# Difference between revisions of "Mathematical Operations"

## Basic and QB64 Numerical Types

Qbasic Number Types
• INTEGER [%]: 2 Byte signed whole number values from -32768 to 32767. 0 to 65535 unsigned. (currently not checked in QB64)
• LONG [&]: 4 byte signed whole number values from -2147483648 to 2147483647. 0 to 4294967295 unsigned.
• SINGLE [!]: 4 byte signed floating decimal point values of up to 7 decimal place accuracy. Cannot be unsigned!
• DOUBLE [#]: 8 byte signed floating decimal point values of up to 15 decimal place accuracy. Cannot be unsigned!
• To get one byte values, can use an ASCII STRING character to represent values from 0 to 255 as in BINARY files.

QB64 Number Types
• _BIT [`]: 1 bit signed whole number values of 0 or -1 signed or 0 or 1 unsigned. _BIT * 8 can hold a signed or unsigned byte value.
• _BYTE [%%]: 1 byte signed whole number values from -128 to 127. Unsigned values from 0 to 255.
• _INTEGER64 [&&]: 8 byte signed whole number values from -9223372036854775808 to 9223372036854775807
• _FLOAT [##]: currently set as 10 byte signed floating decimal point values up to 1.1897E+4932. Cannot be unsigned!

Signed and Unsigned Integer Values

Negative (signed) numerical values can affect calculations when using any of the BASIC operators. SQR cannot use negative values! There may be times that a calculation error is made using those negative values. The SGN function returns the sign of a value as -1 for negative, 0 for zero and 1 for unsigned positive values. ABS always returns an unsigned value.

• SGN(n) returns the value's sign as -1, 0 or 1
• ABS(n) changes negative values to positive ones
• QB64: _UNSIGNED in a DIM, AS or _DEFINE statement for only positive INTEGER values.

_UNSIGNED integer, byte and bit variable values can use the ~ suffix before the type suffix to define the type.

## Mathematical Operation Symbols

Most of the BASIC math operators are ones that require no introduction. The addition, subtraction, multplication and division operators are ones commonly used as shown below:

Symbol Procedure Example Usage
+ Addition c = a + b
- Subtraction c = a - b
- Negation c = - a
* Multiplication c = a * b
/ Division c = a / b

BASIC can also use two other operators for INTEGER division. Integer division returns only whole number values. MOD remainder division returns a value only if an integer division cannot divide a number exactly. Returns 0 if a value is exactly divisible.

Symbol Procedure Example Usage
\ Integer division c = a \ b
MOD Remainder division c = a MOD b

It is an error to divide by zero or to take the remainder modulo zero.

There is also an operator for exponential calculations. The exponential operator is used to raise a number's value to a designated exponent of itself. In QB the exponential return values are DOUBLE values. The SQR function can return a number's Square Root. For other exponential roots the operator can be used with fractions such as (1 / 3) designating the cube root of a number.

Symbol Procedure Example Usage
^ Exponent c = SQR(a ^ 2 + b ^ 2)

Note: Exponent fractions should be parenthesized in order to be treated as a fraction rather than a division operation!

## Basic's Order of Operations

When a normal calculation is made, BASIC works from left to right, but it does certain calculations in the following order:

1. Exponential and exponential Root calculations including SQR.
2. Negation (Note that this means that - 3 ^ 2 is treated as -(3 ^ 2) and not as (-3) ^ 2.)
3. Multiplication and Division calculations

Defining the Operation Order

Sometimes a calculation may need BASIC to do them in another order or the calculation will return bad results. BASIC allows the programmer to decide the order of operations by using parenthesis around parts of the equation. BASIC will do the calculations inside of the parenthesis brackets first and the others from left to right in the normal operation order.

