[CS61C FA20] Lecture 02.2 - Number Representation: Conversions — Transcript

Full transcript

1. 0:00and welcome back now let's see
2. 0:03how to work with binary decimal and
3. 0:06hexadecimal numbers and how to convert
4. 0:08among that triangle binary decimal to
5. 0:10hexadecimal
6. 0:11so let's begin at the beginning we're in
7. 0:13first grade we're learning about base 10
8. 0:15numbers
9. 0:16and you're thinking to yourself uh i
10. 0:18paid money for this let's go
11. 0:20no it makes it makes it's important to
12. 0:22start from the beginning to see the
13. 0:23analogies of how you were thinking about
14. 0:25numbers in the first place
15. 0:26so if you remember the teacher taught
16. 0:28you that you had ten different
17. 0:29characters
18. 0:30zero through nine we're in base ten by
19. 0:32the way the question of why are we in
20. 0:34base ten
21. 0:35ten fingers uh if we were
22. 0:38in simpson's land we'd be in base eight
23. 0:41because they have four fingers
24. 0:42see simpsons thing so if i give you the
25. 0:45number 3271
26. 0:48by the way from now on starting now
27. 0:51in general if i don't write any base
28. 0:52it's the default is going to be base 10
29. 0:55okay uh in fact i i i'm even going to
30. 0:58write the letter out of t e n
31. 1:00there because base 1 0 is ambiguous but
32. 1:02we'll talk about that in a couple slides
33. 1:05so 3271 that means is the same as 3271
34. 1:09subscript 10. okay i can even grab a pen
35. 1:12here
36. 1:13and just annotate that so i'm going to
37. 1:14write a subscript here to indicate
38. 1:17that we're in base 10. what that really
39. 1:19meant is
40. 1:21three thousands plus two hundreds
41. 1:25plus seven tens plus one
42. 1:28one okay and we add them all up this is
43. 1:30like a coefficient times
44. 1:32times this term so it's three thousands
45. 1:34two hundreds
46. 1:35seven tens and one that gives you 3271
47. 1:38there by the way i could have said
48. 1:40look i've got three i got a lot of dash
49. 1:42marks
50. 1:43i've got a lot of when you're in prison
51. 1:45not that i would know about
52. 1:46being in prison but if you're in prison
53. 1:48you might have lots of dashboards how
54. 1:49many days you've been there every day
55. 1:50you
56. 1:51scratch something into the wall well i
57. 1:52might have that number of dash marks
58. 1:54just and that's actually base one you've
59. 1:56just got dash marks it's called
60. 1:58uh tally counting i think so if i want
61. 2:00to convert that to base ten i'd have to
62. 2:02count them up and then figure out how
63. 2:03many
64. 2:03ten how many thousands and how many tens
65. 2:05it's actually an interesting process of
66. 2:07counting from tally
67. 2:08marks to base 10.
68. 2:11base 2 is exactly the same thing as base
69. 2:1310 just subtle
70. 2:15where i said you had 10 characters
71. 2:16represented 10 digits
72. 2:18now i have 2 0 and 1.
73. 2:22and in fact we're always going to start
74. 2:23from 0 and count the way up when we get
75. 2:25past 10 we'll see how to do this in
76. 2:26hexadecimal by actually leaking into the
77. 2:27letters
78. 2:29so if i have 1 1 0 1 now i now need to
79. 2:32indicate that 1 1 0 1 is not in base 10.
80. 2:35it's in base 2.
81. 2:36so i would either write 0 b in front of
82. 2:38it okay if i see 0 b
83. 2:40that means i'm in a binary number or i
84. 2:42can write it base
85. 2:442 subscript 2 or i can say 1 1 0 1
86. 2:47in binary okay and i'm saying translate
87. 2:491101 and bind it
88. 2:511 101 from base 10 to binary that's what
89. 2:53i'm really asking you to do
90. 2:55well how do we do this the same way what
91. 2:58we do
92. 2:59is if you recall before i'm going to go
93. 3:01back to this slide this 10 to the 3 this
94. 3:03is the base
95. 3:04to some power look at that 10 to the 0
96. 3:0710 to the 1
97. 3:0810 to the 2 10 to the 3. the same thing
98. 3:10here let's take the base
99. 3:12to the different powers and this in a
100. 3:14way gives
101. 3:15like a column heading you can sometimes
102. 3:17i even do this sometimes i say
103. 3:19draw it like this and i say 1
104. 3:222 what's 2 squared 4.
