## Question 15: Trigonometry.

QUESTION 15.

(TRIGONOMETRIC).

T^1 = √(A/B) = 10^15 fs

k = y – M/(73/6) = 4623

sin(M(73/6) × 360) = -0.3050305188087

cos(M(73/6) × 360) = 0.95234257627983

a. What are the magnitudes of T^-1, T^2 and T^-2?

b. Workout θ.

c. Workout M/(73/6) and the other decimal remainders and integers (k) of the months, days, hours, minutes and seconds.

d. What is the total value of y?

e. What are the values of t, A and B?

a.

T^-1 = √(B/A) = 10^-15 Ps

T^2 = A/B = 10^30 fs²

T^-2 = B/A = 10^-30 Ps²

b.

sin^-1(-0.3050305188087) = -17.76°

cos^-1(0.95234257627983) = 17.76°

θ = 360 – 17.76 = 342.24° = 8556/25°

c.

M/(73/6) = (8556/25)/360 = 713/750

M = M/(73/6) × (73/6) = 52049/4500

k = M – d/30 = 11

d/30 = M – k = 2549/4500

d = (M – k) × 30 = 2549/150

k = d – h/24 = 16

h/24 = d – k = 149/150

h = (d – k) × 24 = 596/24

k = h – m/60 = 23

m/60 = h – k = 21/25

m = (h – k) × 60 = 252/5

k = m – s/60 = 50

s/60 = m – k = 2/5

s = (m – k) × 60 = 24

d.

y = k + M/(73/6) = 3467963/750

e.

t = A/T^1 = y × 31536000 = 1.45820908224 × 10^11s

A = tT^1 = X × 10^11 × 10^15 = X × 10^26 fs

B = A/T^2 = X × 10^26 / 10^30 = X × 10^-4 Ps

f.

y – M/(73/6) = t / 31536000 – M/(73/6) = 4623

M – d/30 = M/(73/6) × (73/6) – d/30 = 11

d – h/24 = d/30 × 30 – h/24 = 16

h – m/60 = h/24 × 24 – m/60 = 23

m – s/60 = m/60 × 60 – s/60 = 50

s – cs/100 = s/60 × 60 – cs/100 = 24

NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

y – M/(73/6) = 145820908224 / 31536000 – M/(73/6) = 4623

M – d/30 = (4623.95066666667 – 4623) × (73/6) – d/30 = 11

d – h/24 = (11.5664444444439 – 11) × 30 – h/24 = 16

h – m/60 = (16.9933333333165 – 16) × 24 – m/60 = 23

m – s/60 = (23.8399999995959 – 23) × 60 – s/60 = 50

s – cs/100 = (50.3999999757542 – 50) × 60 – cs/100 = 24

t = 4623 years 11 months 16 days 23:50:24

Hand written example: ## Question 14: Trigonometry.

QUESTION 14.

(TRIGONOMETRIC).

T^1 = √(A/B) = 10^6 μs

k = y – M/(73/6) = 3972

sin(θ) = -0.901681645086

cos(θ) = 0.43240052140917

a. What are the magnitudes of T^-1, T^2 and T^-2?

b. Workout θ.

c. Workout the decimal remainders and integers (k) of the months, days, hours, minutes and seconds.

d. What is the total value of y?

e. What are the values of t, A and B?

f. Check if remainder days and months match.

a.

T^-1 = √(B/A) = 10^-6 Ms

T^2 = A/B = 10^12 μs²

T^-2 = B/A = 10^-12 Ms²

b.

sin^-1(-0.901681645086) = -64.38°

cos^-1(0.43240052140917) = 64.38°

θ = 360 – 64.38 = 295.62° = 14781/50°

c.