## Basic's Mathematical Functions

Function Description
ABS(n) returns the absolute (positive) value of n: ABS(-5) = 5
ATN(angle*) returns the arctangent of an angle in radians: π = 4 * ATN(1)
COS(angle*) returns the cosine of an angle in radians. (horizontal component)
EXP(n) returns ex, (n <= 88.02969): e = EXP(1) ' (e = 2.718281828459045)
LOG(n) returns the base e natural logarithm of n. (n > 0)
SGN(n) returns -1 if n < 0, 0 if n = 0, 1 if n > 0: SGN(-5) = -1
SIN(angle*) returns the sine of an angle in radians. (vertical component)
SQR(n) returns the square root of a number. (n >= 0)
TAN(angle*) returns the tangent of an angle in radians

Degree to Radian Conversion: FUNCTION Radian (degrees) Radian = degrees * (4 * ATN(1)) / 180 END FUNCTION FUNCTION Degree (radians) Degree = radians * 180 / (4 * ATN(1)) END FUNCTION Logarithm to base n FUNCTION LOGN (X, n) IF n > 0 AND X > 0 THEN AND LOG(n) <> 0 THEN LOGN = LOG(X) / LOG(n) ELSE BEEP END FUNCTION FUNCTION LOG10 (X) 'base 10 logarithm IF X > 0 THEN LOG10 = LOG(X) / LOG(10.#) ELSE BEEP END FUNCTION

The numerical value of n in the LOG(n) evaluation MUST be a positive value!
The numerical value of n in the EXP(n) evaluation MUST be LESS than or equal to 88.02969!
The numerical value of n in the SQR(n) evaluation can NOT be a negative value!

## Derived Mathematical Functions

The following Trigonometric functions can be derived from the BASIC Mathematical Functions listed above. Each function checks that certain values can be used without error or a BEEP will notify the user that a value could not be returned. An error handling routine can be substituted if desired. Note: Functions requiring π use 4 * ATN(1) for SINGLE accuracy. Use ATN(1.#) for DOUBLE accuracy.

FUNCTION SEC (x) 'Secant IF COS(x) <> 0 THEN SEC = 1 / COS(x) ELSE BEEP END FUNCTION FUNCTION CSC (x) 'CoSecant IF SIN(x) <> 0 THEN CSC = 1 / SIN(x) ELSE BEEP END FUNCTION FUNCTION COT (x) 'CoTangent IF TAN(x) <> 0 THEN COT = 1 / TAN(x) ELSE BEEP END FUNCTION FUNCTION ARCSIN (x) 'Inverse Sine IF x < 1 THEN ARCSIN = ATN(x / SQR(1 - (x * x))) ELSE BEEP END FUNCTION FUNCTION ARCCOS (x) ' Inverse Cosine IF x < 1 THEN ARCCOS = (2 * ATN(1)) - ATN(x / SQR(1 - x * x)) ELSE BEEP END FUNCTION FUNCTION ARCSEC (x) ' Inverse Secant IF x < 1 THEN ARCSEC = ATN(x / SQR(1 - x * x)) + (SGN(x) - 1) * (2 * ATN(1)) ELSE BEEP END FUNCTION FUNCTION ARCCSC (x) ' Inverse CoSecant IF x < 1 THEN ARCCSC = ATN(1 / SQR(1 - x * x)) + (SGN(x)-1) * (2 * ATN(1)) ELSE BEEP END FUNCTION FUNCTION ARCCOT (x) ' Inverse CoTangent ARCCOT = (2 * ATN(1)) - ATN(x) END FUNCTION FUNCTION SINH (x) ' Hyperbolic Sine IF x <= 88.02969 THEN SINH = (EXP(x) - EXP(-x)) / 2 ELSE BEEP END FUNCTION FUNCTION COSH (x) ' Hyperbolic CoSine IF x <= 88.02969 THEN COSH = (EXP(x) + EXP(-x)) / 2 ELSE BEEP END FUNCTION FUNCTION TANH (x) ' Hyperbolic Tangent or SINH(x) / COSH(x) IF 2 * x <= 88.02969 AND EXP(2 * x) + 1 <> 0 THEN TANH = (EXP(2 * x) - 1) / (EXP(2 * x) + 1) ELSE BEEP END IF END FUNCTION FUNCTION SECH (x) ' Hyperbolic Secant or (COSH(x)) ^ -1 IF x <= 88.02969 AND (EXP(x) + EXP(-x)) <> 0 THEN SECH = 2 / (EXP(x) + EXP(-x)) ELSE BEEP END FUNCTION FUNCTION CSCH (x) ' Hyperbolic CoSecant or (SINH(x)) ^ -1 IF x <= 88.02969 AND (EXP(x) - EXP(-x)) <> 0 THEN CSCH = 2 / (EXP(x) - EXP(-x)) ELSE BEEP END FUNCTION FUNCTION COTH (x) ' Hyperbolic CoTangent or COSH(x) / SINH(x) IF 2 * x <= 88.02969 AND EXP(2 * x) - 1 <> 0 THEN COTH = (EXP(2 * x) + 1) / (EXP(2 * x) - 1) ELSE BEEP END IF END FUNCTION FUNCTION ARCSINH (x) ' Inverse Hyperbolic Sine IF (x * x) + 1 >= 0 AND x + SQR((x * x) + 1) > 0 THEN ARCSINH = LOG(x + SQR(x * x + 1)) ELSE BEEP END IF END FUNCTION FUNCTION ARCCOSH (x) ' Inverse Hyperbolic CoSine IF x >= 1 AND x * x - 1 >= 0 AND x + SQR(x * x - 1) > 0 THEN ARCCOSH = LOG(x + SQR(x * x - 1)) ELSE BEEP END IF END FUNCTION FUNCTION ARCTANH (x) ' Inverse Hyperbolic Tangent IF x < 1 THEN ARCTANH = LOG((1 + x) / (1 - x)) / 2 ELSE BEEP END FUNCTION FUNCTION ARCSECH (x) ' Inverse Hyperbolic Secant IF x > 0 AND x <= 1 THEN ARCSECH = LOG((SGN(x) * SQR(1 - x * x) + 1) / x) ELSE BEEP END FUNCTION FUNCTION ARCCSCH (x) ' Inverse Hyperbolic CoSecant IF x <> 0 AND x * x + 1 >= 0 AND (SGN(x) * SQR(x * x + 1) + 1) / x > 0 THEN ARCCSCH = LOG((SGN(x) * SQR(x * x + 1) + 1) / x) ELSE BEEP END IF END FUNCTION FUNCTION ARCCOTH (x) ' Inverse Hyperbolic CoTangent IF x > 1 THEN ARCCOTH = LOG((x + 1) / (x - 1)) / 2 ELSE BEEP END FUNCTION

