Clojure Math/sqrt precision on big integers

Question

Why does the following Clojure code print "true"?

(please notice that the numbers differ in the last digit)

(== (Math/sqrt 10252519345963644753026N)
    (Math/sqrt 10252519345963644753025N))

Not sure whether this question is only about Clojure or if it also applies to other languages (Java's BigInteger?).

Printing them results in:

(str (Math/sqrt 10252519345963644753026N) " "
     (Math/sqrt 10252519345963644753025N))

1.01254725055E11 1.01254725055E11

Answer 1

The Java Math.sqrt function

• takes a double argument and
• returns a doubleresult.

As various comments suggest, the conversions that this induces lose the precision to distinguish between the numbers. In fact, conversion to double does so:

(= (double 10252519345963644753026N)
   (double 10252519345963644753025N))
;true

---

A double has a 53 bit precision. Since 10 bits is about 3 decimal digits, this is about 16 decimal digits precision. Your numbers are 23 digits long, so the last few digits are lost in conversion.