M/(73/6) = (14781/50)/360 = 4927/6000

M = M/(73/6) × (73/6) = 359671/36000

k = M – d/30 = 9

d/30 = M – k = 35671/36000

d = (M – k) × 30 = 35671/1200

k = d – h/24 = 29

h/24 = d – k = 871/1200

h = (d – k) × 24 = 871/50

k = h – m/60 = 17

m/60 = h – k = 21/50

m = (h – k) × 60 = 126/5

k = m – s/60 = 25

s/60 = m – k = 1/5

s = (m – k) × 60 = 12

d.

y = k + M/(73/6) = 23836927/6000

e.

t = A/T^1 = y × 31536000 = 1.25286888312 × 10^11s

A = tT^1 = X × 10^11 × 10^6 = X × 10^17 μs

B = A/T^2 = X × 10^17 / 10^12 = X × 10^5 Ms

f.

y – d/365 = t / 31536000×- d/365 = 3972

d – h/24 = d/365 × 365 – h/24 = 299

h – m/60 = h/24 × 24 – m/60 = 17

m – s/60 = m/60 × 60 – s/60 = 25

s – cs/100 = s/60 × 60 – cs/100 = 12

For example:

d = m × 30 + d

d = 9 × 30 + 29 = 299

NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

y – d/365 = 125286888312 / 31536000 – d/365 = 3972

d – h/24 = (3972.82116666667 – 3972) × 365 – h/24 = 299

h – m/60 = (299.725833333314 – 299) × 24 – m/60 = 17

m – s/60 = (17.4199999995326 – 17) × 60 – s/60 = 25

s – cs/100 = (25.1999999719555 – 25) × 60 – cs/100 = 12

t = 3972 years 299 days 17:25:12

Hand written example: ## Question 13: Trigonometry reverse.

QUESTION 13.

(Trigonometry reverse).

T^1 = √(A/B) = 10^9 ns

k = y – M/(73/6) = 4248

sin(θ) = 0.46719680793178

cos(θ) = -0.8841533479314

a. What are the magnitudes of T^-1, T^2 and T^-2?

b. Workout θ and M/(73/6).

c. What is the total value of y?

d. What are the values of t, A and B?

e. Workout the decimal remainders and integers (k) of the months, days, hours, minutes and seconds.

f. Check if remainder days and months match.

a.

T^-1 = √(B/A) = 10^-9 Gs

T^2 = A/B = 10^18 ns²

T^-2 = B/A = 10^-18 Gs²

b.

sin^-1(0.46719680793178) = 27.852488584475°

cos^-1(-0.8841533479314) = 152.147511415521° = 6664061/43800°

θ = 152.147511415521° = 6664061/43800°

M/(73/6) = (6664061/43800)/360 = 6664061/15768000

c.

y = k + M/(73/6) = 66989128061/15768000

d.

t = A/T^1 = y × 31536000 = 1.33978256122 × 10^11s

A = tT^1 = X × 10^11 × 10^9 = X × 10^20 ns

B = A/T^2 = X × 10^20 / 10^18 = X × 10^2 Gs

e.

M = M/(73/6) × (73/6) = 6664061/1296000

k = M – d/30 = 5

d/30 = M – k = 184061/1296000

d = (M – k) × 30 = 184061/43200

k = d – h/24 = 4

h/24 = d – k = 11261/43200

h = (d – k) × 24 = 11261/1800

k = h – m/60 = 6

m/60 = h – k = 461/1800

m = (h – k) × 60 = 461/30

k = m – s/60 = 15

s/60 = m – k = 11/30

s = (m – k) × 60 = 22

f.

y – d/365 = t / 31536000 – d/365 = 4248

d – h/24 = d/365 × 365 – h/24 = 154

h – m/60 = h/24 × 24 – m/60 = 6

m – s/60 = m/60 × 60 – s/60 = 15

s – cs/100 = s/60 × 60 – cs/100 = 22

For example:

d = m × 30 + d

d = 5 × 30 + 4 = 154

NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

y – d/365 = 133978256122 / 31536000 – d/365 = 4248

d – h/24 = (4248.42263197615 – 4248) × 365 – h/24 = 154

h – m/60 = (154.260671296452 – 154) × 24 – m/60 = 6

m – s/60 = (6.25611111483886 – 6) × 60 – s/60 = 15

s – cs/100 = (15.3666668903315 – 15) × 60 – cs/100 = 22

t = 4248 years 154 days 06:15:22

Hand written example: ## Question 12: Dates and times trigonometry.

QUESTION 12.

(DATES AND TIMES TRIGONOMETRY).

