Time in Powers of Ten

The following sections are included:

• Contents
• Foreword
• Acknowledgements
• Natural Phenomena and Their Timescales
• Large and Small Numbers

The word ‘second’ is derived from the Latin word secundus or gradus secundus, which means ‘second step’ or ‘next step’. The Romans divided the daylight time into 12 hours. As a further division, an hour was first split into 60 minutes, and as a second step, each minute divided into 60 seconds.

A time span of 10 seconds is also known as a decasecond. Deca is derived from the Greek word deka, which means 10. The prefix deci, on the other hand, comes from the Greek word decimus, which means ‘a tenth’. In the official numbering system the words deca, deci and hecto have been eliminated — only powers of 1,000 are named.

100 seconds are also called one hectosecond. The prefix ‘hector’ to indicate 100 is not common anymore. It is derived from the Greek hekaton, which means ‘a hundred’. The word ‘minute’ comes from the Latin pars minuta prima or ‘the first small part’. This signifies the first small part of an hour or, more specifically, 1/60th. The second small part is then the second, as described in Chapter 1.

The division of the hour into 60 minutes can probably be ascribed to the Babylonians. Babylon was situated in what used to be Mesopotamia, currently better known as Iraq. The Babylonians — or maybe even their predecessors, the Sumerians — introduced the sexagesimal (base-60) system more than 5,000 years ago. Not only did they divide the hour into 60 minutes and then 60 seconds, but degrees in geometry and astronomy were also divided into 60 arcminutes, and every arcminute into 60 arcseconds.

A circle, then, encompasses an angle of 6×60 = 360 degrees. The fact that sexagesimal systems were introduced is possibly explained by the fact that the number 60 has so many divisors: 2×30, 3×20, 4×15, 5×12 and 6×10. The Romans already used the number 10 as the basis of their counting system, but the current decimal (base-10) system, where fractions can be written with numbers ranging from 0 to 9 after a period (such as 0.5 for a half), was introduced much later by the mathematician Simon Stevin.

1,000 seconds makes 1 kilosecond — that is a little over 15 minutes. At some universities, mainly in Europe, this is known as the ‘academic quarter’: this stems from the time when the ringing of the church bells was the general method of timekeeping and when the bells rang, you had 15 minutes to get to class.

In modern times, this academic quarter is often interpreted by students to mean that if the lecturer has not shown up within that time, the class has been cancelled and they get the morning or afternoon off — often to the chagrin of the professor, who might simply have found it difficult to get to his class on time.

10,000 seconds, or almost 3 hours, is the duration of many concerts, films and other spectacles. It is about the maximum amount of time anyone would like to spend on any single activity. You are often allowed 3 hours to complete a university exam. In our Western society it is unacceptable to be 3 hours late for an appointment, but in many countries this is actually quite common.

This chapter covers one of the periods in time which is most familiar to us, that is 100,000 seconds or 27.78 hours: a little longer than a full day — 1 day and 4 hours to be precise.

1 megasecond is the same as 1 million seconds. We are venturing into periods of between one and two weeks. The week consists of 7 days, which is 6.048 × 10⁵ seconds. The concept of a week was introduced in the calendar by the Babylonians, who named each day after a deity. The creation of the world by God also took a week, of which the last day was a day of rest, the Sabbath. A year consists of 52 weeks, one or two of which might be spent on holiday.

The duration of the Apollo 11 mission in 1969 was 8 days, 3 hours and 22 minutes, including the 21.5 hours that the crew spent landing — and famously walking — on the Moon.

In this chapter, we digress from the integer powers of 10. However, as so many elements of our daily existence are related in one way or another to the period of a month, we decided to dedicate a chapter to this time span anyway. In terms of powers, we are somewhere between the sixth and seventh power of 10. A month of 30 days consists of exactly 2,592,000 seconds = 2.592 × 10⁶ seconds, thus almost 2.6 million seconds. We can also denote this as 10^6.41 seconds.

A month is a twelfth of a year. As the word suggests, its original meaning was derived from the orbital period of the moon. Just like a ‘day’, there is a sidereal or star-month as well as a synodic month (discussed in more details in the main text). The first calendars were based on the phases of the Moon, but the problem arose that the Moon, as seen from the Sun, makes a little more than 12 rotations around the earth each year. Therefore, a new system was devised using the Sun as the basis for the calendar. As a consequence, the months had to last a little bit longer: 30 or 31 days — with the exception of February having 28 or 29 days.

Lunar calendars are still based on the synodic month. The Islamic festival of Ramadan lasts exactly one lunar calendar month (see also Chapter 7). Some Christian holidays are also based on the lunar calendar — Easter for example, falls on the first Sunday after the first full Moon in spring.