105. 3:26what's 2 to the 3 8 and then i say well
106. 3:30what's the value in that and so i kind
107. 3:32of say i write my number one
108. 3:34one zero one and basically that's
109. 3:37it's the the column heading times the
110. 3:39value below it so you see this here it's
111. 3:41one
112. 3:41eight one four no twos and one one
113. 3:46it's almost like have an analogy we're
114. 3:47going to see in a second if i got boxes
115. 3:49from amazon
116. 3:50and the boxers labeled one egg in this
117. 3:52one
118. 3:53here's an empty box i guess it's an
119. 3:54empty box this is the two box
120. 3:56this is full of eggs it's four and this
121. 3:58is full of eggs it's eight
122. 4:00how many total eggs do i have so i think
123. 4:01about that sometimes the boxes come
124. 4:02empty or full but they're either only
125. 4:04empty or full in here
126. 4:05in this particular case so eight and
127. 4:07four and one is thirteen therefore one
128. 4:08one zero one
129. 4:09is 13. pretty cool
130. 4:13base 16. you got to be warmed up let's
131. 4:16go let's rock and roll
132. 4:17this is called hexadecimal so now you're
133. 4:19going to say well dan i have
134. 4:2016 digits but i can only i can probably
135. 4:24use the first 10 from decimal that's
136. 4:25right so what do i do
137. 4:27you're going to leak into the letters so
138. 4:30we're going to go a through f
139. 4:31so all of a sudden a is now a number b
140. 4:34is a number
141. 4:35and the way that works is a is 10. just
142. 4:37keep locked at it
143. 4:38b is 11. c is 12 d is 13 e is 14
144. 4:41f is 15. so i would like you to memorize
145. 4:44that actually i really would like you to
146. 4:46memorize that i'd like to be proud to
147. 4:47say all of my 621c graduates
148. 4:49know that and know how to connect that
149. 4:50so i'm going to ask you here we go a5 i
150. 4:52made a5
151. 4:53in hexadecimal written in hexadecimal
152. 4:57dollars today how much money did i make
153. 4:59that's pretty cool now
154. 5:00you could also have a number 1101 is a
155. 5:03valid hexadecimal number so again we
156. 5:04still have to indicate what our base is
157. 5:06when i'm giving you a number
158. 5:07so i can either say a5 a hexadecimal
159. 5:10number a5 what's that in decimal that's
160. 5:12what i'm asking doing here
161. 5:13or i can precede it with 0x just like 0b
162. 5:170x that tells you unambiguously that
163. 5:21that's a hexadecimal number
164. 5:23or again you can write a subscript 16.
165. 5:26we do our same thing we write our
166. 5:27columns here's our columns
167. 5:29the first column is always one the
168. 5:31second column is always b this is this
169. 5:33is by the way
170. 5:33this is always base to the zero base to
171. 5:36the one so
172. 5:37base to the one is 16 base to the 2 is
173. 5:39256
174. 5:41etc so what have i got well
175. 5:44a5 i write a5 what are right over here
176. 5:48zero there are always leading zeros we
177. 5:51just ignore them right there's all
178. 5:52if i said i made 12 dollars in in
179. 5:54decimal 12 today
180. 5:55i also made 0 12. i also made 0 0 12. we
181. 5:57just don't say the zeros
182. 5:59okay so a5 let's do it so a times 16.
