furnace/doc/1-intro/hex.md

hexadecimal

the hexadecimal numeral system differs from the decimal system by having 16 digits rather than 10:

hex| decimal
---|---------
 0 | 0
 1 | 1
 2 | 2
 3 | 3
 4 | 4
 5 | 5
 6 | 6
 7 | 7
 8 | 8
 9 | 9
 A | 10
 B | 11
 C | 12
 D | 13
 E | 14
 F | 15

when there is more than one digit, these are multiplied by 16, 256, 4096 and so on rather than 10, 100, 1000:

hex | decimal
----|---------
 00 | 0
 04 | 4
 08 | 8
 0F | 15
 10 | 16
 11 | 17
 12 | 18
 13 | 19
 20 | 32
 30 | 48
 40 | 64

hex to decimal

for example, take hexadecimal number AA:

          2nd digit -\   /- 1st digit
                     A   A
 16^1*10 = 16*10 = 160 + 10 = 170

now for hexadecimal number 4C5F:

      3rd digit -\    /- 2nd digit
 4th digit -\    |    |    /- 1st digit
            4    C    5    F
            |    |    |    |
            |    |    |    15 =        15 =    15 +
            |    |    \16^1*5 =   16 * 5  =    80
            |    \--- 16^2*12 =  256 * 12 =  3072
            \--------- 16^3*4 = 4096 * 4  = 16384
                                           -------
                                          = 19551

decimal to hex

if it's less than 16, just memorize the table at the top of this document.

otherwise find the power of 16 that is closest to the number you want to convert, but no larger than the number. then divide, and take the remainder. divide the remainder with the previous power of 16, until the divider is 1.

for example, for the decimal number 220:

220 ÷ 16 = 13 (r = 12)    D
 12 ÷  1 = 12 (stop here) C

= DC

now for decimal number 69420:

69420 ÷ 65536 =  1 (r = 3884)  1
 3884 ÷  4096 =  0 (r = 3884)  0
 3884 ÷   256 = 15 (r =   44)  F
   44 ÷    16 =  2 (r =   12)  2
   12 ÷     1 = 12 (stop here) C

= 10F2C

bitmask

a bitmask is a value that actually represents a set of individual binary bits to be toggled, some of which may be grouped to form small binary numbers. these are used by adding up the value of each bit and converting the result to hex. in macros, these are shown in decimal.

bit	binary	decimal
bit 7	1000 0000	128
bit 6	0100 0000	64
bit 5	0010 0000	32
bit 4	0001 0000	16
bit 3	0000 1000	8
bit 2	0000 0100	4
bit 1	0000 0010	2
bit 0	0000 0001	1

for example, to turn bits 7 and 5 on, and put 110 (decimal 6) in bits 1 to 3:

bit 7 = `1... ....` = 128
bit 6 = `.0.. ....` =   0
bit 5 = `..1. ....` =  32
bit 4 = `...0 ....` =   0
bit 3 = `.... 1...` =   8
bit 2 = `.... .1..` =   4
bit 1 = `.... ..0.` =   0
bit 0 = `.... ...0` =   0

added together, the resulting binary is 1010 1100, which converts to hex as AC and to decimal as 172.

hex to decimal table

hex	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
00	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
10	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31
20	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47
30	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63
40	64	65	66	67	68	69	70	71	72	73	74	75	76	77	78	79
50	80	81	82	83	84	85	86	87	88	89	90	91	92	93	94	95
60	96	97	98	99	100	101	102	103	104	105	106	107	108	109	110	111
70	112	113	114	115	116	117	118	119	120	121	122	123	124	125	126	127
80	128	129	130	131	132	133	134	135	136	137	138	139	140	141	142	143
90	144	145	146	147	148	149	150	151	152	153	154	155	156	157	158	159
A0	160	161	162	163	164	165	166	167	168	169	170	171	172	173	174	175
B0	176	177	178	179	180	181	182	183	184	185	186	187	188	189	190	191
C0	192	193	194	195	196	197	198	199	200	201	202	203	204	205	206	207
D0	208	209	210	211	212	213	214	215	216	217	218	219	220	221	222	223
E0	224	225	226	227	228	229	230	231	232	233	234	235	236	237	238	239
F0	240	241	242	243	244	245	246	247	248	249	250	251	252	253	254	255