11/2/1544 09:16:32 – 15/7/5792 15:31:54

T^1 = √(A/B) = 10^3 ms

a. What are the integers (k) of the years, months, days, hours, minutes and seconds between the two dates and times?

b. What are the values of s, m, h, d, M and y and what are the angles of their remainders or decimals in degrees and radians? Give sin(θ) and cos(θ).

c. What are the magnitudes of T^-1, T^2 and T^-2?

d. What are the values of t, A and B?

e. Reverse or undo t back into y/M/d/h/m/s integers (k) format.

f. Check that the remainder months add up to the remainder days.

a.

k = y – M/(73/6) = 5792 – 1544 = 4248

k = M – d/30 = 7 – 2 = 5

k = d – h/24 = 15 – 11 = 4

k = h – m/60 = 15 – 9 = 6

k = m – s/60 = 31 – 16 = 15

k = s – cs/100 = 54 – 32 = 22

Because there is no obvious consensus on precisely how many days are in month, that is 28 – 31, therefore, when we are using months, in order to get the months and days to match and corroborate we must state that we are using 30 days in a month, therefore, we are using conversion factor of 73/6 months in a year. However, you could use a different whole number (a number without a decimal) for the conversion factor (or the amount of days in a month) such as 365/31.

Note: to convert a remainder decimal of a unit time into an angle of degrees and radians we times the decimal by 360 and 2π respectively.

For example:

s/60 = 11/30 × 360 = 132° Or s/60 = 11/30 × 2π = 11π/15

b.

s/60 = 22/60 = 11/30

θ = 132° or 11π/15

sin(θ) = 0.74314482547739

cos(θ) = -0.6691306063589

m = k + s/60 = 461/30

m/60 = (461/30)/60 = 461/1800

θ = 461/5° or 922π/5

sin(θ) = 0.99926291641062

cos(θ) = -0.0383878090875

h = k + m/60 = 11261/1800

h/24 = (11261/1800)/24 = 11261/43200

θ = 11261/120° or 11261π/21600

sin(θ) = 0.99775300871239

cos(θ) = -0.0669995045159

d = k + h/24 = 184061/43200

d/30 = (184061/43200)/30 = 184061/1296000

θ = 184061/3600° or 184061π/648000

sin(θ) = 0.77855054474696

cos(θ) = 0.62758190642674

M = k + d/30 = 6664061/1296000

M/(73/6) = (6664061/1296000)/(73/6) = 6664061/15768000

θ = 6664061/43800° or 6664061π/7884000

sin(θ) = 0.46719680793178

cos(θ) = -0.8841533479314

y = k + M/(73/6) = 66989128061/15768000

c.

T^-1 = √(B/A) = 10^-3 ks

T^2 = A/B = 10^6 ms²

T^-2 = B/A = 10^-6 ks²

d.

t = A/T^1 = y × 31536000 = 1.33978256122 × 10^11 s

X = 1.33978256122

A = tT^1 = X × 10^11 × 10^3 = X × 10^14 ms

B = A/T^2 = X × 10^14 / 10^6 = X × 10^8 ks

e.

y – M/(73/6) = t / 31536000 – M/(73/6) = 4248

M – d/30 = M/(73/6) × (73/6) – d/30 = 5

d – h/24 = d/30 × 30 – h/24 = 4

h – m/60 = h/24 × 24 – m/60 = 6

m – s/60 = m/60 × 60 – s/60 = 15

s – cs/100 = s/60 × 60 – cs/100 = 22

NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

y – M/(73/6) = 133978256122 / 31536000 – M/(73/6) = 4248

M – d/30 = (4248.42263197615 – 4248) × (73/6) – d/30 = 5

d – h/24 = (5.14202237654839 – 5) × 30 – h/24 = 4

h – m/60 = (4.26067129645161 – 4) × 24 – m/60 = 6

m – s/60 = (6.25611111483865 – 6) × 60 – s/60 = 15

s – cs/100 = (15.3666668903188 – 15) × 60 – cs/100 = 22

Then insert the integers (k) into the y/M/d/h/m/s format:

t = 4248 years 5 months 4 days 06:15:22

f.

y – d/365 = t / 31536000×- d/365 = 4248

d – h/24 = d/365 × 365 – h/24 = 154

h – m/60 = h/24 × 24 – m/60 = 6

m – s/60 = m/60 × 60 – s/60 = 15

s – cs/100 = s/60 × 60 – cs/100 = 22

For example:

d = m × 30 + d

d = 5 × 30 + 4 = 154

NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

y – d/365 = 133978256122 / 31536000 – d/365 = 4248

d – h/24 = (4248.42263197615 – 4248) × 365 – h/24 = 154

h – m/60 = (154.260671296452 – 154) × 24 – m/60 = 6

m – s/60 = (6.25611111483886 – 6) × 60 – s/60 = 15

s – cs/100 = (15.3666668903315 – 15) × 60 – cs/100 = 22

t = 4248 years 154 days 06:15:22

Hand written example: ## Question 11: Dates and times (long).

