Dates and Times II

Time spans and time zones.

Last Time

1. A date-time counts seconds since 1970 (Unix time).
2. Parsing turns text into a date-time.
3. Getters and setters read and change its parts.
4. Some parts, like months, are circular.

Key Ideas for Today

1. A duration is a fixed span in seconds.
2. A period is a span in human units.
3. UTC is the world’s reference clock.
4. A time zone is an offset from UTC.

Concept Acquisition

1. Subtracting two dates gives a difftime, whose units vary.
2. A duration measures a span in exact seconds, so adding one ignores daylight saving time and leap years.
3. A period measures a span in human units (days, weeks, years) whose length in seconds varies.
4. UTC is the global time standard; each time zone is an offset from it, identified in R by an Olson name.
5. Changing a time zone can either keep the instant and change how it prints, or keep the clock reading and change the instant.

Tool Acquisition

1. today(), ymd(), and ymd_hms(tz =).
2. as.duration() and the duration constructors dseconds() through dyears().
3. as.period() and the period constructors seconds() through years().
4. interval().
5. Sys.timezone(), OlsonNames(), with_tz(), and force_tz().

Concept Application

1. Compute an age as a duration and as a period, and the age one year from now.
2. Predict the result of adding one day to 1 am on the day daylight saving time starts, as a duration and as a period.
3. Sketch the tidy and the raw data frames behind “The City that Sleeps,” which compares transit in two time zones.

Please answer the following two questions. I will call on someone randomly after the timer is up.

1. How long is a day?
2. How long is a year?

00:30

Agenda

1. How does a computer Store dates and times?
2. How do we Create dates and times?
3. How do we Get and Set them?

4. How do you specify Time Spans?
5. How do you account for Time Zones?

Time Spans

How old am I?

library(lubridate)
a <- today()
b <- ymd("1983-02-28")
class(a)

[1] "Date"

class(b)

[1] "Date"

a - b

Time difference of 15948 days

Durations

Subtraction of two Date vectors -> difftime vector.

class(a - b)

[1] "difftime"

On the pinned board, draw a timeline from 1970, mark b and a, and label the span in days and in seconds. Return to it at the end of Time Spans to add the distinction between calendar time and physical time.

A difftime can be recorded in many different units, so better to use lubridate::as.duration() to always record it in seconds.

my_age_s <- as.duration(a - b)
class(my_age_s)

[1] "Duration"
attr(,"package")
[1] "lubridate"

my_age_s

[1] "1377907200s (~43.66 years)"

Duration

In lubridate, a time span with a fixed length in seconds. It tracks physical time, so it ignores irregularities in the calendar like daylight saving time and leap years.

Make one by coercing a difftime with as.duration() or with a constructor function:

dseconds(15)

[1] "15s"

dminutes(10)

[1] "600s (~10 minutes)"

dhours(c(12, 24))

[1] "43200s (~12 hours)" "86400s (~1 days)"  

ddays(0:3)

[1] "0s"                "86400s (~1 days)"  "172800s (~2 days)"
[4] "259200s (~3 days)"

dweeks(3)

[1] "1814400s (~3 weeks)"

dyears(1)

[1] "31557600s (~1 years)"

Some Wrinkles

one_am <- ymd_hms("2026-03-08 01:00:00", tz = "America/New_York")
one_am

[1] "2026-03-08 01:00:00 EST"

What should the following yield?

one_am + ddays(1)

[1] "2026-03-09 02:00:00 EDT"

Daylight savings time!

Please answer the following two questions. I will call on someone randomly after the timer is up.

1. How long is a day?
2. How long is a year?

00:30

This is a revisit of the opening question.

Point out that actually “days” don’t have a single fixed length. Most days are 24 hours but at the edge of daylight savings time, its shorter or longer. Same with leap years. So if you talk about the average day or year length, it’s not quite 24 hrs or 365 days.

Period

In lubridate, a time span that is like a duration except it doesn’t have a fixed length in seconds. It tracks human units (day, week) that are responsive to irregularities.