This is the timescale of the seasons. Ten million seconds is almost four months, and one year is 3.1536 × 10⁷ seconds. The period of three months is also referred to as a quarter — many companies report their financial results ‘quarterly’. In higher education in continental Europe, a course is often taught for one-quarter of the year.

Within three years, a child grows from a newborn to a toddler. After about a year the baby has already learnt how to laugh, walk and he may even babble a few well-chosen words. After that he masters eating with a spoon and perhaps works out how to pile up a few building blocks without them toppling over. At around the age of three, many kids are potty trained and go to kindergarten.

Three years is also the typical duration of a Bachelor's degree program at university. It took us about three years to write this book. Also in three years, the five ships in the fleet of Ferdinand Magellan circumnavigated the globe for the first time ever (1519–1522). The Portuguese explorer, who perished during this voyage, gave his name to a vital waterway through Chile that links the Atlantic and Pacific oceans, two lunar craters and one on Mars, and a Patagonian penguin.

The period of 10⁹ seconds is also called a gigasecond. In one gigasecond, more than 30 years will pass. ‘Giga’ comes from the Greek gigas, which means giant or gigantic. In IT, the term ‘giga’ is often used in the word ‘gigabyte’.

The name ‘gigabyte’ is not strictly accurate. A gigabyte is really 2³⁰ = 1,073,741,824 bytes, which is not exactly the same as 10⁹. The term gigabyte is therefore not used in the correct manner, which is why the alternative term of gibibyte was introduced to indicate a storage capacity of 2³⁰ bytes.

The computer, by the way, has changed significantly in one gigasecond. The first Apple saw the light of day in 1976 (see picture on the right), containing a RAM memory of four kilobytes — or rather four kibibytes = 4,096 bytes. As little as one gigasecond later, most personal computers have a memory of at least 64 gibibytes — and quite a bit more by the time you are reading this book.

As we reach 9 powers of 10, we encounter potential communication problems: depending on where you come from, you might consider 10⁹ to be a billion, a thousand million or even a milliard. In a separate section after the introduction of this book, we have summarized the long and short scale nomenclature; see also the beginning of Chapter 17.

Three hundred years ago, the end of the ‘Golden Century’ was nearing for Holland — the time of great statesmen and Dutch colonialism. The Netherlands reigned over what is now called Indonesia for more than three centuries, from 1602 to 1949. Within its own borders, it was also the time of great Dutch scientists, writers and painters. The famous philosopher Spinoza died in 1677, scientist Christiaan Huygens in 1695, and the painters Vermeer and Rembrandt in 1675 and 1669 respectively. Joost van den Vondel wrote his renowned play Lucifer in 1654, and the philosopher and scientist René Descartes resided in the Netherlands from 1628 to 1649.

The English physicist Isaac Newton, considered by many to be the greatest scientist of all time, also lived contiguously with these masters of the 17th century. He wrote his best known work, the Principia, in 1687, thus more than 10¹⁰ seconds ago. His masterpiece includes his theory on gravity and describes the laws of classical mechanics. The works of these leading historical figures continue to be of great significance for society today.

Three thousand years brings us back to the ages before the beginning of the classical period of the great Greek and Roman civilizations, which began to flourish around 700 BC. Before then, Egypt and Mesopotamia reigned, but also the first Chinese dynasties and other ancient cultures existed. In about 3,300 BC, the first forms of script came into existence in Sumer, in what is now southeastern Iraq. Agriculture and irrigation already existed and the first cities arose. The Jewish people originated about 4,000 years ago. The oldest archeological finds from the city of David, the original Jerusalem, are dated to 3,700 BC.

Tera is the prefix that means 10¹² or a thousand billion, or a trillion. So one terasecond is one trillion seconds. The word tera is derived from the Greek teras, which means monster. The timescale discussed in this chapter corresponds to the development of the first anatomically modern humans in Europe.

Homo sapiens sapiens came to Europe approximately 40,000 years ago. The Neanderthals already roamed this part of the world, but became extinct. It is believed that Homo sapiens migrated north from Africa; recent studies appear to indicate that certain Homo sapiens skulls are more than 195,000 years old.

In the preceding chapter, we looked at the migration of Homo sapiens to Europe and other parts of the world. The cradle of our species is found in Africa, where about 200,000 years ago it came into existence within the genus homo. This has been determined with the aid of fossil records and DNA research. Homo sapiens is the only remaining, living species of this genus. A timeline of the predecessors of Homo sapiens and other hominids, such as Neanderthals, will be shown in the next chapter. As its name suggests — sapiens in Latin means wise or rational — the Homo sapiens has a well-developed brain, which he uses to think and to communicate with fellow humans. This enables him to solve relatively easy problems. Because he stands up straight, he can use both hands to pick up objects and to construct tools.