183. 6:03wait what well i have to look up so a is
184. 6:0510
185. 6:0610 times 16 is 160 plus 1 times
186. 6:095 times 1 is 5. 160 plus 5 is 165. so if
187. 6:13i made a5
188. 6:14a hexadecimal hex a5 dollars i made 165
189. 6:18in decimal dollars pretty cool right
190. 6:22every base is base 10. here's an alien
191. 6:26and here's an astronaut bail
192. 6:27alien says there are 10 rocks astronaut
193. 6:30says
194. 6:31oh yeah you must be using base four uh
195. 6:34uh see i use base ten and alien says no
196. 6:37no i use base ten what's base four
197. 6:40because in every base there's a ten
198. 6:43and in fact every base ten is their base
199. 6:48base two remember binary one zero is two
200. 6:51base decimal one zero is ten hexadecimal
201. 6:54one
202. 6:54zero is sixteen so every base has a ten
203. 6:58and every base is that base all your
204. 7:01base
205. 7:02are belong to us so now let's convert
206. 7:04we've now converted from
207. 7:06decimal converted from binary and hex to
208. 7:09decimal
209. 7:10let's now convert from decimal to binary
210. 7:13and to hex
211. 7:15so let's let's reverse it don't tell
212. 7:17them this is the same number we
213. 7:18converted before okay
214. 7:1913. so this is an hour this is actually
215. 7:22the first algorithm that's i mean that
216. 7:23was an algorithm too but it was a little
217. 7:24easy
218. 7:24this is an algorithm that has some ifs
219. 7:26then we have to make a decision here
220. 7:28the way i think of this my analogy for
221. 7:30this this has worked well for
222. 7:31students when i've taught them to think
223. 7:34of the following
224. 7:34i've got these boxes to ship back to
225. 7:36amazon all labeled
226. 7:38in those bases i've got a box labeled
227. 7:40one box labeled two a box labeled four
228. 7:42box labeled eight okay
229. 7:45i want to reduce my postage amazon will
230. 7:48only take
231. 7:49a full box back remember it's either
232. 7:50full or empty i won't take a full box
233. 7:52back
234. 7:53so i first start by this i say i got 13
235. 7:57and the first the algorithm says this is
236. 7:59the column number by the way you know to
237. 8:01the left of this is a 16
238. 8:03this i should write this better there's
239. 8:04a 16 there's a 32
240. 8:07etc on the left of that so i can start
241. 8:09there let's start with the 16. this just
242. 8:10just to be just to go rogue for a second
243. 8:13i say
244. 8:14is the column number less than or equal
245. 8:16the number i have left the number of
246. 8:18eggs i have left to you know
247. 8:19boy those chickens were productive so i
248. 8:22got to ship 13
249. 8:23eggs back to amazon i got a box said 16
250. 8:25and i know amazon will
251. 8:26take a full box so can i fit 13 in a
252. 8:30box of 16 and fill it up i can't so i
253. 8:33write a zero here
254. 8:34boop pretty cool go to the next guy
255. 8:37so let's try this then i go to the eight
256. 8:39and i've actually animated this for you
257. 8:41so i say can i fit and fill an
258. 8:44eight box with whatever eggs i've got
259. 8:46left which is thirteen i say yes
260. 8:49then i ask how many of those
261. 8:52boxes can i fill well here only one in
262. 8:55fact in binary it's only gonna
263. 8:56be one or zero but so we'll see in hex
264. 8:57it's actually more we'll have to say in
265. 8:59general be more than just one
266. 9:01so once i filled eight how many do i
267. 9:03have left cross it off i say i've got
268. 9:05five left
269. 9:06let's keep going i go to my next box can
270. 9:09i fill a four
271. 9:10yes i can how many times can i fill the
272. 9:13four only once
273. 9:14so therefore i fill it up i take i fill
274. 9:17the four how many is left after i fill
275. 9:18the four box
276. 9:19one left i've got the next box we've got
277. 9:22a two can i fill the two with only one
278. 9:24egg
279. 9:24i cannot then i can't use it so a zero
280. 9:27in that one
281. 9:28go to the next one can i fill a box of
282. 9:30one that box labeled one with one
283. 9:32egg yes i can how many times can i fill
284. 9:34it only once there's one and i've seen
285. 9:36now
286. 9:37and i just by the way i stopped when my
287. 9:38my sum is zero so once i cross it off
288. 9:40and i get my zero i stop and i just put
289. 9:42zeros in the rest of these guys if i
290. 9:43have if i happen to let's say i had
291. 9:44eight i'd put a one for eight and put
292. 9:46all the zeros i could just