Example period constructors:

seconds(15)

[1] "15S"

minutes(10)

[1] "10M 0S"

hours(c(12, 24))

[1] "12H 0M 0S" "24H 0M 0S"

days(0:3)

[1] "0S"          "1d 0H 0M 0S" "2d 0H 0M 0S" "3d 0H 0M 0S"

weeks(3)

[1] "21d 0H 0M 0S"

years(1)

[1] "1y 0m 0d 0H 0M 0S"

Working with periods

one_am

[1] "2026-03-08 01:00:00 EST"

one_am + days(1)

[1] "2026-03-09 01:00:00 EDT"

What will the following yield?

ymd("2024-01-01") + dyears(1)

[1] "2024-12-31 06:00:00 UTC"

ymd("2024-01-01") + years(1) # accounts for leap year

[1] "2025-01-01"

How old will I be…

In a year?

my_age_s

[1] "1377907200s (~43.66 years)"

my_age_s + dyears(1) # adding average seconds in a year

[1] "1409464800s (~44.66 years)"

my_age_s + years(1) # adding a calendar year

Error:
! Incompatible classes: <Duration> + <Period>

Interval

In lubridate, a time span with a specific start and end. Because it knows where it starts, it can be measured in seconds or in calendar units.

leap <- interval(ymd("2024-01-01"),
                 ymd("2025-01-01"))
leap

[1] 2024-01-01 UTC--2025-01-01 UTC

as.duration(leap)

[1] "31622400s (~1 years)"

as.period(leap)

[1] "1y 0m 0d 0H 0M 0S"

How old will I be in a year?

my_age_i <- interval(b, a)
as.period(my_age_i)

[1] "43y 8m 0d 0H 0M 0S"

as.period(my_age_i) + years(1)

[1] "44y 8m 0d 0H 0M 0S"

Time Spans recap

Base R

• difftime: inconsistent units

lubridate

• duration: fixed span in seconds
• period: variable span in human units
• interval: span with a start and an end

On the pinned board, return to the timeline from “Durations”. Label the span between b and a as an interval, and add the distinction between physical time (durations, counted in seconds) and calendar time (periods, counted in human units).

The City that Sleeps

1. Sketch one data frame indicating the rows and cols (and their type) to make this plot in ggplot2.

2. Sketch two data frames that represent the original raw data as it likely is recorded by the Kyoto transit authority and the New York transit authority.

04:00

Time Zones

UTC

Coordinated Universal Time (UTC) is the primary global time standard. Successor to GMT.

Offsets: each time zone has a particular offset from UTC.

• London: +0 (+1 in summer)
• New York: -5 (-4 in summer)
• Kyoto: +9

Sys.timezone() # What R says my timezone is

[1] "Etc/UTC"

ymd_hms("2024-06-01 12:00:00", tz = "UTC")

[1] "2024-06-01 12:00:00 UTC"

ymd_hms("2024-06-01 12:00:00", tz = "America/New_York")

[1] "2024-06-01 12:00:00 EDT"

Olson Names

OlsonNames()