The first footprint of a human species, literally, is believed to date from 3.5 million years ago. As the result of continuous evolution, three new forms developed out of Australopithecus afarensis (a human species on two legs). Two died out quite quickly. The third kind, Homo habilis (the ‘handy’ man) must have lived around two million years ago and developed into Homo erectus (upright man). Homo habilis developed sharp stones into tools, for example as hammers or cutters, and launched us into the Stone Age. At present, we hardly use stone any more to manufacture tools, but turn to iron or other metals instead. The Stone Age lasted for about 2.5 million years and made way for the Bronze Age, followed by the Iron Age.

In English, 10¹⁵ is a ‘quadrillion’. In Dutch, this number is called ‘biljard’. 10¹² is called a ‘billion’ in most countries in continental Europe, while 10⁹ is called a ‘billion’ in most English and Arabic-speaking countries. The variation in names for these large numbers is due to the fact that most continental European countries use what is called the ‘long scale’ naming system, and the UK and the US the so-called ‘short scale’ (to add to the confusion, the UK officially changed to the short scale in 1974, although even today many British people commonly favor the long scale).

Large numbers up to 10⁹ have the same names in both systems, but beginning with 10⁹, names for large numbers start to differ. In the short scale, each new term greater than a million is 1,000 times the previous terms. ‘Billion’, therefore, means a thousand millions (10⁹). ‘Trillion’ means a thousand billions (10¹²), a quadrillion is 10¹⁵, and so on. In the long scale naming system, however, each new term greater than a million is 1,000,000 times the previous term: a billion means a million millions (10¹²), a trillion a million billions (10¹⁸), and so on. The short scale jumps with 3 powers of 10, while the long scale jumps with 6 powers of 10. We have summarized the long and short scales in a separate section after the introduction. Throughout this book we use the short scale, even though the original (written in Dutch) used the long scale.

Our Earth has changed significantly over the last 10 quadrillion seconds. More than 200 million years ago, all continents were connected, forming a giant continent, Pangaea — after the Greek ‘all’ (pan) and ‘earth’ (gaia). Large parts of the supercontinent were desert-like and unsuitable for any life. As a result of plate tectonics, Pangaea broke up into smaller pieces and became the continents we know today. As more land came into contact with water, the biodiversity of the Earth blossomed.

With a panorama over billions of years, we leave behind geological periods and the evolution of life, and witness the maturing of the Earth as a planet. It took our world a few billion years to become what it is today. At 3.17 billion years in the past, we find ourselves in the midst of astronomic and cosmological developments, such as the emergence of solar systems, galaxies and stars, and, somewhat longer ago, the origin of the universe itself. That said, some events on an atomic scale occur in this timeframe as well, which we will discuss in more detail below.

10¹⁸ is also referred to as a quintillion, or one million to the third power. One quintillion seconds is also 3.17 × 10¹⁰ years, or 31.7 gigayears. This is a time span that is very difficult to imagine, but still, interesting things happen during these large timescales.

There is not much to say about extremely large timescales — after 10³² seconds, more than 10²⁴ (a septillion) years. Galaxies will have broken apart or collapsed almost completely into black holes. Most of them will be so far away, in the outer limits of the universe, that they are far beyond the horizon of what we can detect, and we will never see or hear from them again. Any remaining planets or stars will have cooled off completely and their temperatures will be approaching absolute zero. Even so, there are much larger timescales about which physicists have something interesting to share. These we discuss briefly in this chapter.

From the longest timescales in the previous chapter we now jump to the very shortest, in which processes that take place go much faster than the blink of an eye, much speedier than lightning, or much quicker than the fastest computers. However, the fundamental physical ideas behind both extremely long and short timespans are linked, and some can only be studied and described by the most modern concepts in present day research. These theories — for example, quantum gravity — are nowhere near completion, and that is why the discussion of our next subjects is rife with speculation.

Moreover, the smallest timescales we are able to actually measure last about 10⁻¹⁶ seconds, which is a long way removed from the minute periods of around 10⁻⁴⁴ seconds that we are about to get into. But the universe works to a schedule that is virtually incomprehensible: it can squeeze a whole history into the shortest time we can imagine. And there are a number of other ultra-fast phenomena of which we are gaining an increasingly detailed understanding.

