Primary Mathematics/Time expressions

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Primary Mathematics
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Time math[edit]

Time is often difficult to perform mathematical operations upon due to all the different units used.

Units[edit]

Seconds[edit]

The second is the basic unit of time. In scientific areas, seconds are often used for even large time measurements, with scientific notation applied in such cases. In common usage, however, many other time intervals are also used.

Seconds are abbreviated "sec" or just "s".

Minutes[edit]

There are 60 seconds per minute.

Minutes are abbreviated "min" or just "m".

Hours[edit]

There are also 60 minutes per hour, which means there are 60×60 or 3600 seconds per hour.

Hours are abbreviated "hr" or just "h".

Days[edit]

There are 24 hours per day, which means there are 60×24 or 1440 minutes per day. This, in turn, means there are 60×60×24 or 86400 seconds per day.

Days are abbreviated "d".

Weeks[edit]

There are 7 days per week. This means there are 7×24 or 168 hours per week.

Weeks are abbreviated "wk".

Months[edit]

There are 4 1/3 weeks per month, on average, and from 28 to 31 days per month, as follows:

28 or 29 days, depending on leap year (see the section below for leap year info):
  • February
30 days:
  • April, June, September, and November.
31 days:
  • All remaining months.

Months are abbreviated "mo"

Memory methods[edit]

Recalling which months have how many days can be tricky. There are some memory methods available to assist you, however:

  • Knuckle method. Place your two closed fists together and use the knuckles and valleys between the knuckles to represent months:
  J   M   M   J    A   O   D
  A   A   A   U    U   C   E
  N   R   Y   L    G   T   C
  _   _   _   _    _   _   _   _
 / \_/ \_/ \_/ \  / \_/ \_/ \_/ \
|               ||               |
|   F   A   J   ||   S   N       |
|   E   P   U   ||   E   O       |
|   B   R   N   ||   P   V       |
|               ||               |
|              /  \              |
|             /    \             |
|            /      \            |
The knuckles represent months with 31 days while the valleys represent months with 30 days (or less, in the case of February). Note that the thumbs and the gap between the hands are not used, and neither is the last valley and last knuckle.
  • Rhyme method.

"30 days hath September, April, June, and November..."

Years[edit]

The length of the year is based on how long it takes the Earth to revolve once around the Sun, which is approximately 365 ¼ days.

Regular years have 365 days, while leap years have 366 (due to the addition of February 29th).

Normal years have 52 weeks and 1 day, while leap years have 52 weeks and 2 days.

All years have 12 months.

Years are abbreviated "yr" or just "y".

What years are leap years ?[edit]

Since there is a fractional number of days in a year, not an integer, some adjustments need to be made to prevent drift from occurring (where January would eventually end up in the summer). There are occasional "leap seconds" added to some years, in addition to leap years, to keep this from happening. Here are the rules for leap years:

  • Every fourth year is a leap year:
  • Except every 100th year, which is not a leap year.
  • Except every 400th year, which is a leap year.
Here's a list of "every 4th year" from 1900 through 2100. All are leap years except for 1900 and 2100:
  • 1900 (Not a leap year, due to the "every 100th year" rule).
  • 1904
  • 1908
  • 1912
  • 1916
  • 1920
  • 1924
  • 1928
  • 1932
  • 1936
  • 1940
  • 1944
  • 1948
  • 1952
  • 1956
  • 1960
  • 1964
  • 1968
  • 1972
  • 1976
  • 1980
  • 1984
  • 1988
  • 1992
  • 1996
  • 2000 (Is a leap year, due to the "every 400th year" rule).
  • 2004
  • 2008
  • 2012
  • 2016
  • 2020
  • 2024
  • 2028
  • 2032
  • 2036
  • 2040
  • 2044
  • 2048
  • 2052
  • 2056
  • 2060
  • 2064
  • 2068
  • 2072
  • 2076
  • 2080
  • 2084
  • 2088
  • 2092
  • 2096
  • 2100 (Not a leap year, due to the "every 100th year" rule).

Longer periods[edit]

  • Decade = 10 years
  • Century = 100 years
  • Millenium = 1000 years
  • Myr = million years
  • Byr = billion years

Systems[edit]

12 hour clock[edit]

In this traditional system, the first 12 hours in the day are called "AM" ("ante meridian", or before noon) and the last 12 are called "PM" ("post meridian", or after noon). One oddity of this system is that no zero hours are used, with 12 (from the previous time period) used in it's place.