  [1] "Africa/Abidjan"                 "Africa/Accra"                  
  [3] "Africa/Addis_Ababa"             "Africa/Algiers"                
  [5] "Africa/Asmara"                  "Africa/Bamako"                 
  [7] "Africa/Bangui"                  "Africa/Banjul"                 
  [9] "Africa/Bissau"                  "Africa/Blantyre"               
 [11] "Africa/Brazzaville"             "Africa/Bujumbura"              
 [13] "Africa/Cairo"                   "Africa/Casablanca"             
 [15] "Africa/Ceuta"                   "Africa/Conakry"                
 [17] "Africa/Dakar"                   "Africa/Dar_es_Salaam"          
 [19] "Africa/Djibouti"                "Africa/Douala"                 
 [21] "Africa/El_Aaiun"                "Africa/Freetown"               
 [23] "Africa/Gaborone"                "Africa/Harare"                 
 [25] "Africa/Johannesburg"            "Africa/Juba"                   
 [27] "Africa/Kampala"                 "Africa/Khartoum"               
 [29] "Africa/Kigali"                  "Africa/Kinshasa"               
 [31] "Africa/Lagos"                   "Africa/Libreville"             
 [33] "Africa/Lome"                    "Africa/Luanda"                 
 [35] "Africa/Lubumbashi"              "Africa/Lusaka"                 
 [37] "Africa/Malabo"                  "Africa/Maputo"                 
 [39] "Africa/Maseru"                  "Africa/Mbabane"                
 [41] "Africa/Mogadishu"               "Africa/Monrovia"               
 [43] "Africa/Nairobi"                 "Africa/Ndjamena"               
 [45] "Africa/Niamey"                  "Africa/Nouakchott"             
 [47] "Africa/Ouagadougou"             "Africa/Porto-Novo"             
 [49] "Africa/Sao_Tome"                "Africa/Timbuktu"               
 [51] "Africa/Tripoli"                 "Africa/Tunis"                  
 [53] "Africa/Windhoek"                "America/Adak"                  
 [55] "America/Anchorage"              "America/Anguilla"              
 [57] "America/Antigua"                "America/Araguaina"             
 [59] "America/Argentina/Buenos_Aires" "America/Argentina/Catamarca"   
 [61] "America/Argentina/Cordoba"      "America/Argentina/Jujuy"       
 [63] "America/Argentina/La_Rioja"     "America/Argentina/Mendoza"     
 [65] "America/Argentina/Rio_Gallegos" "America/Argentina/Salta"       
 [67] "America/Argentina/San_Juan"     "America/Argentina/San_Luis"    
 [69] "America/Argentina/Tucuman"      "America/Argentina/Ushuaia"     
 [71] "America/Aruba"                  "America/Asuncion"              
 [73] "America/Atikokan"               "America/Atka"                  
 [75] "America/Bahia"                  "America/Bahia_Banderas"        
 [77] "America/Barbados"               "America/Belem"                 
 [79] "America/Belize"                 "America/Blanc-Sablon"          
 [81] "America/Boa_Vista"              "America/Bogota"                
 [83] "America/Boise"                  "America/Cambridge_Bay"         
 [85] "America/Campo_Grande"           "America/Cancun"                
 [87] "America/Caracas"                "America/Cayenne"               
 [89] "America/Cayman"                 "America/Chicago"               
 [91] "America/Chihuahua"              "America/Ciudad_Juarez"         
 [93] "America/Coral_Harbour"          "America/Costa_Rica"            
 [95] "America/Coyhaique"              "America/Creston"               
 [97] "America/Cuiaba"                 "America/Curacao"               
 [99] "America/Danmarkshavn"           "America/Dawson"                
[101] "America/Dawson_Creek"           "America/Denver"                
[103] "America/Detroit"                "America/Dominica"              
[105] "America/Edmonton"               "America/Eirunepe"              
[107] "America/El_Salvador"            "America/Ensenada"              
[109] "America/Fort_Nelson"            "America/Fortaleza"             
[111] "America/Glace_Bay"              "America/Goose_Bay"             
[113] "America/Grand_Turk"             "America/Grenada"               
[115] "America/Guadeloupe"             "America/Guatemala"             
[117] "America/Guayaquil"              "America/Guyana"                
[119] "America/Halifax"                "America/Havana"                
[121] "America/Hermosillo"             "America/Indiana/Indianapolis"  
[123] "America/Indiana/Knox"           "America/Indiana/Marengo"       
[125] "America/Indiana/Petersburg"     "America/Indiana/Tell_City"     
[127] "America/Indiana/Vevay"          "America/Indiana/Vincennes"     
[129] "America/Indiana/Winamac"        "America/Inuvik"                
[131] "America/Iqaluit"                "America/Jamaica"               
[133] "America/Juneau"                 "America/Kentucky/Louisville"   
[135] "America/Kentucky/Monticello"    "America/Kralendijk"            
[137] "America/La_Paz"                 "America/Lima"                  
[139] "America/Los_Angeles"            "America/Lower_Princes"         
[141] "America/Maceio"                 "America/Managua"               
[143] "America/Manaus"                 "America/Marigot"               
[145] "America/Martinique"             "America/Matamoros"             
[147] "America/Mazatlan"               "America/Menominee"             
[149] "America/Merida"                 "America/Metlakatla"            
[151] "America/Mexico_City"            "America/Miquelon"              
[153] "America/Moncton"                "America/Monterrey"             
[155] "America/Montevideo"             "America/Montreal"              
[157] "America/Montserrat"             "America/Nassau"                
[159] "America/New_York"               "America/Nipigon"               
[161] "America/Nome"                   "America/Noronha"               
[163] "America/North_Dakota/Beulah"    "America/North_Dakota/Center"   
[165] "America/North_Dakota/New_Salem" "America/Nuuk"                  
