Immutable, arbitrary-precision signed decimal numbers. A BigDecimal consists of an arbitrary precision integer unscaled value and a 32-bit integer scale. If the scale is zero or positive, the scale is the number of digits to the right of the decimal point. If the scale is negative, the unscaled value of the number is multiplied by ten to the power of the negation of the scale. The value of the number represented by the BigDecimal is therefore (unscaledValue × 10-scale).

The BigDecimal class provides operations for arithmetic, scale manipulation, rounding, comparison, hashing, and format conversion. The toString method provides a canonical representation of a BigDecimal.

The BigDecimal class gives its user complete control over rounding behavior. If no rounding mode is specified and the exact result cannot be represented, an ArithmeticException is thrown; otherwise, calculations can be carried out to a chosen precision and rounding mode by supplying an appropriate MathContext object to the operation. In either case, eight rounding modes are provided for the control of rounding. Using the integer fields in this class (such as ROUND_HALF_UP) to represent rounding mode is deprecated; the enumeration values of the RoundingMode enum, (such as RoundingMode#HALF_UP) should be used instead.

When a MathContext object is supplied with a precision setting of 0 (for example, MathContext#UNLIMITED), arithmetic operations are exact, as are the arithmetic methods which take no MathContext object. As a corollary of computing the exact result, the rounding mode setting of a MathContext object with a precision setting of 0 is not used and thus irrelevant. In the case of divide, the exact quotient could have an infinitely long decimal expansion; for example, 1 divided by 3. If the quotient has a nonterminating decimal expansion and the operation is specified to return an exact result, an ArithmeticException is thrown. Otherwise, the exact result of the division is returned, as done for other operations.

When the precision setting is not 0, the rules of BigDecimal arithmetic are broadly compatible with selected modes of operation of the arithmetic defined in ANSI X3.274-1996 and ANSI X3.274-1996/AM 1-2000 (section 7.4). Unlike those standards, BigDecimal includes many rounding modes. Any conflicts between these ANSI standards and the BigDecimal specification are resolved in favor of BigDecimal.

Since the same numerical value can have different representations (with different scales), the rules of arithmetic and rounding must specify both the numerical result and the scale used in the result's representation. The different representations of the same numerical value are called members of the same cohort. The natural order of BigDecimal considers members of the same cohort to be equal to each other. In contrast, the equals method requires both the numerical value and representation to be the same for equality to hold. The results of methods like scale() and unscaledValue() will differ for numerically equal values with different representations.

In general the rounding modes and precision setting determine how operations return results with a limited number of digits when the exact result has more digits (perhaps infinitely many in the case of division and square root) than the number of digits returned. First, the total number of digits to return is specified by the MathContext's precision setting; this determines the result's precision. The digit count starts from the leftmost nonzero digit of the exact result. The rounding mode determines how any discarded trailing digits affect the returned result.

For all arithmetic operators, the operation is carried out as though an exact intermediate result were first calculated and then rounded to the number of digits specified by the precision setting (if necessary), using the selected rounding mode. If the exact result is not returned, some digit positions of the exact result are discarded. When rounding increases the magnitude of the returned result, it is possible for a new digit position to be created by a carry propagating to a leading "9" digit. For example, rounding the value 999.9 to three digits rounding up would be numerically equal to one thousand, represented as 100×10¹. In such cases, the new "1" is the leading digit position of the returned result.

For methods and constructors with a MathContext parameter, if the result is inexact but the rounding mode is UNNECESSARY, an ArithmeticException will be thrown.

Besides a logical exact result, each arithmetic operation has a preferred scale for representing a result. The preferred scale for each operation is listed in the table below.

Preferred Scales for Results of Arithmetic Operations

Operation	Preferred Scale of Result
Add	max(addend.scale(), augend.scale())
Subtract	max(minuend.scale(), subtrahend.scale())
Multiply	multiplier.scale() + multiplicand.scale()
Divide	dividend.scale() - divisor.scale()
Square root	ceil(radicand.scale()/2.0)
nth root	ceil((double) radicand.scale()/n)

These scales are the ones used by the methods which return exact arithmetic results; except that an exact divide may have to use a larger scale since the exact result may have more digits. For example, 1/32 is 0.03125.

Before rounding, the scale of the logical exact intermediate result is the preferred scale for that operation. If the exact numerical result cannot be represented in precision digits, rounding selects the set of digits to return and the scale of the result is reduced from the scale of the intermediate result to the least scale which can represent the precision digits actually returned. If the exact result can be represented with at most precision digits, the representation of the result with the scale closest to the preferred scale is returned. In particular, an exactly representable quotient may be represented in fewer than precision digits by removing trailing zeros and decreasing the scale. For example, rounding to three digits using the floor rounding mode,
19/100 = 0.19 // integer=19, scale=2
but
21/110 = 0.190 // integer=190, scale=3

Note that for add, subtract, and multiply, the reduction in scale will equal the number of digit positions of the exact result which are discarded. If the rounding causes a carry propagation to create a new high-order digit position, an additional digit of the result is discarded than when no new digit position is created.