24 hour clock[edit]

In this system, sometimes called military time, the first number is zero, and the full 24 hours is counted:

Common name           12-hour clock   24-hour clock
=================     =============   =============
"Midnight"            12:00 AM         0:00
"1 in the morning"     1:00 AM         1:00
"2 in the morning"     2:00 AM         2:00
"3 in the morning"     3:00 AM         3:00
"4 in the morning"     4:00 AM         4:00
"5 in the morning"     5:00 AM         5:00
"6 in the morning"     6:00 AM         6:00
"7 in the morning"     7:00 AM         7:00
"8 in the morning"     8:00 AM         8:00
"9 in the morning"     9:00 AM         9:00
"10 in the morning"   10:00 AM        10:00
"11 in the morning"   11:00 AM        11:00
"Noon"                12:00 PM        12:00 
"1 in the afternoon"   1:00 PM        13:00
"2 in the afternoon"   2:00 PM        14:00
"3 in the afternoon"   3:00 PM        15:00
"4 in the afternoon"   4:00 PM        16:00
"5 in the afternoon"   5:00 PM        17:00
"6 in the evening"     6:00 PM        18:00
"7 in the evening"     7:00 PM        19:00
"8 in the evening"     8:00 PM        20:00
"9 in the evening"     9:00 PM        21:00
"10 in the evening"   10:00 PM        22:00
"11 in the evening"   11:00 PM        23:00

In the military, the time isn't necessarily reset at midnight. For example, a two-day operation may go up to 48:00 (said "forty-eight hundred"). This is done to make time math simpler. So, for example, the time between 20:00 and 40:00 is 20 hours.

Decimal time[edit]

There was an attempt at decimal time during the French Revolution, but it never caught on. Days were divided into 10 "hours", which were divided into 100 "minutes" each. "Weeks" were also 10 days long. No metric time system has ever been established.

Operations[edit]

Addition[edit]

Addition of a time interval to a starting time (or of two time intervals) works like normal addition, except that you may want to convert to different units.

Examples[edit]
  • 43 seconds + 2 seconds = 45 seconds
  • 43 seconds + 22 seconds = 65 seconds, which you may want to convert to 1 minute and 5 seconds
  • 43 seconds + 11:59:22 AM = 12:00:05 PM
  • 43 seconds + 12:00:05 PM = 12:00:48 PM

Subtraction[edit]

Subtraction of a time interval from an ending time (or a time interval from another) works like normal subtraction, except that you may want to convert to different units.

Examples[edit]
  • 45 seconds - 2 seconds = 43 seconds
  • 1 minute and 5 seconds - 43 seconds = 65 seconds - 43 seconds = 22 seconds.
  • 12:00:05 PM - 43 seconds = 11:59:22 AM

Multiplication[edit]

You can multiply a time interval by a scalar (unitless value), but it doesn't yet make sense to multiply specific time with anything else or two time intervals together. As before, you may want to change units to something that makes sense or perform a different operation.

In Physics, you may see time intervals being multiplied. However, this is beyond the scope of elementary math and should be reserved until you are ready for the course (which requires knowledge of algebra.)

Examples[edit]
  • 43 seconds × 2 = 86 seconds = 1 minute and 26 seconds
  • 1 day, 21 hours, 45 minutes, and 43 seconds × 2 = 2 days, 42 hours, 90 minutes, and 86 seconds = 2 days, 42 hours, 91 minutes, and 26 seconds = 2 days, 43 hours, 31 minutes, and 26 seconds = 3 days, 19 hours, 31 minutes, and 26 seconds


Division[edit]

You can divide a time interval by a scalar (unitless value), but it doesn't make sense to divide one time by another. As before, you may want to change units.

You may also see a scalar divided by a time interval. This can be used to show frequency (how often something occurs), or velocity (how fast something moves.)

Repeated division by a time interval appears in physics and is used to denote acceleration. However, this concept is beyond the scope of this book.

Examples[edit]
  • 1 minute and 25 seconds / 2 = 85 seconds / 2 = 42.5 seconds
  • 30 / 5 seconds = 6 / 1 second (indicates something happens six times every second)
  • 50 km / 2 hours = 25 km / 1 hour (indicates somthing moving at 25km/h).


Primary Mathematics
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