[167] "America/Ojinaga"                "America/Panama"                
[169] "America/Pangnirtung"            "America/Paramaribo"            
[171] "America/Phoenix"                "America/Port_of_Spain"         
[173] "America/Port-au-Prince"         "America/Porto_Acre"            
[175] "America/Porto_Velho"            "America/Puerto_Rico"           
[177] "America/Punta_Arenas"           "America/Rainy_River"           
[179] "America/Rankin_Inlet"           "America/Recife"                
[181] "America/Regina"                 "America/Resolute"              
[183] "America/Rio_Branco"             "America/Santa_Isabel"          
[185] "America/Santarem"               "America/Santiago"              
[187] "America/Santo_Domingo"          "America/Sao_Paulo"             
[189] "America/Scoresbysund"           "America/Shiprock"              
[191] "America/Sitka"                  "America/St_Barthelemy"         
[193] "America/St_Johns"               "America/St_Kitts"              
[195] "America/St_Lucia"               "America/St_Thomas"             
[197] "America/St_Vincent"             "America/Swift_Current"         
[199] "America/Tegucigalpa"            "America/Thule"                 
[201] "America/Thunder_Bay"            "America/Tijuana"               
[203] "America/Toronto"                "America/Tortola"               
[205] "America/Vancouver"              "America/Virgin"                
[207] "America/Whitehorse"             "America/Winnipeg"              
[209] "America/Yakutat"                "America/Yellowknife"           
[211] "Antarctica/Casey"               "Antarctica/Davis"              
[213] "Antarctica/DumontDUrville"      "Antarctica/Macquarie"          
[215] "Antarctica/Mawson"              "Antarctica/McMurdo"            
[217] "Antarctica/Palmer"              "Antarctica/Rothera"            
[219] "Antarctica/Syowa"               "Antarctica/Troll"              
[221] "Antarctica/Vostok"              "Arctic/Longyearbyen"           
[223] "Asia/Aden"                      "Asia/Almaty"                   
[225] "Asia/Amman"                     "Asia/Anadyr"                   
[227] "Asia/Aqtau"                     "Asia/Aqtobe"                   
[229] "Asia/Ashgabat"                  "Asia/Atyrau"                   
[231] "Asia/Baghdad"                   "Asia/Bahrain"                  
[233] "Asia/Baku"                      "Asia/Bangkok"                  
[235] "Asia/Barnaul"                   "Asia/Beirut"                   
[237] "Asia/Bishkek"                   "Asia/Brunei"                   
[239] "Asia/Chita"                     "Asia/Choibalsan"               
[241] "Asia/Chongqing"                 "Asia/Colombo"                  
[243] "Asia/Damascus"                  "Asia/Dhaka"                    
[245] "Asia/Dili"                      "Asia/Dubai"                    
[247] "Asia/Dushanbe"                  "Asia/Famagusta"                
[249] "Asia/Gaza"                      "Asia/Harbin"                   
[251] "Asia/Hebron"                    "Asia/Ho_Chi_Minh"              
[253] "Asia/Hong_Kong"                 "Asia/Hovd"                     
[255] "Asia/Irkutsk"                   "Asia/Istanbul"                 
[257] "Asia/Jakarta"                   "Asia/Jayapura"                 
[259] "Asia/Jerusalem"                 "Asia/Kabul"                    
[261] "Asia/Kamchatka"                 "Asia/Karachi"                  
[263] "Asia/Kashgar"                   "Asia/Kathmandu"                
[265] "Asia/Khandyga"                  "Asia/Kolkata"                  
[267] "Asia/Krasnoyarsk"               "Asia/Kuala_Lumpur"             
[269] "Asia/Kuching"                   "Asia/Kuwait"                   
[271] "Asia/Macau"                     "Asia/Magadan"                  
[273] "Asia/Makassar"                  "Asia/Manila"                   
[275] "Asia/Muscat"                    "Asia/Nicosia"                  
[277] "Asia/Novokuznetsk"              "Asia/Novosibirsk"              
[279] "Asia/Omsk"                      "Asia/Oral"                     
[281] "Asia/Phnom_Penh"                "Asia/Pontianak"                
[283] "Asia/Pyongyang"                 "Asia/Qatar"                    
[285] "Asia/Qostanay"                  "Asia/Qyzylorda"                
[287] "Asia/Riyadh"                    "Asia/Sakhalin"                 
[289] "Asia/Samarkand"                 "Asia/Seoul"                    
[291] "Asia/Shanghai"                  "Asia/Singapore"                
[293] "Asia/Srednekolymsk"             "Asia/Taipei"                   
[295] "Asia/Tashkent"                  "Asia/Tbilisi"                  
[297] "Asia/Tehran"                    "Asia/Tel_Aviv"                 
[299] "Asia/Thimphu"                   "Asia/Tokyo"                    
[301] "Asia/Tomsk"                     "Asia/Ulaanbaatar"              
[303] "Asia/Urumqi"                    "Asia/Ust-Nera"                 
[305] "Asia/Vientiane"                 "Asia/Vladivostok"              
[307] "Asia/Yakutsk"                   "Asia/Yangon"                   
[309] "Asia/Yekaterinburg"             "Asia/Yerevan"                  
[311] "Atlantic/Azores"                "Atlantic/Bermuda"              
[313] "Atlantic/Canary"                "Atlantic/Cape_Verde"           
[315] "Atlantic/Faroe"                 "Atlantic/Jan_Mayen"            
[317] "Atlantic/Madeira"               "Atlantic/Reykjavik"            
[319] "Atlantic/South_Georgia"         "Atlantic/St_Helena"            
[321] "Atlantic/Stanley"               "Australia/Adelaide"            
[323] "Australia/Brisbane"             "Australia/Broken_Hill"         
[325] "Australia/Canberra"             "Australia/Currie"              
[327] "Australia/Darwin"               "Australia/Eucla"               
[329] "Australia/Hobart"               "Australia/Lindeman"            
[331] "Australia/Lord_Howe"            "Australia/Melbourne"           