Hyperbolic Function Relationships: COSH(-x) = COSH(x) SINH(-x) = -SINH(x) SECH(-x) = SECH(x) CSCH(-x) = -CSCH(x) TANH(-x) = -TANH(x) COTH(-x) = -COTH(x) Inverse Hyperbolic Function Relatonships: ARSECH(x) = ARCOSH(x) ^ -1 ARCSCH(x) = ARSINH(x) ^ -1 ARCOTH(x) = ARTANH(x) ^ -1 Hyperbolic sine and cosine satisfy the Pythagorean trig. identity: (COSH(x) ^ 2) - (SINH(x) ^ 2) = 1

Microsoft's Derived BASIC Functions (KB 28249)

## Mathematical Logical operators

The following logical operators compare numerical values using bitwise operations. The two numbers are compared by the number's Binary bits on and the result of the operation determines the value returned in decimal form. NOT checks one value and returns the opposite. It returns 0 if a value is not 0 and -1 if it is 0. See Binary for more on bitwise operations.

Truth table of the 6 BASIC Logical Operators

The results of the bitwise logical operations, where A and B are operands, and T and F indicate that a bit is set or not set:
Operands Operations
A B NOT B A AND B A OR B A XOR B A EQV B A IMP B
T T F T T F T T
T F T F T T F F
F T F F T T F T
F F T F F F T T
Relational Operations return negative one (-1, all bits set) and zero (0, no bits set) for true and false, respectively.
This allows relational tests to be inverted and combined using the bitwise logical operations.

BASIC can accept any + or - value that is not 0 to be True when used in an evaluation.

