 17 digits. I don't even know anymore :(. How'd you get this question by the way? – Cristian Sanchez Commented Sep 22, 2010 at 2:31
• @Daniel: it just popped into my head... I use something like that (with a letter prepended) to create random id attributes for temporary dom elements sometimes, and I wondered how big the numbers could get. – Dagg Nabbit Commented Sep 22, 2010 at 3:09
• Depends on the precision of the float number. Although you can see 0.00012365682050585747 with 20 decimal places, is it possible you could get 0.00012365682050585747 + 0.00000000000000000001? Read more here: en.wikipedia/wiki/… – eumiro Commented Sep 22, 2010 at 10:13
• This all depends on how the implementation does its rounding – Josh Stodola Commented Sep 22, 2010 at 20:01

2 Answers 2

It depends on how the random number is generated, and how the number will be converted to string. The ECMAScript spec doesn't specify both of these.

In practice, the number will have at most 17 significant figures, so the maximum should be at most 10¹⁷.

The spec does specify that a number will be displayed in decimal form (instead of scientific form) when the exponent is between -6 and 20 (10^-6 ≤ x < 10²¹), so we just need to restrict our attention on numbers in [10^-6, 1) when trying to seek the maximum exhaustively.

However, in this range a number must be representable as s × 2^e, where 1 ≤ s ≤ 2 − 2^-52 with a precision of Δs = 2^-52 and -20 ≤ e ≤ -1. The spec remends that ToNumber(ToString(x)) == x, so the number should be precise down to 2^-52+e for a given e. Thus the "17-digit" number with (2 − n × 2^-52) × 2^e with the smallest n will be the biggest number representable with a given e, after chopping the initial 0..

             v                 
(-20) 0.0000019073486328124998
(-19) 0.0000038146972656249996
(-18) 0.0000076293945312499975 (n=3)
(-17)  0.000015258789062499998
(-16)  0.000030517578124999997
(-15)  0.000061035156249999986 (n=2)
(-14)   0.00012207031249999999
(-13)   0.00024414062499999997
(-12)   0.00048828124999999995
(-11)    0.0009765624999999999 (always 16-digit?)
(-10)    0.0019531249999999998
(-9)     0.0039062499999999996
(-8)     0.0078124999999999965 (n=4)
(-7)      0.015624999999999998
(-6)      0.031249999999999997
(-5)      0.062499999999999986 (n=2)
(-4)       0.12499999999999999
(-3)       0.24999999999999997
(-2)       0.49999999999999994
(-1)        0.9999999999999999 (always 16-digit?)

From here we know that the absolute maximum is 78,124,999,999,999,965.

Math.random() can return any nonnegative numbers in the interval [0, 1), so the safe minimum is -324 from 5e-324 (the smallest subnormal number in double precision is 4.94 × 10^-324).

For me the highest number is 1/0 (===Infinity) and the lowest obviously -1/0 (in Chromium browser).

Edit: You can also try parse a number from string to see which evaluates to Infinity.

var a = "1";
while(parseInt(a)!==Infinity) a=a+"0";
alert("Length of the highest number is: " + (a.length-1));

309 for me.