# Math

 This document is better viewed at https://docs.openzeppelin.com/contracts/api/math

These are math-related utilities.

## Libraries

### `SafeMath`

Wrappers over Solidity’s arithmetic operations with added overflow checks.

Arithmetic operations in Solidity wrap on overflow. This can easily result in bugs, because programmers usually assume that an overflow raises an error, which is the standard behavior in high level programming languages. `SafeMath` restores this intuition by reverting the transaction when an operation overflows.

Using this library instead of the unchecked operations eliminates an entire class of bugs, so it’s recommended to use it always.

#### `add(uint256 a, uint256 b) → uint256`internal

Returns the addition of two unsigned integers, reverting on overflow.

Counterpart to Solidity’s `+` operator.

Requirements:

#### `sub(uint256 a, uint256 b) → uint256`internal

Returns the subtraction of two unsigned integers, reverting on overflow (when the result is negative).

Counterpart to Solidity’s `-` operator.

Requirements:

• Subtraction cannot overflow.

#### `sub(uint256 a, uint256 b, string errorMessage) → uint256`internal

Returns the subtraction of two unsigned integers, reverting with custom message on overflow (when the result is negative).

Counterpart to Solidity’s `-` operator.

Requirements:

• Subtraction cannot overflow.

#### `mul(uint256 a, uint256 b) → uint256`internal

Returns the multiplication of two unsigned integers, reverting on overflow.

Counterpart to Solidity’s `*` operator.

Requirements:

• Multiplication cannot overflow.

#### `div(uint256 a, uint256 b) → uint256`internal

Returns the integer division of two unsigned integers. Reverts on division by zero. The result is rounded towards zero.

Counterpart to Solidity’s `/` operator. Note: this function uses a `revert` opcode (which leaves remaining gas untouched) while Solidity uses an invalid opcode to revert (consuming all remaining gas).

Requirements:

• The divisor cannot be zero.

#### `div(uint256 a, uint256 b, string errorMessage) → uint256`internal

Returns the integer division of two unsigned integers. Reverts with custom message on division by zero. The result is rounded towards zero.

Counterpart to Solidity’s `/` operator. Note: this function uses a `revert` opcode (which leaves remaining gas untouched) while Solidity uses an invalid opcode to revert (consuming all remaining gas).

Requirements:

• The divisor cannot be zero.

#### `mod(uint256 a, uint256 b) → uint256`internal

Returns the remainder of dividing two unsigned integers. (unsigned integer modulo), Reverts when dividing by zero.

Counterpart to Solidity’s `%` operator. This function uses a `revert` opcode (which leaves remaining gas untouched) while Solidity uses an invalid opcode to revert (consuming all remaining gas).

Requirements:

• The divisor cannot be zero.

#### `mod(uint256 a, uint256 b, string errorMessage) → uint256`internal

Returns the remainder of dividing two unsigned integers. (unsigned integer modulo), Reverts with custom message when dividing by zero.

Counterpart to Solidity’s `%` operator. This function uses a `revert` opcode (which leaves remaining gas untouched) while Solidity uses an invalid opcode to revert (consuming all remaining gas).

Requirements:

• The divisor cannot be zero.

### `SignedSafeMath`

Signed math operations with safety checks that revert on error.

#### `mul(int256 a, int256 b) → int256`internal

Returns the multiplication of two signed integers, reverting on overflow.

Counterpart to Solidity’s `*` operator.

Requirements:

• Multiplication cannot overflow.

#### `div(int256 a, int256 b) → int256`internal

Returns the integer division of two signed integers. Reverts on division by zero. The result is rounded towards zero.

Counterpart to Solidity’s `/` operator. Note: this function uses a `revert` opcode (which leaves remaining gas untouched) while Solidity uses an invalid opcode to revert (consuming all remaining gas).

Requirements:

• The divisor cannot be zero.

#### `sub(int256 a, int256 b) → int256`internal

Returns the subtraction of two signed integers, reverting on overflow.

Counterpart to Solidity’s `-` operator.

Requirements:

• Subtraction cannot overflow.

#### `add(int256 a, int256 b) → int256`internal

Returns the addition of two signed integers, reverting on overflow.

Counterpart to Solidity’s `+` operator.

Requirements:

### `Math`

Standard math utilities missing in the Solidity language.

#### `max(uint256 a, uint256 b) → uint256`internal

Returns the largest of two numbers.

#### `min(uint256 a, uint256 b) → uint256`internal

Returns the smallest of two numbers.

#### `average(uint256 a, uint256 b) → uint256`internal

Returns the average of two numbers. The result is rounded towards zero.