[333] "Australia/Perth"                "Australia/Sydney"              
[335] "Australia/Yancowinna"           "CET"                           
[337] "CST6CDT"                        "EET"                           
[339] "EST"                            "EST5EDT"                       
[341] "Etc/GMT"                        "Etc/GMT-0"                     
[343] "Etc/GMT-1"                      "Etc/GMT-10"                    
[345] "Etc/GMT-11"                     "Etc/GMT-12"                    
[347] "Etc/GMT-13"                     "Etc/GMT-14"                    
[349] "Etc/GMT-2"                      "Etc/GMT-3"                     
[351] "Etc/GMT-4"                      "Etc/GMT-5"                     
[353] "Etc/GMT-6"                      "Etc/GMT-7"                     
[355] "Etc/GMT-8"                      "Etc/GMT-9"                     
[357] "Etc/GMT+0"                      "Etc/GMT+1"                     
[359] "Etc/GMT+10"                     "Etc/GMT+11"                    
[361] "Etc/GMT+12"                     "Etc/GMT+2"                     
[363] "Etc/GMT+3"                      "Etc/GMT+4"                     
[365] "Etc/GMT+5"                      "Etc/GMT+6"                     
[367] "Etc/GMT+7"                      "Etc/GMT+8"                     
[369] "Etc/GMT+9"                      "Etc/GMT0"                      
[371] "Etc/Greenwich"                  "Etc/UCT"                       
[373] "Etc/Universal"                  "Etc/UTC"                       
[375] "Etc/Zulu"                       "Europe/Amsterdam"              
[377] "Europe/Andorra"                 "Europe/Astrakhan"              
[379] "Europe/Athens"                  "Europe/Belfast"                
[381] "Europe/Belgrade"                "Europe/Berlin"                 
[383] "Europe/Bratislava"              "Europe/Brussels"               
[385] "Europe/Bucharest"               "Europe/Budapest"               
[387] "Europe/Busingen"                "Europe/Chisinau"               
[389] "Europe/Copenhagen"              "Europe/Dublin"                 
[391] "Europe/Gibraltar"               "Europe/Guernsey"               
[393] "Europe/Helsinki"                "Europe/Isle_of_Man"            
[395] "Europe/Istanbul"                "Europe/Jersey"                 
[397] "Europe/Kaliningrad"             "Europe/Kirov"                  
[399] "Europe/Kyiv"                    "Europe/Lisbon"                 
[401] "Europe/Ljubljana"               "Europe/London"                 
[403] "Europe/Luxembourg"              "Europe/Madrid"                 
[405] "Europe/Malta"                   "Europe/Mariehamn"              
[407] "Europe/Minsk"                   "Europe/Monaco"                 
[409] "Europe/Moscow"                  "Europe/Nicosia"                
[411] "Europe/Oslo"                    "Europe/Paris"                  
[413] "Europe/Podgorica"               "Europe/Prague"                 
[415] "Europe/Riga"                    "Europe/Rome"                   
[417] "Europe/Samara"                  "Europe/San_Marino"             
[419] "Europe/Sarajevo"                "Europe/Saratov"                
[421] "Europe/Simferopol"              "Europe/Skopje"                 
[423] "Europe/Sofia"                   "Europe/Stockholm"              
[425] "Europe/Tallinn"                 "Europe/Tirane"                 
[427] "Europe/Tiraspol"                "Europe/Ulyanovsk"              
[429] "Europe/Vaduz"                   "Europe/Vatican"                
[431] "Europe/Vienna"                  "Europe/Vilnius"                
[433] "Europe/Volgograd"               "Europe/Warsaw"                 
[435] "Europe/Zagreb"                  "Europe/Zurich"                 
[437] "Factory"                        "GMT"                           
[439] "HST"                            "Indian/Antananarivo"           
[441] "Indian/Chagos"                  "Indian/Christmas"              
[443] "Indian/Cocos"                   "Indian/Comoro"                 
[445] "Indian/Kerguelen"               "Indian/Mahe"                   
[447] "Indian/Maldives"                "Indian/Mauritius"              
[449] "Indian/Mayotte"                 "Indian/Reunion"                
[451] "MET"                            "MST"                           
[453] "MST7MDT"                        "Pacific/Apia"                  
[455] "Pacific/Auckland"               "Pacific/Bougainville"          
[457] "Pacific/Chatham"                "Pacific/Chuuk"                 
[459] "Pacific/Easter"                 "Pacific/Efate"                 
[461] "Pacific/Fakaofo"                "Pacific/Fiji"                  
[463] "Pacific/Funafuti"               "Pacific/Galapagos"             
[465] "Pacific/Gambier"                "Pacific/Guadalcanal"           
[467] "Pacific/Guam"                   "Pacific/Honolulu"              
[469] "Pacific/Johnston"               "Pacific/Kanton"                
[471] "Pacific/Kiritimati"             "Pacific/Kosrae"                
[473] "Pacific/Kwajalein"              "Pacific/Majuro"                
[475] "Pacific/Marquesas"              "Pacific/Midway"                
[477] "Pacific/Nauru"                  "Pacific/Niue"                  
[479] "Pacific/Norfolk"                "Pacific/Noumea"                
[481] "Pacific/Pago_Pago"              "Pacific/Palau"                 
[483] "Pacific/Pitcairn"               "Pacific/Pohnpei"               
[485] "Pacific/Port_Moresby"           "Pacific/Rarotonga"             
[487] "Pacific/Saipan"                 "Pacific/Samoa"                 
[489] "Pacific/Tahiti"                 "Pacific/Tarawa"                
[491] "Pacific/Tongatapu"              "Pacific/Wake"                  
[493] "Pacific/Wallis"                 "Pacific/Yap"                   
[495] "PST8PDT"                        "UTC"                           
[497] "WET"                           
attr(,"Version")
[1] "2026c"