293. 9:47you know short circuit it and go up
294. 9:48right right to the end
295. 9:50so converting 13 to binary
296. 9:531101 is the equivalent by going down
297. 9:56thinking of the amazon egg boxes you got
298. 9:59it
299. 9:59now let's do this in hexadecimal
300. 10:04165 again don't tell the students but
301. 10:06this is the same number we had before
302. 10:08let's go 165 which is a
303. 10:11decimal number i want to convert that to
304. 10:14hexadecimal
305. 10:15so start with my columns in fact i've
306. 10:17added some extra ones here now
307. 10:194096. so i asked the question
308. 10:22can i fill the 4096 box probably 16 to
309. 10:24the three is 4096
310. 10:25another nice thing to memorize but i
311. 10:27wouldn't worry about that one too much
312. 10:30i cannot zero let's go to the next guy
313. 10:33can i fill the 256
314. 10:35labeled box with 165 i cannot zero
315. 10:39can i fill a 16 box yes i can
316. 10:43how many see this is the first time it's
317. 10:45come up how many
318. 10:4716s how many 16 boxes can i fill with my
319. 10:50165 eggs sitting all
320. 10:52on my floor 10 of them do i write
321. 10:5510 no i look up in that base
322. 10:58what the character the digit that
323. 11:01represents that is
324. 11:02and it's going to be an a i would list
325. 11:03an a not a 10 i'd list an
326. 11:05a so far once i have a or
327. 11:0810 of these 16 boxes that's 160
328. 11:13how many is left five and i keep going
329. 11:16same algorithm
330. 11:17this is the same algorithm for every
331. 11:18base isn't that pretty cool so now you
332. 11:20know
333. 11:20i could actually ask you an exam to
334. 11:22convert to some base you haven't heard
335. 11:23of before
336. 11:24and you should be able to do that same
337. 11:25algorithm okay so
338. 11:27five left i've got boxes with one how
339. 11:29many can i fill
340. 11:31uh you can can i fill a box with one yes
341. 11:32i can how many of those boxes can i fill
342. 11:34five i fill the five i'm down to zero
343. 11:37and i'm done
344. 11:38it's pretty cool this is fun right so
345. 11:41now
346. 11:42of the triangle here's a decimal binary
347. 11:45hex
348. 11:46i can now go back and forth that's
349. 11:47pretty cool i can't go this way yet
350. 11:49until this slide
351. 11:51boom binder to hex so i've got some
352. 11:54binary number
353. 11:55how do i convert it to to hex here's the
354. 11:57key don't make this mistake
355. 12:00you have to left pad it which means add
356. 12:03zeros to maybe you always add zeros to
357. 12:05the left
358. 12:05you have to left pad it so that it's all
359. 12:07the length of that binary digit
360. 12:10always is a multiple of four if you do
361. 12:13this
362. 12:13it'll work out trivially so if i give
363. 12:15you one one one one
364. 12:17four ones and a zero convert it to hex
365. 12:21what have i got one one one one
366. 12:24one one one one zero to hex well it's
367. 12:27only five
368. 12:28it's five let's add three more now i've
369. 12:30got
370. 12:32eight characters three zeros four ones
371. 12:35and a zero you see it right here okay
372. 12:37now this guy is eight
373. 12:41binary digits or bits by the way i
374. 12:43didn't mention binary digits
375. 12:45bi and the ts from digits becomes bits
376. 12:50that's why we call them bits so look
377. 12:52that up now i have this
378. 12:53look that up here's the table this table
379. 12:56right here boom
380. 12:58boom memorize this table
381. 13:01it will save you in time and will make
382. 13:03you feel like
383. 13:04oh stand a little taller 61c i know i
384. 13:07now know i'm fluent with hexadecimal
385. 13:09you can ask me boom what's uh i don't
386. 13:12know
387. 13:12let's look at it by the way the way you
388. 13:14do this then you look it up by
389. 13:15fours one of the things you're going to
390. 13:17notice is
391. 13:19four binary digits remember this from
392. 13:21before the first video
393. 13:22is two to the four sixteen things
394. 13:26well sixteen things is this number of
395. 13:28hexadecimal
396. 13:30options zero through f one accidental
397. 13:32digit is
398. 13:3316 things so there's my four there's my
399. 13:3616 binary bits i just looked them up i
400. 13:39say
401. 13:40what's that i just look it up boom
402. 13:43e so i write e
403. 13:46what's 0 i look it up 1
404. 13:50answer is 1e easy you could also
405. 13:53not do the padding but start from the
