Java supports integer and floating-point arithmetic directly in the
language. Higher-level math operations are supported through the
`java.lang.Math`

class. As
you may have seen by now, wrapper classes for primitive data types allow
you to treat them as objects. Wrapper classes also hold some methods for
basic conversions.

First, a few words about built-in arithmetic in Java. Java handles
errors in integer arithmetic by throwing an `ArithmeticException`

:

`int`

`zero`

`=`

`0`

`;`

`try`

`{`

`int`

`i`

`=`

`72`

`/`

`zero`

`;`

`}`

`catch`

`(`

`ArithmeticException`

`e`

`)`

`{`

`// division by zero`

`}`

To generate the error in this example, we created the intermediate
variable `zero`

. The compiler is somewhat
crafty and would have caught us if we had blatantly tried to perform
division by a literal zero.

Floating-point arithmetic expressions, on the other hand, don’t throw exceptions. Instead, they take on the special out-of-range values shown in Table 11-1.

The following example generates an infinite result:

`double`

`zero`

`=`

`0.0`

`;`

`double`

`d`

`=`

`1.0`

`/`

`zero`

`;`

`if`

`(`

`d`

`==`

`Double`

`.`

`POSITIVE_INFINITY`

`)`

`System`

`.`

`out`

`.`

`println`

`(`

`"Division by zero"`

`);`

The special value `NaN`

(not a
number) indicates the result of dividing zero by zero. This value has the
special mathematical distinction of not being equal to itself (`NaN != NaN`

evaluates to `true`

). Use `Float.isNaN()`

or `Double.isNaN()`

to test for `NaN`

.

The `java.lang.Math`

class is Java’s math library. It holds a suite of static methods
covering all of the usual mathematical operations like `sin()`

, `cos()`

, and `sqrt()`

. The `Math`

class isn’t very object-oriented (you
can’t create an instance of `Math`

).
Instead, it’s really just a convenient holder for static methods that
are more like global functions. As we saw in Chapter 6, it’s possible to use the static import
functionality to import the names of static methods and constants like
this directly into the scope of our class and use them by their simple,
unqualified names.

Table 11-2 summarizes the
methods in `java.lang.Math`

.

Table 11-2. Methods in java.lang.Math

Method | Argument type(s) | Functionality |
---|---|---|

| Absolute value | |

| Arc cosine | |

| Arc sine | |

| Arc tangent | |

| | Angle part of rectangular-to-polar coordinate transform |

| Smallest whole number greater than or
equal to | |

| Cube root of | |

| | Cosine |

| Hyperbolic cosine | |

| | |

| Largest whole number less than or
equal to | |

| Precision calculation of the | |

| | Natural logarithm of |

| | Log base 10 of |

| | The value |

| The value | |

| | |

| Random-number generator | |

| Converts double value to integral value in double format | |

| Rounds to whole number | |

| Get the sign of the number at 1.0, –1.0, or 0 | |

| Sine | |

| | Hyperbolic sine |

| Square root | |

| Tangent | |

| | Hyperbolic tangent |

| Convert radians to degrees | |

| Convert degrees to radians |

`log()`

, `pow()`

, and `sqrt()`

can throw a runtime `ArithmeticException`

.
`abs()`

, `max()`

, and `min()`

are overloaded for all the scalar
values, `int`

, `long`

, `float`

, or `double`

, and return the corresponding type.
Versions of `Math.round()`

accept
either `float`

or `double`

and return `int`

or `long`

, respectively. The rest of the methods
operate on and return `double`

values:

`double`

`irrational`

`=`

`Math`

`.`

`sqrt`

`(`

`2.0`

`);`

`// 1.414...`

`int`

`bigger`

`=`

`Math`

`.`

`max`

`(`

`3`

`,`

`4`

`);`

`// 4`

`long`

`one`

`=`

`Math`

`.`

`round`

`(`

`1.125798`

`);`

`// 1`

For convenience, `Math`

also
contains the static final double values `E`

and `PI`

:

`double`

`circumference`

`=`

`diameter`

`*`

`Math`

`.`

`PI`

`;`

If the `long`

and
`double`

types are not large or precise
enough for you, the `java.math`

package
provides two classes, `BigInteger`

and
`BigDecimal`

, that
support arbitrary-precision numbers. These full-featured classes have a
bevy of methods for performing arbitrary-precision math and precisely
controlling rounding of remainders. In the following example, we use
`BigDecimal`

to add two very large
numbers and then create a fraction with a 100-digit result:

`long`

`l1`

`=`

`9223372036854775807L`

`;`

`// Long.MAX_VALUE`

`long`

`l2`

`=`

`9223372036854775807L`

`;`

`System`

`.`

`out`

`.`

`println`

`(`

`l1`

`+`

`l2`

`);`

`// -2 ! Not good.`

`try`

`{`

`BigDecimal`

`bd1`

`=`

`new`

`BigDecimal`

`(`

`"9223372036854775807"`

`);`

`BigDecimal`

`bd2`

`=`

`new`

`BigDecimal`

`(`

`9223372036854775807L`

`);`

`System`

`.`

`out`

`.`

`println`

`(`

`bd1`

`.`

`add`

`(`

`bd2`

`)`

`);`

`// 18446744073709551614`

`BigDecimal`

`numerator`

`=`

`new`

`BigDecimal`

`(`

`1`

`);`

`BigDecimal`

`denominator`

`=`

`new`

`BigDecimal`

`(`

`3`

`);`

`BigDecimal`

`fraction`

`=`

`numerator`

`.`

`divide`

`(`

`denominator`

`,`

`100`

`,`

`BigDecimal`

`.`

`ROUND_UP`

`);`

`// 100 digit fraction = 0.333333 ... 3334`

`}`

`catch`

`(`

`NumberFormatException`

`nfe`

`)`

`{`

`}`

`catch`

`(`

`ArithmeticException`

`ae`

`)`

`{`

`}`

If you implement cryptographic or scientific algorithms for fun,
`BigInteger`

is crucial. Other than
that, you’re not likely to need these classes.