QUESTION 11.

(DATES AND TIMES LONG).

30/12/1776 08:16:52 – 7/3/7321 19:48:18

T^1 = √(A/B) = 10^21 zs

a. What are the whole numbers of the years, months, days, hours, minutes and seconds between the two dates and times?

b. What is the value of y?

c. What are the magnitudes of T^-1, T^2 and T^-2?

d. What are the values of A, B and t?

e. Reverse or undo t back into y/M/d/h/m/s whole numbers format.

f. Check that the remainder months add up to the remainder days.

a.

w = y – M/(73/6) = 7321 – 1776 – 1 = 5544

w = M – d/30 = 12 – 12 + 3 – 1 = 2

w = d – h/24 = 30 – 30 + 7 = 7

w = h – m/60 = 19 – 8 = 11

w = m – s/60 = 48 – 16 – 1 = 31

w= s – cs/100 = 60 – 52 – 18 = 26

Because there is no obvious consensus on precisely how many days are in month, that is 28 – 31, therefore, when we are using months, in order to get the months and days to match and corroborate we must state that we are using 30 days in a month, therefore, we are using conversion factor of 73/6 months in a year. However, you could use a different whole number (a number without a decimal) for the conversion factor (or the amount of days in a month) such as 365/31.

b.

s/60 = 26/60 = 13/30

m = w + s/60 = 943/30

m/60 = (943/30)/60 = 943/1800

h = w + m/60 = 20743/1800

h/24 = (20743/1800)/24 = 20743/43200

d = w + h/24 = 323143/43200

d/30 = (323143/43200)/30 = 323143/1296000

M = w + d/30 = 2915143/1296000

M/(73/6) = (2915143/1296000)/(73/6) = 2915143/15768000

y = w + M/(73/6) = 87420707143/15768000

c.

T^-1 = √(B/A) = 10^-21 Zs

T^2 = A/B = 10^42 zs²

T^-2 = B/A = 10^-42 Zs²

d.

t = A/T^1 = y × 31536000 = 1.74841414286 × 10^11 s

X = 1.74841414286

A = tT^1 = X × 10^11 × 10^21 = X × 10^32 zs

B = A/T^2 = X × 10^32 / 10^42 = X × 10^-10 Zs

e.

y – M/(73/6) = B / (31536000 × T^-1) – M/(73/6) = 5544

M – d/30 = M/(73/6) × (73/6) – d/30 = 2

d – h/24 = d/30 × 30 – h/24 = 7

h – m/60 = h/24 × 24 – m/60 = 11

m – s/60 = m/60 × 60 – s/60 = 31

s – cs/100 = s/60 × 60 – cs/100 = 26

NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

y – M/(73/6) = 1.74841414286E-10 / (31536000 × T^-1) – M/(73/6) = 5544

M – d/30 = (5544.18487715627 – 5544) × (73/6) – d/30 = 2

d – h/24 = (2.24933873456272 – 2) × 30 – h/24 = 7

h – m/60 = (7.48016203688166 – 8) × 24 – m/60 = 11

m – s/60 = (11.5238888851599 – 11) × 60 – s/60 = 31

s – cs/100 = (31.4333331095922 – 31) × 60 – cs/100 = 26

Then insert the whole numbers into the y/M/d/h/m/s format:

t = 5544 years 2 months 8 days 11:31:26

f.

