manual:chapter3:reals

A number in newRPL is represented as exact or approximate. To enter a number as approximate use a trailing dot. To enter a number as exact omit any trailing dot. For routine arithmetic this distinction doesn't matter much, but it does come into play when evaluating symbolic expressions.

Here are some examples:

Approximate Numbers Exact Numbers
1. 1
1.007. 1.007
1.007.e-10 1.007e-10

Arithmetic performed on numbers takes into account the operands (exact or approximate) and results are displayed accordingly.

For example,

2:                             1
1:                             3
……………………………………………………………………………………
/

results in 0.333333333. (approximate), whereas

2:                             1
1:                             2
……………………………………………………………………………………
/

results in 0.5 (exact).

Numbers in different bases can be entered by preceding the value with a # and a trailing a letter to indicate the base (b = binary, o = octal, d = decimal, h = hexadecimal; note that the trailing letter is case insensitive and that even if the #d syntax for decimal numbers is accepted, they are nevertheless displayed without leading # and trailing d).

A quicker method to enter these numbers consists in using the base cycler: simply start keying the number in the preferred base then press RShold-3 and the number on the edit line will cycle between the different bases.

The base cycler is smart enough to skip bases if the keyed number contains illegal digits. For example, if there's 123 on the edit line the base cycler will rotate between #123o, #123h and 123, skipping an illegal #123b.

If the number is already on the stack the base cycler will display all four bases in succession, provided the number is integer.

Another option is the BASE menu (RS-3): in its first page the soft key will give access to the aforementioned base cycler and four non-programmable base conversion tools which act on the number on the edit line or the first level of the stack.

Arithmetic can be done on numbers in different bases with the result displayed as the base of the first argument. Only integer numbers in the range $-(2^{63}-1)$ to $2^{63}-1$ can be expressed in multiple bases. Numbers can be either exact or approximate; outside this range the base will be automatically switched to decimal.

In other words, it is assumed that non base-10 numbers are signed, 64-bits integers. Of these 64 bits, the least significant 63 bits store the magnitude and the 64th most significant bit stores the sign (0 means positive, 1 means negative). Negative numbers are stored internally in two's complement but are shown as positive numbers with a leading minus sign.

For example:

2:                         #125o
1:                          #EEh
……………………………………………………………………………………
-

yields -#231o.

Here are more examples of arithmetic operations in bases other than 10:

2:                        #1101b
1:                          #FFh
……………………………………………………………………………………
+

yields #100001100b.

2:                           256
1:                        #FFFFh
……………………………………………………………………………………
-

yields -65279.

2:                         #355o
1:                    #11010101b
……………………………………………………………………………………
*

yields #142461o.

2:                           #7h
1:                             2
……………………………………………………………………………………
/

yields 3.5.

2:                           #2h
1:                            63
……………………………………………………………………………………
^

yields 9.223372E18.

If there is a need to work with different word sizes newRPL provides a dedicated set of bit operations: these command always honor the selected word size when returning the result.

The word size can be set using the command STWS (STore Word Size). Valid ranges are 1 to 63 (not including the sign bit). So, for example, to work with 32-bit signed numbers, set the word size to 31. To view the currently set word size, use RCWS (ReCall Word Size). /Note that setting too small a word size can lead to overflow and hence unexpected results, such as this:

« 7 STWS
120 4 BMUL
»

The result is -32, not 480 as expected with a larger word size.

Command Purpose Example
BADD Add #11001b #100000b BADD
BSUB Subtract #371o #250 BSUB
BMUL Multiply #1A4h #7h BMUL
BDIV Divide 12 5 BDIV
BNEG Negate (Two's Complement) #11001b BNEG
BAND Bitwise AND #1101010b #1100010b BAND
BOR Bitwise OR #1101010b #1100010b BOR
BXOR Bitwise XOR #1101010b #1100010b BXOR
BNOT Bitwise NOT (One's Complement) #11001b BNOT
BLSL Bitwise Logical Shift Left #11001b 4 BLSL
BLSR Bitwise Logical Shift Right #11001b 2 BLSR
BASR Bitwise Arithmetic Shift Right #215o 2 BASR
BRL Bitwise Rotate Left #3FFh 4 BRL
BRR Bitwise Rotate Right #3FFh 1 BRR

Real numbers form an ordered field and its elements can be compared: newRPL provides a full set of arithmetic and logic operators plus some specialized ones.

All these operators (except CMP) return 1 if the test is true and 0 if the test is false. Actually any value different from 0 is considered true, but conventionally the operators return always 1 when the test succeeds.

The classical arithmetic test operators are == (equality), ≠ (not equality), < (less than), > (greater than), ≤ (less than or equal), ≥ (greater than or equal).

2:                          #10h
1:                       #10000b
……………………………………………………………………………………
==

yields 1.

2:                          #11o
1:                           10.
……………………………………………………………………………………
<

yields 0.

Logical test operators are AND, OR, XOR and NOT.

These operators must not be confused with the similar named bitwise operators BAND, BOR, BXOR and BNOT described earlier: while the latter operate on a bit-by-bit basis, the logical operators consider the truth value of their arguments. Some examples will help to clarify:

2:                        #1000b
1:                       #11000b
……………………………………………………………………………………
BAND

yields #1000b, but

2:                        #1000b
1:                       #11000b
……………………………………………………………………………………
AND

yields 1. Similarly

2:                        #1000b
1:                           #0b
……………………………………………………………………………………
BOR

yields #1000b, but

2:                        #1000b
1:                           #0b
……………………………………………………………………………………
AND

yields 0.

Finally, specialized test operators perform particular tests: they are CMP, SAME and ISTRUE.

CMP returns the SIGN of the difference between its arguments:

2:                            -2
1:                             8
……………………………………………………………………………………
CMP

yields -1.

For real numbers

• SAME and == are equivalent;
• ISTRUE and « 0 ≠ » are equivalent.

Operations on real numbers aren't always defined on the whole field of real numbers: a notable example is the division, which is not defined if the divisor is 0. newRPL handles these occurrances defining two special symbols: ∞ (infinity) and NaN (Not-a-Number).

The ∞ symbol is returned when attempting, as anticipated, a division by 0 or when the result of a function (e.g. TAN) diverges. Albeit ∞ must not be considered a number, it obeys to a number of rules. The following examples assume flag -103 is cleared.

• ∞ + or - a finite number yields ∞;
• ∞ * or / a finite number yields ∞;
• the sign rule in multiplication or division applies, therefore -∞ is a valid object;
• sum, multiplication or power of ∞ with itself yields ∞.

The NaN symbol is newRPL way of informing the user that the attempted calculation returns an undefined result: typical examples include the calculation of the difference ∞ - ∞ or the sine of ∞. In general any function that includes NaN among its arguments will yield NaN.

• manual/chapter3/reals.txt