Changing time

Two flavors:

Change the instant in time.

x1 <- ymd_hms("2024-06-01 12:00:00", tz = "America/New_York")
x1

[1] "2024-06-01 12:00:00 EDT"

force_tz(x1, tzone = "Asia/Tokyo")

[1] "2024-06-01 12:00:00 JST"

Keep the same instant but change the way it’s printed:

with_tz(x1, tzone = "Asia/Tokyo")

[1] "2024-06-02 01:00:00 JST"

Guess the Metaphor

Time zones are like the chrono chaos ruins: one clock tower stands upright while the world around it splinters into shards that don’t line up. UTC is the tower; every time zone is a shard set at its own angle, and daylight saving time knocks some of them askew twice a year.

Working with Timezones

Advice: During your data cleaning step, convert date-time data to use UTC. Run all computations on UTC. Consider during the Explanatory Data Analysis phase if local is more useful.

Key Ideas for Today

1. A duration is a fixed span in seconds.
2. A period is a span in human units.
3. UTC is the world’s reference clock.
4. A time zone is an offset from UTC.

For Next Time

1. Finish PS 4, due tomorrow (Saturday 10/17) at 11:59 pm.
2. Take Quiz 4 on Monday or Tuesday (10/19-10/20).
3. Project Studio 3 is next week.