y – d/365 = B / (31536000 × T^-1) – d/365 = 5544

d – h/24 = d/365 × 365 – h/24 = 67

h – m/60 = h/24 × 24 – m/60 = 11

m – s/60 = m/60 × 60 – s/60 = 31

s – cs/100 = s/60 × 60 – cs/100 = 26

For example:

d = m × 30 + d

d = 2 × 30 + 7 = 67

NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the integers (k) and multiply the decimal by (73/6), 30, 24 or 60. We do not write or type long numbers. We only have to type the first division below and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

y – d/365 = 1.74841414286E-10 / (31536000 × T^-1) – d/365 = 5544

d – h/24 = (5544.18487715627 – 5544) × 365 – h/24 = 67

h – m/60 = (67.4801620368817 – 67) × 24 – m/60 = 11

m – s/60 = (11.52388888516 – 11) × 60 – s/60 = 31

s – cs/100 = (31.4333331095986 – 31) × 60 – cs/100 = 26

Hand written example: ## Question 10: Mayan time.

Question 10.

(MAYAN TIME)

A = BT^2 = 7.1683012 × 10^16 ms

X = 7.1683012

a. What are the magnitudes of T^1, T^-1, T^2 and T^-2?

b. What are the values of B and t?

c. Using t, what are the whole numbers of the Mayan baktun, katun, tun, uinal and kin? (Show the equivalent Western time).

a.

T^1 = √(A/B) = 10^3 ms

T^-1 = √(B/A) = 10^-3 ks

T^2 = A/B = 10^6 ms²

T^-2 = B/A = 10^-6 ks²

b.

B = A/T^2 = X × 10^16 / 10^6 = X × 10^10 ks

t = BT^1 = X × 10^10 × 10^3 = X × 10^13 s

To convert the total seconds (t) to baktun, katun, tun, uinal and kin format we can use this formula.

Note: 28800/73, 1440/73 and 72/73 etc are conversion factors, as in 1 Mayan baktun is 28800/73 Western Years.

Note: we are not using months for this exercise.

Note: a Mayan time is given then the equivalent Western time, and so on.

c.

baktun – katun/20 = t / 12441600000 – katun/20 = 5761

y – d/365 = baktun × 28800/73 – d/365 = 227053

katun – tun/20 = katun/20 × 20 – tun/20 = 11

y – d/365 = katun × 1440/73 – d/365 = 220

tun – uinal/18 = tun/20 × 20 – uinal/18 = 3

y – d/365 = tun × 72/73 – d/365 = 3

d – h/24 = tun × 360 – h/24 = 1290

uinal – kin/20 = uinal/18 × 18 – kin/20 = 10

d – h/24 = uinal × 20 – h/24 = 210

kin – h/24 = kin/20 × 20 – h/24 = 10

h – m/60 = h/24 × 24 – m/60 = 17

m – s/60 = m/60 × 60 – s/60 = 46

s – cs/100 = s/60 × 60 – cs/100 = 40

NOTE: obviously we do NOT literally minus the decimal such as katun/20 on the calculator, we simply minus the whole number and multiply the decimal by 20 or 18 etc. We do not write or type long numbers. We only have to type the first division below such as baktun – katun/20 and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

baktun – katun/20 = 71683012000000 / 12441600000 – katun/20 = 5761

y – d/365 = 5761.55896347737 × 28800/73 – d/365 = 227053

katun – tun/20 = (5761.55896347737 – 5761) × 20 – tun/20 = 11

y – d/365 = 11.1792695473196 × 1440/73 – d/365 = 220

tun – uinal/18 = (11.1792695473196 – 11) × 20 – uinal/18 = 3

y – d/365 = 3.58539094639127 × 72/73 – d/365 = 3

d – h/24 = 3.58539094639127 × 360 – h/24 = 1290

uinal – kin/20 = (3.58539094639127 – 3) × 18 – kin/20 = 10

d – h/24 = 10.5370370350429 × 20 – h/24 = 210

kin – h/24 = (10.5370370350429 – 10) × 20 – h/24 = 10

h – m/60 = (10.7407407008577 – 10) × 24 – m/60 = 17

m – s/60 = (17.7777768205851 – 17) × 60 – s/60 = 46

s – cs/100 = (46.6666092351079 – 46) × 60 – cs/100 = 40

Then insert the whole numbers into the baktun/katun/tun/uinal/kin format:

t = 5761 baktun 11 katun 3 tun 10 uinal 10 kin 17:46:40

Hand written example: ## Question 9: Martian time.

