The Sexagesimal Number System
A lighthouse on the Malecón in Santo Domingo flashes every $18$ seconds, and another one flashes every $45$ seconds. If they flash together right now, when will they flash together again? To answer, you have to calculate with minutes and seconds, and that means working in base $60$. On this page you learn to add, subtract, multiply, and divide times and angles, and you solve the mystery of the lighthouses.
The base-60 system
A number written in the sexagesimal numeration system uses base $60$. It is positional: each position is worth $60$ times the one to its right. You use it every day without realizing it: on the clock and in angles.
A measurement written with several units, such as $2\text{ h } 15\text{ min } 30\text{ s}$, is in compound form. If you express it with a single unit, it is in simple form.
To go from simple to compound form you do the opposite: you divide by $60$ and the remainders are the smaller units.
The Babylonians and base 60
About $4{,}000$ years ago, the Babylonians were already writing numbers in base $60$. They used two wedges: $\langle$ was worth $10$ and $\curlyvee$ was worth $1$. Each position was written with the wedges needed, grouped together.
| Value | Wedges |
|---|---|
| $12$ | $\langle\ \curlyvee\curlyvee$ |
| $45$ | $\langle\langle\langle\langle\ \curlyvee\curlyvee\curlyvee\curlyvee\curlyvee$ |
| $18$ | $\langle\ \curlyvee\curlyvee\curlyvee\curlyvee\curlyvee\curlyvee\curlyvee\curlyvee$ |
When you calculate the total, each position weighs $60$ times more than the one to its right.
The sexagesimal number system is positional and has a base of $60$. State whether the following statements are true or false: a) In the sexagesimal system, $60$ units of…
The sexagesimal number system is positional and has a base of $60$. State whether the following statements are true or false:
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a) In the sexagesimal system, $60$ units of one order form one unit of the next higher order.
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b) The sexagesimal system is currently used to measure temperature.
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c) One hour has $3600$ seconds.
An architect is designing a plan and needs to work with very precise angles. If they have an angle of $45^\circ$ and $20'$, express it entirely in minutes (single-unit…
An architect is designing a plan and needs to work with very precise angles. If they have an angle of $45^\circ$ and $20'$, express it entirely in minutes (single-unit form).
The ancient Babylonians used a positional base-$60$ number system. To represent their astronomical calculations, they used symbols for $10$ ($\langle$) and for $1$…
The ancient Babylonians used a positional base-$60$ number system. To represent their astronomical calculations, they used symbols for $10$ ($\langle$) and for $1$ ($\curlyvee$). If an astronomer recorded the observation time of a star as the sexagesimal number $(12, 45, 18)$, where the first value is hours, the second minutes and the third seconds:
- a) Represent this number using Babylonian symbols (grouping the wedge marks for each position).
- b) Calculate how many total seconds the observation lasted, expressing the result in standard form.
Adding sexagesimal measurements
You add column by column, as with natural numbers, but when a column reaches $60$ or more, you "carry" one unit to the next column.
In an athletics race, the first-place runner reached the finish line with a time of $1$ min $42$ s. The runner in second place arrived $15$ seconds later. What time did…
In an athletics race, the first-place runner reached the finish line with a time of $1$ min $42$ s. The runner in second place arrived $15$ seconds later. What time did the second athlete record? (Express the result in minutes and seconds).
In an esports championship, the first game of a final lasted $45$ min and $38$ s. After a short break, the second game lasted $52$ min and $45$ s. Calculate the total…
In an esports championship, the first game of a final lasted $45$ min and $38$ s. After a short break, the second game lasted $52$ min and $45$ s. Calculate the total duration of both games combined and express the result in an appropriate time format.
Subtracting: borrowing 60 at a time
When you subtract column by column, if the top digit is smaller than the bottom one, you "borrow" from the column to the left. In base $60$ each borrow is worth $60$, not $10$.
When there are more than two addends and a final subtraction, as in the problem about video clips that must fit a time limit, first add all the clips and then subtract the total from the limit. If there are hundredths, remember that $1\text{ s} = 100$ hundredths, so you "carry" as soon as you go past $100$.
Lucía is practicing with her telescope and has turned the lens through an angle of $120^\circ$. If she then decides to reverse the turn by $35^\circ$ $15'$, what angle…
Lucía is practicing with her telescope and has turned the lens through an angle of $120^\circ$. If she then decides to reverse the turn by $35^\circ$ $15'$, what angle does her telescope show now?
In a technical drawing class, we are asked to draw the complementary angle of a given angle. If the original angle measures $34^\circ 52' 18''$, carry out the necessary…
In a technical drawing class, we are asked to draw the complementary angle of a given angle. If the original angle measures $34^\circ 52' 18''$, carry out the necessary calculations to find the exact measure of the angle needed to complete the right angle ($90^\circ$).
A video editor is working on a short film for social media. The project has a strict time limit of exactly $10$ minutes. They have four clips with the following…
A video editor is working on a short film for social media. The project has a strict time limit of exactly $10$ minutes. They have four clips with the following durations:
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Clip 1: $2$ min $14$ s $45$ hundredths.
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Clip 2: $155$ s $80$ hundredths.
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Clip 3: $3$ min $12$ s.
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Clip 4: $1$ min $58$ s $95$ hundredths.