As we mentioned in Chapter 4, Java uses the IEEE 754 standard to
represent floating-point numbers (float and double types) internally.
Those of you familiar with how floating-point math works will already
know that “decimal” numbers are represented in binary in this standard
by separating the number into three components: a sign (positive or
negative), an exponent representing the *magnitude* in powers of 2 of the
number, and a *mantissa* using up most of the bits to
represent the precise value irrespective of its magnitude. While for
most applications the precision of float and double-type floating-point
numbers is sufficient enough that we don’t need to worry about running
into limitations, there are times when specialized apps may wish to work
with the floating-point values more directly.

By definition, floating-point numbers trade off precision and
scale. Even the smallest Java floating-point type, `float`

, can represent
(literally) astronomical numbers ranging from negative
10^{–45} to positive
10^{38}. This is accomplished, put in decimal
terms, by having the mantissa part of the floating-point value represent
a fixed number of “digits” and the exponent tell us where to put the
decimal point. As the numbers get larger in magnitude, the “precision”
therefore gets shifted to the “left” as more digits appear to the left
of the decimal point. What this means is that floating-point numbers can
very precisly (with a large number of digits) represent small values
like pi, but for bigger numbers (in the billions and trillions)
those digits will be taken up with the more signifcant digits.
Therefore, the gap between any two consecutive numbers that can be
represented by a floating-point value grows larger as the numbers get
bigger.

For some applications, knowing the limitations may be important.
The `java.lang.Math`

class therefore
provides a few methods for interrogating floats and doubles about their
precision. The `Math.ulp()`

method
retrieves the “unit of least precision” for a given floating-point
number, which is the smallest value that bits in the mantissa represent
at their current exponent. Another way to say this is that the `ulp()`

is the approximate distance from the
floating-point number to the next closest higher or lower floating-point
number that can be represented. Adding positive values smaller than half
the ULP to a float will not yield a new number. Adding values between
half and the full ULP will result in the value plus the ULP. The
`Math.nextUp()`

method is
a convenience that will take a float and tell you the next number that
can be represented by adding the ULP.

`float`

`trillionish`

`=`

`(`

`float`

`)`

`1`

`e12`

`;`

`// trillionish ~= 999,999,995,904`

`float`

`ulp`

`=`

`Math`

`.`

`ulp`

`(`

`f`

`);`

`// ulp = 65536`

`float`

`next`

`=`

`Math`

`.`

`nextUp`

`(`

`f`

`);`

`// next ~= 1000000061440`

`trillionish`

`+=`

`32767`

`;`

`// trillionish still ~= 999,999,995,904. No change!`

Additionally, the `java.lang.Math`

class contains the method
`getExponent()`

, which
retrieves the exponent part of a floating-point number (and from there
one could determine the mantissa by division). It is also possible to
get the raw bits of a float or double using their corresponding wrapper
class methods `floatToIntBits()`

and
`doubleTo`

`RawLongBits()`

and pick out the (IEEE
standard) bits yourself.

You can use the `java.util.Random`

class
to generate random values. It’s a pseudorandom-number generator that is
initialized with a 48-bit seed.^{[31]} Because it’s a pseudorandom algorithm, you’ll get the same
series of values every time you use the same seed value. The default
constructor uses the current time to produce a seed, but you can specify
your own value in the constructor:

`long`

`seed`

`=`

`mySeed`

`;`

`Random`

`rnums`

`=`

`new`

`Random`

`(`

`seed`

`);`

After you have a generator, you can ask for one or more random values of various types using the methods listed in Table 11-3.

Table 11-3. Random-number methods

Method | Range |
---|---|

| |

–2147483648 to 2147483647 | |

| 0 to (n – 1) inclusive |

–9223372036854775808 to 9223372036854775807 | |

0.0 inclusive to 1.0 exclusive | |

0.0 inclusive to 1.0 exclusive | |

Gaussian distributed double with mean 0.0 and standard deviation of 1.0 |

By default, the values are uniformly distributed. You can use the
`nextGaussian()`

method to create a
Gaussian (bell curve) distribution of `double`

values, with a mean of 0.0 and a
standard deviation of 1.0. (Lots of natural phenomena follow a Gaussian
distribution rather than a strictly uniform random one.)

The `static`

method `Math.random()`

retrieves a random `double`

value. This method initializes a
`private`

random-number generator in
the `Math`

class the first time it’s
used, using the default `Random`

constructor. Thus, every call to `Math.random()`

corresponds to a call to
`nextDouble()`

on that random-number
generator.

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