Numeral Systems

Most people just do calculations in the decimal numeral system. But there are also other numeral systems. The mostly used ones are binary, hexagonal and octal. In SpeQ you can specify values in these numeral systems by adding a prefix to a value. There are also functions available to get a value in a specific numeral system.

Overview

SpeQ handles the following numeral systems.

Numeral System	Base	Prefix	Symbols	Example
Binary	Base-2	0b	01	0b1101 Ans = 13 Bin(13) Ans = 0b1101
Decimal	Base-10	(no prefix)	0123456789	12 Ans = 12 Dec(0xF3) Ans = 243
Hexadecimal	Base-16	0x	0123456789ABCDEF	0xF3 Ans = 243 Hex(243) Ans = 0xF3
Octal	Base-8	0o	01234567	0o23 Ans = 19 Oct(19) Ans = 0o23

Important for the notation of numbers in other numeral systems is the system variable Bytes and the option Show leading zeros.

Binary numbers

Why should we bother about other numeral systems? Is the decimal system not enough? Well, for example computers do work with binary values. A computer can only store bits in it's memory. A bit can have a value 0 or 1, a binary value. You can write out a binary value as follows.

0101 = 0*2³ + 1*2² + 0*2¹ + 1*2⁰
1011 = 1*2³ + 0*2² + 1*2¹ + 1*2⁰
1111 = 1*2³ + 1*2² + 1*2¹ + 1*2⁰
etc...

With a binary number of four digits you can represent the decimal numbers 0 to 15.

Decimal	Binary
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111

Decimal	Binary
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111

To use binary values in SpeQ you have to add a prefix "0b" to the value, for example "0b1011".

Bytes and negative values

In computers bits are grouped in bytes: one byte contains 8 bits. With one byte you can represent 2⁸ = 256 values. With two bytes you can represent 2¹⁶ = 65536 values.

To be able to represent negative values the Two's complement rule is used. For example the value range for one byte is not chosen from 0 to 255, but -128 to 127. For a binary value of one byte (8 digits) the value representation is as follows:

Decimal	Binary representation with 1 byte
127	0111 1111
126	0111 1110
...	...
2	0000 0010
1	0000 0001
0	0000 0000
-1	1111 1111
-2	1111 1110
...	...
-127	1000 0001
-128	1000 0000

As you can see, the representation of the negative values depends on the number of bytes that are available for the value. For example with one byte the number 0b11111111 represents -1, but with two bytes it represents 255.

With SpeQ you can specify the number of bytes you want to work with to 1, 2 or 4. You can do this in the Settings window or from within the Workarea by adjusting the system variable Bytes. There is also an option to show or hide leading zeros in numbers, as described on the page Settings. For example if Bytes=1, there are 8 digits available for binary numbers. With this option checked, the answer of Bin(23) will be 0b00010111 instead of 0b10111.

Hexadecimal numbers

With the hexadecimal representation you can display numbers in a much shorter way than binary. The hexadecimal system is base-16 and has the symbols

0 1 2 3 4 5 6 7 8 9 A B C D E F

where the value of A=10, B=11 C=12, D=13, E=14 and F=15. Each four digits in a binary number can be represented by one hexadecimal digit. The conversion is very easy. For example

Binary	1101	0011	0101	0001
Hexadecimal	D	3	5	1	= D351

Hexadecimal numbers are for example used to represent twenty-four-bit colors in HTML files in the format #RRGGBB. Here RR specifies the value of the Red component of the color, GG the Green component and BB the Blue component. For example red=#FF0000, green=#008000 and blue=#0000FF.

To use hexadecimal values in SpeQ you have to add a prefix "0x" to the value, for example "0xD351".

octal numbers

The octal numeral system is base-8 and has the symbols

0 1 2 3 4 5 6 7

Each three digits in a binary number can be represented by one octal digit. For example

Binary	001	001	001	101	010	001
Octal	1	1	1	5	2	1	= 111521

To use octal values in SpeQ you have to add a prefix "0o" to the value, for example "0o111521".

Examples

'Examples of using different numeral systems

' A binary number
0b1101001101010001
       Ans = 54097

' A hexagonal number
0xE3DD1
       Ans = 933329

' An octal number
0o1773
       Ans = 1019

' Specify the numeral system by enclosing
' the line with a function Bin, Dec, Hex or Oct
Bin(3 + 18 * 2)
       Ans = 0b100111

' Convert a binary value to a hexadecimal value
Hex(0b1101001101010001)
       Ans = 0xD351

Hex(26)
       Ans = 0x1A
Hex(15)
       Ans = 0xF
Hex(255)
       Ans = 0xFF
Hex(65340)
       Ans = 0xFF3C
0xFF3C
       Ans = 65340
Hex(-12)
       Ans = 0xFFFFFFF4

Oct(13)
       Ans = 0o15
Oct(120 + 5)
       Ans = 0o175
0o175
       Ans = 125

' The specification of a the numeral system is lost when 
' you perform an operation after the conversion.
Bin(24 + 2)
       Ans = 0b11010
Bin(24) + 2
       Ans = 26

' Negative values. First specify the number of bytes for the values
Bytes = 1;
0b11111111
       Ans = -1
Bytes = 2;
0b11111111
       Ans = 255

' Set Bytes to the maximum available
Bytes = 4;