406. 13:54right okay if you start from the right
407. 13:56it doesn't matter so i
408. 13:57go here there's that 1 1 1 0. that's my
409. 14:00e
410. 14:01this is my one with some leading zeros
411. 14:03okay here's here's an important thing
412. 14:05if you go from the left you will get it
413. 14:07wrong
414. 14:09one one one one that would be f
415. 14:12zero and i can't pad to the right
416. 14:14because that changes the number
417. 14:16so it is not f zero it is not f zero
418. 14:19very important that you see that okay so
419. 14:21start from the right or
420. 14:22pad it so that's always multiple to four
421. 14:24and just that doesn't matter which way
422. 14:25you go
423. 14:27hex to binary piece of cake let's look
424. 14:29them up
425. 14:30literally look them up concatenate it
426. 14:32all together and then drop the leading
427. 14:33zeros
428. 14:34so one e so what's your e
429. 14:37there's your e we did this already
430. 14:40what's your one
431. 14:41drop leading zeros cross those off what
432. 14:43do you get
433. 14:44one one one one zero okay binary hex
434. 14:47that's the triangle
435. 14:48huh with the thing they made the
436. 14:50triangle okay
437. 14:53so four bits
438. 14:57okay let's start for the eight i start
439. 14:58with the second bullet point here eight
440. 14:59bits is called a byte
441. 15:00just historically it's called a byte
442. 15:03okay
443. 15:04now four bits half of that as you see
444. 15:06here this is this is my
445. 15:08four bit column that's called a nibble
446. 15:11get it a bite
447. 15:12see a bite and then a a nibble
448. 15:16you also you also have i've i've tried
449. 15:19to advocate that two bits should be like
450. 15:20a
451. 15:21taste you know i don't know nobody's
452. 15:23ever bought under there
453. 15:24one hex digit is 16 different things
454. 15:26we've seen that before
455. 15:28two hex digits okay here we go two hex
456. 15:30digits how many things can i do
457. 15:32if you remember before zero to zero zero
458. 15:35to f
459. 15:36that's that's that's two hex digits well
460. 15:38what's ff let's do the math
461. 15:40oh it's 255. starting from zero that
462. 15:43means 256 things
463. 15:44okay so or n hex digits is
464. 15:4816 to the n there's another way to think
465. 15:49about that now
466. 15:51turns out back in the days of thinking
467. 15:54of making displays
468. 15:55making computer displays okay they
469. 15:57realized that
470. 15:59we need just about eight digits eight
471. 16:01digits eight bits
472. 16:02to represent a single color partly
473. 16:04because the human eye can't see more
474. 16:05than if i had
475. 16:06not by the way they now have high-end
476. 16:08tvs that have more of those bits which
477. 16:10is interesting because they can do
478. 16:10dynamic range there's a long longer
479. 16:12conversation there
480. 16:13but the point of this is early tvs used
481. 16:16eight bits for each of the red
482. 16:18green and blue channels they concatenate
483. 16:20them together to make a 24-bit
484. 16:22color you need to know this for your
485. 16:24first project
486. 16:25let's see how that is illusion yeah
487. 16:30so zero to 255 red zero 255
488. 16:34green 55 blue you concatenate them
489. 16:37together
490. 16:38what you get is six hexadecimal digits
491. 16:42and that's a color and if you've ever
492. 16:44seen web programming or seen some color
493. 16:47on a mac
494. 16:47by the way i could go rogue and show you
495. 16:49this this is probably we'll show you
496. 16:50this in a second
497. 16:51this is a color represented as six
498. 16:55hexadecimal digits i'm gonna go rogue
499. 16:57because i love doing this demo
500. 16:59here we go i love this demo and the show
501. 17:02keep my imitation discovery watch this
502. 17:04so here we go ready apple
503. 17:06system preferences desktop and screen
504. 17:09saver
505. 17:11desktop
506. 17:15i can choose a color i can customize the
507. 17:18color there's a long long way to get
508. 17:20there
509. 17:20i could have done this i could have done
510. 17:21this by the way in powerpoint
511. 17:25look at this this is rgb sliders
512. 17:30that and here is the hex value so all
513. 17:33white
514. 17:34is all f's if i
515. 17:37take these values down look at this
516. 17:39watch how everything changes color
517. 17:41as i move this color around watch this
518. 17:45ff000 is all red