## Basic's Rounding Functions

Rounding is used when the program needs a certain number value or type. There are 4 INTEGER or LONG Integer functions and one function each for closest SINGLE and closest DOUBLE numerical types. Closest functions use "bankers" rounding which rounds up if the decimal point value is over one half. Variable types should match the return value.
Name Description
INT(n) rounds down to lower Integer value whether positive or negative
FIX(n) rounds positive values lower and negative to a less negative Integer value
CINT(n) rounds to closest Integer. Rounds up for decimal point values over one half.
CLNG(n) rounds Integer or Long values to closest value like CINT.(values over 32767)
CSNG(n) rounds Single values to closest last decimal point value.
CDBL(n) rounds Double values to closest last decimal point value.
_ROUND rounds to closest numerical value in QB64 only.
Note: Each of the above functions define the value's type in addition to rounding the values.

## Base Number Systems

Comparing the INTEGER Base Number Systems Decimal (base 10) Binary (base 2) Hexadecimal (base 16) Octal (base 8) &B &H HEX\$(n) &O OCT\$(n) 0 0000 0 0 1 0001 1 1 2 0010 2 2 3 0011 3 3 4 0100 4 4 5 0101 5 5 6 0110 6 6 7 0111 7 7 -- maxed 8 1000 8 10 maxed-- 9 1001 9 11 10 1010 A 12 11 1011 B 13 12 1100 C 14 13 1101 D 15 14 1110 E 16 15 ------------- 1111 <--- Match ---> F ---------------- 17 -- max 2 16 10000 10 20 When the Decimal value is 15, the other 2 base systems are all maxed out! The Binary values can be compared to all of the HEX value digit values so it is possible to convert between the two quite easily. To convert a HEX value to Binary just add the 4 binary digits for each HEX digit place so: F A C E &HFACE = 1111 + 1010 + 1100 + 1101 = &B1111101011001101 To convert a Binary value to HEX you just need to divide the number into sections of four digits starting from the right(LSB) end. If one has less than 4 digits on the left end you could add the leading zeros like below: &B101011100010001001 = 0010 1011 1000 1000 1001 hexadecimal = 2 + B + 8 + 8 + 9 = &H2B889 See the Decimal to Binary conversion function that uses HEX\$ on the &H page.

VAL converts string numbers to Decimal values.
VAL reads the string from left to right and converts numerical string values, - and . to decimal values until it finds a character other than those 3 characters. Commas are NOT read either! However HEXadecimal and OCTal base values can be read with &H or &O.
The OCT\$ string function return can be converted to a decimal value using VAL("&O" + OCT\$(n)).
The HEX\$ string function return can be converted to a decimal value using VAL("&H" + HEX\$(n)).

STR\$ converts numerical values to string characters for PRINT or variable strings. It also removes the right number PRINT space.

## Bits and Bytes

BITS
• The MSB is the most significant(largest) bit value and LSB is the least significant bit of a binary or register memory address value. The order in which the bits are read determines the binary or decimal byte value. There are two common ways to read a byte:
• "Big-endian": MSB is the first bit encountered, decreasing to the LSB as the last bit by position, memory address or time.
• "Little-endian": LSB is the first bit encountered, increasing to the MSB as the last bit by position, memory address or time.

Offset or Position: 0 1 2 3 4 5 6 7 Example: 11110000 ---------------------------------- -------- Big-Endian Bit On Value: 128 64 32 16 8 4 2 1 240 Little-Endian Bit On Value: 1 2 4 8 16 32 64 128 15

The big-endian method compares exponents of 2 7 down to 2 0 while the little-endian method does the opposite.
BYTES
• INTEGER values consist of 2 bytes called the HI and LO bytes. Anytime that the number of binary digits is a multiple of 16 (2bytes, 4 bytes, etc.) and the HI byte's MSB is on(1), the value returned will be negative. Even with SINGLE or DOUBLE values!

16 BIT INTEGER OR REGISTER AH (High Byte Bits) AL (Low Byte Bits) BIT: 15 14 13 12 11 10 9 8 | 7 6 5 4 3 2 1 0 ---------------------------------------|-------------------------------------- HEX: 8000 4000 2000 1000 800 400 200 100 | 80 40 20 10 8 4 2 1 | DEC: -32768 16384 8192 4096 2048 1024 512 256 | 128 64 32 16 8 4 2 1

The HI byte's MSB is often called the sign bit! When all 16 of the integer binary bits are on, the decimal return is -1.