Question 9:

(MARTIAN TIME)

Letters with an M in front of it such as My or Md are Martian time, as in Martian Year and Martian Day. Just y and d etc are Earth time.

t = A/T^1 = 2.601973 × 10^10 s

X = 2.601973

T^1 = √(A/B) = 10^3 ms

a. What are the magnitudes of T^-1, T^2 and T^-2?

b. What are the values of A and B?

c. Using A, what are the whole numbers of the Martian years, days, hours, minutes and seconds? (Show the equivalent Earth time).

a.

T^-1 = √(B/A) = 10^-3 ks

T^2 = A/B = 10^6 ms²

T^-2 = B/A = 10^-6 ks²

b.

A = tT^1 = X × 10^10 × 10^3 = X × 10^13 ms

B = A/T^2 = X × 10^13 / 10^6 = X × 10^7 ks

To convert the total milliseconds (A) to Martian years (My), Martian days (Md), Martian hours (Mh), Martian minutes (Mm) and Martian seconds (Ms) format we can use the formula below.

Note: 1.88213623601231 and 1.0274912517 are conversion factors, as in 1 Martian Year is 1.88213623601231 Earth Years. 59355048.3388842 is how many seconds in a Martian year and 668.5991 is how many sols (Martian Days) in a Martian year.

Note: we are not using months for this exercise.

Note: a Martian time is given then the equivalent Earth time, and so on.

c.

My – Md/668.5991 = A / (59355048.3388842 × T^1) – Md/668.5991 = 438

y – d/365 = My × 1.8821 – d/365 = 825

Md – Mh/24 = Md/668.5991 × 668.5991 – Mh/24 = 250

d – h/24 = Md × 1.027 – h/24 = 257

Mh – Mm/60 = Mh/24 × 24 – Mm/60 = 6

h – m/60 = Mh × 1.027 – m/60 = 6

Mm – Ms/60 = Mm/60 × 60 – Ms/60 = 45

m – s/60 = Mm × 1.027 – s/60 = 47

Ms – Mcs/100 = Ms/60 × 60 – Mcs/100 = 47

s – cs/100 = Ms × 1.027 – cs/100 = 48

NOTE: obviously we do NOT literally minus the decimal such as d/668.5991 on the calculator, we simply minus the whole number and multiply the decimal by 668.5991, 24 or 60. We do not write or type long numbers. We only have to type the first division below such as My – Md/668.5991 and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

My – Md/668.5991 = 26019730000000 / (59355048.3388842 × T^1) – Md/668.5991 = 438

y – d/365 = 438.37433762065 × 1.8821 – d/365 = 825

Md – Mh/24 = (438.37433762065 – 438) × 668.5991 – Mh/24 = 250

d – h/24 = 250.28179626251 × 1.027 – h/24 = 257

Mh – Mm/60 = (250.28179626251 – 250) × 24 – Mm/60 = 6

h – m/60 = 6.76311030024476 × 1.027 – m/60 = 6

Mm – Ms/60 = (6.76311030024476 – 6) × 60 – Ms/60 = 45

m – s/60 = 45.7866180146857 × 1.027 – s/60 = 47

Ms – Mcs/100 = (45.7866180146857 – 45) × 60 – Mcs/100 = 47

s – cs/100 = 47.197080881142 × 1.027 – cs/100 = 48

Then insert the whole numbers into the My/Md/Mh/Mm/Ms format:

t = 438 solar years 250 sols 06:45:47

Hand written example: ## Question 8: Dates and times (long).

Question 8:

(DATES AND TIMES LONG).

07/02/1066 09:45:49 19/06/4216 14:27:13

T^2 = A/B = 10^12 μs²

a. What are the whole numbers of the years, months, days, hours, minutes and seconds between the two dates and times?

b. What is the value of y?

c. What are the magnitudes of T^1, T^-1 and T^-2?

d. What are the values of A, B and t?

e. Reverse or undo t back into y/M/d/h/m/s whole numbers format.

f. Check that the remainder months add up to the remainder days.

a.