Calculate the exact time left over or needed to reach the limit of $10$ minutes. (Remember that $1$ second has $100$ hundredths).
Multiplying and dividing by a number
To multiply a sexagesimal measurement by a whole number, you multiply each unit separately and then convert whatever goes past $60$.
To divide, you start with the largest unit. You convert the remainder to the next unit (by multiplying by $60$), add it to what was already there, and keep dividing.
When the division does not come out exact in minutes, you turn the remainder into seconds. For example, for an angle of $145^\circ\ 25'$ split into $6$ parts, first you convert everything to minutes: $145 \times 60 + 25 = 8{,}725'$. Then $8{,}725 : 6 = 1{,}454$ with remainder $1'$, which is $60''$ to share out.
A professional diving team has an air supply that allows them to stay underwater for $3$ hours and $15$ minutes. If they must divide this time equally to explore $5$…
A professional diving team has an air supply that allows them to stay underwater for $3$ hours and $15$ minutes. If they must divide this time equally to explore $5$ different sectors of a reef, how much time should they spend observing each sector?
In the design of a part for a high-precision engine, an angle of $145^\circ \ 25'$ must be divided into 6 equal sectors in order to place some bearings. Due to machine…
In the design of a part for a high-precision engine, an angle of $145^\circ \ 25'$ must be divided into 6 equal sectors in order to place some bearings. Due to machine limitations, the measure of each sector must be a whole number of minutes (no seconds) and as large as possible. When carrying out the division, the leftover angle is shared equally between two lateral safety margins.
The drawing is not to scale: the margins are so small that they cannot be seen.
- a) Calculate the measure of each of the 6 sectors.
- b) Calculate the leftover angle and the measure of the angle of each safety margin.
A surveillance drone carries out a security sweep by rotating on its axis. It starts at a position of $0^\circ$ and makes the following consecutive movements: 1. Turns…
A surveillance drone carries out a security sweep by rotating on its axis. It starts at a position of $0^\circ$ and makes the following consecutive movements:
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Turns to the right through an angle of $125^\circ \ 45' \ 52''$.
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Turns to the left by $87^\circ \ 50' \ 15''$.
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Makes a third turn to the right that is exactly three times the angle resulting from subtracting the previous two movements.
Calculate the final position of the drone relative to its starting point in degrees, minutes and seconds.
From decimal to sexagesimal
Sometimes a measurement comes with decimals: $2.35\text{ h}$ or $0.72$ of a turn. To write it in compound form, you convert the decimal part by multiplying by $60$ (or by $360$ if it is a fraction of a full turn).
For fractions of a full turn, you first convert to degrees: $0.35$ of a turn is $0.35 \times 360^\circ = 126^\circ$. If decimals come out, you convert them to minutes and seconds as before.
To split a time into equal parts when there are decimals, you can also divide in decimal and then convert. For example, $7\text{ h} : 4 = 1.75\text{ h} = 1\text{ h } 45\text{ min}$.
A telecommunications satellite orbits the Earth. On their control panel, a technician observes that the satellite has completed $4.72$ orbits in the last $24$ hours. a)…
A telecommunications satellite orbits the Earth. On their control panel, a technician observes that the satellite has completed $4.72$ orbits in the last $24$ hours.
- a) Express the decimal part of the orbits ($0.72$) in compound unit form using degrees, minutes and seconds (knowing that a complete orbit is $360^\circ$).
- b) If the satellite takes the same time for each orbit, what is the exact time it takes to complete a single turn? Express the result in hours, minutes and seconds.
Periodic coincidences
If several signals repeat every certain number of seconds, knowing when they all coincide is a problem about common multiples. Go back to the beginning: the lighthouses of $18$ s and $45$ s.
Before calculating, convert all the periods to the same unit, usually seconds: $1\text{ min } 15\text{ s} = 75\text{ s}$.
A lighthouse emits three different light signals to guide ships: The red light flashes every $18$ seconds. The green light flashes every $45$ seconds. The white light…
A lighthouse emits three different light signals to guide ships:
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The red light flashes every $18$ seconds.
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The green light flashes every $45$ seconds.
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The white light flashes every $1$ minute and $15$ seconds.
If at $22:00:00$ at night all three lights flash together for the first time:
- a) How much time will pass until they flash together again? Express the result in minutes and seconds.
- b) At what exact time will they flash together for the second time?
- c) How many times will all three lights flash together from $22:00:00$ to $00:00:00$, both times included?
Angles of a polygon
The sum of the interior angles of a polygon with $n$ sides is $S = (n-2) \times 180^\circ$. If you know all the angles but one, you add the known ones and subtract the total from $S$, all in compound form.
In an irregular hexagon, the measurements of five of its interior angles are known: $A = 110^\circ \ 15' \ 20''$ $B = 125^\circ \ 40'$ $C = 98^\circ \ 25''$…
In an irregular hexagon, the measurements of five of its interior angles are known:
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$A = 110^\circ \ 15' \ 20''$
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$B = 125^\circ \ 40'$
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$C = 98^\circ \ 25''$
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$D = 130^\circ \ 12' \ 15''$
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$E = 115^\circ \ 50' \ 30''$
Given that the sum of the interior angles of a $n$-sided polygon is $S = (n - 2) \times 180^\circ$, calculate the measure of the sixth angle $F$ in compound form.