519. 17:48huh with the thing if i add this color
520. 17:51in
521. 17:53ffff00 it's yellow
522. 17:56is that cool so i can now adjust this
523. 17:58and play with this and i encourage you
524. 17:59to
525. 18:00find one of these things one of these
526. 18:01tools on the web to this
527. 18:03to play with the colors to see actually
528. 18:04colors here's d0367f
529. 18:08uh as a color here on the bottom pretty
530. 18:12cool
531. 18:14which base do we use now you've got me
532. 18:15six guys which one do i use well decimal
533. 18:17is obviously great for humans especially
534. 18:19when doing arithmetic we know how to do
535. 18:20that very easily
536. 18:21hexadecimal terrible for arithmetic just
537. 18:24terrible
538. 18:25but if you're looking at a string of
539. 18:26binary digits i could either give you
540. 18:28a crazy long number by digits or i could
541. 18:30compact them down to a hexadecimal
542. 18:32equivalent by the way would you prefer
543. 18:33to give you
544. 18:34that six hexadecimal digit color or 24
545. 18:38bits you don't want the 24 bits right
546. 18:40so i don't want that in binary
547. 18:43what computers use you're going to be
548. 18:45able to do multiplication division
549. 18:47subtraction all the operations you think
550. 18:49of in all the numbers computers always
551. 18:51use binary because that's what they can
552. 18:52do
553. 18:52we'll talk about that a little later in
554. 18:54the class transistors only know two
555. 18:56states on or off
556. 18:57current flowing or not flowing so
557. 18:58they're going to use binary because of
558. 18:59that
559. 19:01regardless of how a number is written 32
560. 19:04base 10
561. 19:04or 32 base 1 0
562. 19:07or 0x 20 or 1 0
563. 19:110 0 000 base 2 or 0b1000
564. 19:16they represent the same number those are
565. 19:18all numerals
566. 19:19representing the same number we'll talk
567. 19:20about numerals in the next next video
568. 19:23please do use the subscripts or the zero
569. 19:25x or the zero b to precede your number
570. 19:27so it's not ambiguous
571. 19:29the computer knows it too this is really
572. 19:32cool actually
573. 19:33here's a little piece of c code we're
574. 19:34gonna see in the next lecture but check
575. 19:36this out i say 1 2 3 4 is my
576. 19:38number right base 10 okay ten
577. 19:42and i say let's just print this out so i
578. 19:44print this out in percent d
579. 19:45for you can see this here in percent d
580. 19:48for
581. 19:49decimal percent d
582. 19:52percent x and percent o for octal
583. 19:55base eight what does it print
584. 19:58124 yeah that's good what else is a
585. 20:02print
586. 20:034d2 this is a problem
587. 20:06it doesn't proceed with 0x or have a
588. 20:08subscript 16.
589. 20:10this is bad this means if it were like
590. 20:12let's say let's say this number was 16
591. 20:14it represents one zero and you wouldn't
592. 20:15know what base it is
593. 20:16so you better add the zero x yourself
594. 20:18when you're printing it to percent x
595. 20:20you got that good percent oh well how
596. 20:22does it do percent oh it turns out that
597. 20:24the way we do the great computers do
598. 20:25octal is you
599. 20:26lead at a leading zero it's a little
600. 20:28ambiguous because you can always have a
601. 20:29leading zero
602. 20:30but by having a leading zero it tells
603. 20:32the computer it's going to be octal
604. 20:33we'll see that as we put these numbers
605. 20:35in
606. 20:36i can also go the other way watch this i
607. 20:39can say
608. 20:40pass the hexadecimal number in look at
609. 20:43this i can
610. 20:43hard code a hexadecimal number in by
611. 20:46saying 0x
612. 20:47if i say percent d that says convert it
613. 20:49to decimal what does it give me
614. 20:50one two three four is that cool same
615. 20:53thing
616. 20:54zero b some huge thing percent d
617. 20:57converts it binary
618. 20:58to decimal and here's my octal number
619. 21:02preceded by a zero and here's percent d
620. 21:04and we get one two three four for all of
621. 21:06them
622. 21:06isn't that cool so i just took these
623. 21:08numbers here and put them there
624. 21:10all right we're done with the series of
625. 21:12we're done with this lecture
626. 21:14the next lecture we're going to talk
627. 21:15about how to represent
628. 21:17negative numbers positive numbers in a
629. 21:19more general way number representation
630. 21:21we'll see you there