Other methods may have slightly different rounding semantics. For example, the result of the pow method using the specified algorithm can occasionally differ from the rounded mathematical result by more than one unit in the last place, one ulp.

Two types of operations are provided for manipulating the scale of a BigDecimal: scaling/rounding operations and decimal point motion operations. Scaling/rounding operations (setScale and round) return a BigDecimal whose value is approximately (or exactly) equal to that of the operand, but whose scale or precision is the specified value; that is, they increase or decrease the precision of the stored number with minimal effect on its value. Decimal point motion operations (movePointLeft and movePointRight) return a BigDecimal created from the operand by moving the decimal point a specified distance in the specified direction.

As a 32-bit integer, the set of values for the scale is large, but bounded. If the scale of a result would exceed the range of a 32-bit integer, either by overflow or underflow, the operation may throw an ArithmeticException.

For the sake of brevity and clarity, pseudo-code is used throughout the descriptions of BigDecimal methods. The pseudo-code expression (i + j) is shorthand for "a BigDecimal whose value is that of the BigDecimal i added to that of the BigDecimal j." The pseudo-code expression (i == j) is shorthand for "true if and only if the BigDecimal i represents the same value as the BigDecimal j." Other pseudo-code expressions are interpreted similarly. Square brackets are used to represent the particular BigInteger and scale pair defining a BigDecimal value; for example [19, 2] is the BigDecimal numerically equal to 0.19 having a scale of 2.

All methods and constructors for this class throw NullPointerException when passed a null object reference for any input parameter.

Relation to IEEE 754 Decimal Arithmetic

Starting with its 2008 revision, the IEEE 754 Standard for Floating-point Arithmetic has covered decimal formats and operations. While there are broad similarities in the decimal arithmetic defined by IEEE 754 and by this class, there are notable differences as well. The fundamental similarity shared by BigDecimal and IEEE 754 decimal arithmetic is the conceptual operation of computing the mathematical infinitely precise real number value of an operation and then mapping that real number to a representable decimal floating-point value under a rounding policy. The rounding policy is called a rounding mode for BigDecimal and called a rounding-direction attribute in IEEE 754-2019. When the exact value is not representable, the rounding policy determines which of the two representable decimal values bracketing the exact value is selected as the computed result. The notion of a preferred scale/preferred exponent is also shared by both systems.

For differences, IEEE 754 includes several kinds of values not modeled by BigDecimal including negative zero, signed infinities, and NaN (not-a-number). IEEE 754 defines formats, which are parameterized by base (binary or decimal), number of digits of precision, and exponent range. A format determines the set of representable values. Most operations accept as input one or more values of a given format and produce a result in the same format. A BigDecimal's scale is equivalent to negating an IEEE 754 value's exponent. BigDecimal values do not have a format in the same sense; all values have the same possible range of scale/exponent and the unscaled value has arbitrary precision. Instead, for the BigDecimal operations taking a MathContext parameter, if the MathContext has a nonzero precision, the set of possible representable values for the result is determined by the precision of the MathContext argument. For example in BigDecimal, if a nonzero three-digit number and a nonzero four-digit number are multiplied together in the context of a MathContext object having a precision of three, the result will have three digits (assuming no overflow or underflow, etc.).

The rounding policies implemented by BigDecimal operations indicated by rounding modes are a proper superset of the IEEE 754 rounding-direction attributes.

BigDecimal arithmetic will most resemble IEEE 754 decimal arithmetic if a MathContext corresponding to an IEEE 754 decimal format, such as decimal64 or decimal128 is used to round all starting values and intermediate operations. The numerical values computed can differ if the exponent range of the IEEE 754 format being approximated is exceeded since a MathContext does not constrain the scale of BigDecimal results. Operations that would generate a NaN or exact infinity, such as dividing by zero, throw an ArithmeticException in BigDecimal arithmetic.

Algorithmic Complexity

Operations on BigDecimal values have a range of algorithmic complexities; in general, those complexities are a function of both the size of the unscaled value as well as the size of the scale. For example, an exact multiply of two BigDecimal values is subject to the same complexity constraints as BigInteger multiply of the unscaled values. In contrast, a BigDecimal value with a compact representation like new BigDecimal(1E-1000000000) has a toPlainString result with over one billion characters.

Operations may also allocate and compute on intermediate results, potentially those allocations may be as large as in proportion to the running time of the algorithm.

Users of BigDecimal concerned with bounding the running time or space of operations can screen out BigDecimal values with unscaled values or scales above a chosen magnitude.