w = y – M/(365/31) = 4216 – 1066 = 3150

w = M – d/31 = 6 – 2 = 4

w = d – h/24 = 19 – 7 = 12

w = h – m/60 = 14 – 9 – 1 = 4

w = m – s/60 = 60 – 45 + 27 – 1 = 41

w= s – cs/100 = 60 – 49 + 13 = 24

Because there is no obvious consensus on precisely how many days are in month, that is 28 – 31, therefore, when we are using months, in order to get the months and days to match and corroborate we must state that we are using 31 days in a month, therefore, we are using conversion factor of 365/31 months in a year. However, you could use a different whole number (a number without a decimal) for the conversion factor (or the amount of days in a month) such as 365/30 = 73/6.

b.

s/60 = 24/60 = 2/5

m = w + s/60 = 41 + 2/5 = 207/5

m/60 = (207/5)/60 = 69/100

h = w + m/60 = 4 + 69/100 = 469/100

h/24 = (469/100)/24 = 469/2400

d = w + h/24 = 12 + 469/2400 = 29269/2400

d/31 = (29269/2400)/31 = 29269/74400

M = w + d/31 = 4 + 29269/74400 = 326869/74400

M/(365/31) = (326869/74400)/(365/31) = 326869/876000

y = w + M/(365/31) = 3150 + 326869/876000 = 2759726869/876000

c.

T^1 = √(A/B) = 10^6 μs

T^-1 = √(B/A) = 10^-6 Ms

T^-2 = B/A = 10^-12 Ms²

d.

t = y × 31536000 = 9.9350167284 × 10^10 s

X = 9.9350167284

A = tT^1 = X × 10^10 × 10^6 = X × 10^16 μs

B = A/T^2 = X × 10^16 / 10^12 = X × 10^4 Ms

e.

y – M/(365/31) = t / 31536000 – M/(365/31) = 3150

M – d/31 = M/(365/31) × (365/31) – d/31 = 4

d – h/24 = d/31 × 31 – h/24 = 12

h – m/60 = h/24 × 24 – m/60 = 4

m – s/60 = m/60 × 60 – s/60 = 41

s – cs/100 = s/60 × 60 – cs/100 = 24

NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the whole number and multiply the decimal by (365/31), 31, 24 or 60. We do not write or type long numbers. We only have to type the first division below such as y – M/(365/31) and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

y – M/(365/31) = 99350167284 / 31536000 – M/(365/31) = 3150

M – d/31 = (3150.37313812785 – 3150) × (365/31) – d/31 = 4

d – h/24 = (4.39340053763537 – 4) × 31 – h/24 = 12

h – m/60 = (12.1954166666965 – 12) × 24 – m/60 = 4

m – s/60 = (4.69000000071512 – 4) × 60 – s/60 = 41

s – cs/100 = (41.400000042907 – 41) × 60 – cs/100 = 24

Then insert the whole numbers into the y/M/d/h/m/s format:

t = 3150 years 4 months 12 days 04:41:24

f.

y – d/365 = t / 31536000 – d/365 = 3150

d – h/24 = d/365 × 365 – h/24 = 136

h – m/60 = h/24 × 24 – m/60 = 4

m – s/60 = m/60 × 60 – s/60 = 41

s – cs/100 = s/60 × 60 – cs/100 = 24

For example:

d = m × 31 + d

d = 4 × 31 + 12 = 136

NOTE: obviously we do NOT literally minus the decimal such as s/60 on the calculator, we simply minus the whole number and multiply the decimal by (365/31), 31, 24 or 60. We do not write or type long numbers. We only have to type the first division below such as y – M/(365/31) and ‘lift’ the whole number, the rest is done by the calculator.

For example:

Note: you do not need to write the below, it is all done on the calculator.

y – d/365 = 99350167284 / 31536000 – d/365 = 3150

d – h/24 = (3150.37313812785 – 3150) × 365 – h/24 = 136

h – m/60 = (136.195416666862 – 136) × 24 – m/60 = 4

m – s/60 = (4.69000000469896 – 4) × 60 – s/60 = 41

s – cs/100 = (41.4000002819375 – 41) × 60 – cs/100 = 24

Hand written example: ## Question 7: Dates and times (short).

Question 7:

(DATES AND TIMES SHORT).

17/12/608 14:22:46 3/4/5733 11:47:17

T^2 = A/B = 10^4 cs²

a. What are the whole numbers of the years, months, days, hours, minutes and seconds between the two dates and times?

b. What is the value of y?

c. What are the magnitudes of T^1, T^-1 and T^-2?

d. What are the values of t, A and B?

a.

w = y – M/(365/31) = 5733 – 608 – 1 = 5124

w = M – d/31 = 12 – 12 + 4 – 1 = 3

w = d – h/24 = 31 -17 + 3 – 1 = 16

w = h – m/60 = 24 – 14 + 11 = 21

w = m – s/60 = 47 – 22 – 1 = 24

w = s – cs/100 = 60 – 46 + 17 = 31

Because there is no obvious consensus on precisely how many days are in month, that is 28 – 31, therefore, when we are using months, in order to get the months and days to match and corroborate we must state that we are using 31 days in a month, therefore, we are using conversion factor of 365/31 months in a year. However, you could use a different whole number (a number without a decimal) for the conversion factor (or the amount of days in a month) such as 365/30 = 73/6.

b.

s/60 = 31/60

m = w + s/60 = 24 + 31/60 = 1471/60

m/60 = (1471/60)/60 = 1471/3600

h = w + m/60 = 21 + 1471/3600 = 77071/3600

h/24 = (77071/3600)/24 = 77071/86400

d = w + h/24 = 16 + 77071/86400 = 1459471/86400

d/31 = (1459471/86400)/31 = 1459471/2678400

M = w + d/31 = 3 + 1459471/2678400 = 9494671/2678400

M/(365/31) = (9494671/2678400)/(365/31) = 9494731/31536000

y = w + M/(365/31) = 5246 + 9494671/31536000 = 161599958671/31536000

c.

T^1 = √(A/B) = 10^2 cs

T^-1 = √(B/A) = 10^-2 hs

T^-2 = B/A = 10^-4 hs²

d.

t = y × 31536000 = 1.61599958671 × 10^11 s

X = 1.61599958671

A = tT^1 = X × 10^11 × 10^2 = X × 10^13 ms

B = A/T^2 = X × 10^13 / 10^4 = X × 10^9 ks

Hand written example: ## Question 6: Dates and times (short).

Question 6:

(DATES AND TIMES SHORT).

12/9/1496 21:37:48 2/3/6743 09:24:17

T^2 = A/B = 10^6 ms²

a. What are the whole numbers of the years, months, days, hours, minutes and seconds between the two dates and times?

b. What is the value of y?

c. What are the magnitudes of T^1, T^-1 and T^-2?

d. What are the values of t, A and B?

a.

w = y – M/(365/31) = 6743 – 1496 – 1 = 5246

w = M – d/31 = 12 – 9 + 3 – 1 = 5

w = d – h/24 = 31 – 12 + 2 – 1 = 20

w = h – m/60 = 24 – 21 + 9 – 1 = 11

w = m – s/60 = 60 – 37 + 24 – 1 = 46

w = s – cs/100 = 60 – 48 + 17 = 29

Because there is no obvious consensus on precisely how many days are in month, that is 28 – 31, therefore, when we are using months, in order to get the months and days to match and corroborate we must state that we are using 31 days in a month, therefore, we are using conversion factor of 365/31 months in a year. However, you could use a different whole number (a number without a decimal) for the conversion factor (or the amount of days in a month) such as 365/30 = 73/6.

b.

s/60 = 29/60

m = w + s/60 = 46 + 29/60 = 2789/60

m/60 = (2789/60)/60 = 2789/3600

h = w + m/60 = 11 + 2789/3600 = 42389/3600

h/24 = (42389/3600)/24 = 42389/86400

d = w + h/24 = 20 + 42389/86400 = 1770389/86400

d/31 = (1770389/86400)/31 = 1770389/2678400

M = w + d/31 = 5 + 1770389/2678400 = 15162389/2678400

M/(365/31) = (15162389/2678400)/(365/31) = 15162389/31536000

y = w + M/(365/31) = 5246 + 15162389/31536000 = 165453018389/31536000

c.

T^1 = √(A/B) = 10^3 ms

T^-1 = √(B/A) = 10^-3 ks

T^-2 = B/A = 10^-6 ks²

d.

t = y × 31536000 = 1.65453018389 × 10^11 s

X = 1.65453018389

A = tT^1 = X × 10^11 × 10^3 = X × 10^14 ms

B = A/T^2 = X × 10^14 / 10^6 = X × 10^8 ks

Hand written example: 