We have arrived in more familiar territory, where science has its foundations on firmer ground. At timescales of 10⁻²⁵ seconds we find the half-lives of elementary particles that we know from the Standard Model. In the Standard Model of elementary particles, all matter consists of elementary building blocks, known as quarks and leptons. There are also additional particles that are responsible for the forces between quarks and leptons. Before discussing the decay times of top quarks and W and Z particles, let us look more closely at the Standard Model.

Yocto is an SI prefix that is derived from the Greek octo (eight). It stands for 10⁻²⁴, or 1/1000⁸. It is the smallest officially-named prefix. The prefix yocto can also be used with other units. For example, a proton weighs 1.6 yoctograms. Along the same lines, a yoctosecond is the typical timescale encountered in the world of quarks and gluons. As mentioned in the previous chapter, there are six different quarks that have odd names with little meaning: up, down, strange, charm, bottom and top.

In the next three chapters we will predominantly encounter the decay processes of elementary particles or light isotopes. These are the timescales during which a number of the most unstable mesons decay. The pion, with its mass of about 135 MeV/c², the lightest meson, has a wave function with a vibration time of about 30 yoctoseconds. This is also the time span within which many strong interaction processes in atomic nuclei occur.

There are quite a number of mesons and baryons. Many elementary particles are symbolized by or named after a letter of the Greek or Latin alphabet. In the table on p.118 we show the most well-known mesons, which consist of up, down and strange quarks and their antiparticles. There are similar figures with mesons that have one or more charm, bottom and top quarks. In the 1960s, more and more new baryons and mesons were discovered, along with the system that became known as quark theory, or ‘Quantum Chromodynamics’ (QCD): baryons consist of three quarks, mesons of a quark and an antiquark, and antibaryons of three antiquarks. During the 1970s, the insight that quarks are held together by gluons was developed. Fluctuations create a background, known in scientific literature as a ‘sea’, of quark-antiquark pairs, which also assert themselves from time to time.

In this chapter, we make the transition from the yocto to the zepto eras. In ‘zepto’ we recognize the Latin term septem, which means ‘seven’. Just as with the yoctosecond, the name comes from powers of a thousand. The zeptosecond is 1/1,000⁷ seconds. Its opposite is the zettasecond, which is 1000⁷ seconds. The opposite of the yoctosecond is the yottasecond, 1000⁸ seconds. As with the previous period, the most important natural phenomena that occur are the decay processes of mesons, but here we see the decay of extremely unstable isotopes as well.

This is the first time we encounter frequencies of electromagnetic radiation. It forms a recurring theme in this book, because frequencies vary from extremely high (far beyond ultraviolet) to extraordinarily low (from infrared light to the longest frequencies of radio signals). Frequencies correlate with timescales, as light and radio waves have a period of one divided by the frequency. The timescale of 10 zeptoseconds corresponds to the wave period of the most energetic gamma rays — more about this at the conclusion of this chapter.

We are slowly but surely reaching timescales that we are able to conceptualize. A good example is X-rays. Wave periods of X-rays can be as short as 10⁻¹⁹ seconds. The corresponding wavelength is then 0.3 Ångström (Å), or 0.3 × 10⁻¹⁰ meters, and that is about the size of small atoms.

The prefix ‘atto’ is derived from the Danish word atten for 18. In one attosecond, light travels a distance of about 0.3 nanometers = 3 Ångströms (1Å = 0.33 × 10⁻¹⁸ light-seconds), or about three hydrogen atoms. Photons of this wavelength have a frequency of 10¹⁸ Hz = 1 exahertz (1 EHz), and therefore an energy of about 4,136 electron volts — or about 4 keV — each.

To put these proportions into perspective: one attosecond is to a second, as a second is to twice the age of the universe.

Electromagnetic waves with a frequency of 10¹⁷ hertz have a wavelength of three nanometers. That also means that light travels three nanometers in 10 attoseconds. For a light wave that propagates with velocity c and frequency f, the wavelength λ obeys the following formula:

Light travels 30 nanometers in this time.

The prefix ‘femto’ is derived from the Danish word femten, which means ‘15’.

In one femtosecond, light travels a distance of 300 nanometers.

Light of this wavelength has an energy quantum of more than four electron volts. The frequency with which each vibration lasts for a femtosecond is 10¹⁵ hertz, or one petahertz (PHz). This is the frequency of ordinary, visible light.

In ten femtoseconds, light travels a distance of 0.003 millimeters, or three micrometers.

A micrometer (μn) used to be called a micron (μ). Before that an even older term was used, a millimillimeter (1/1,000th of 1/1,000th of a meter). These are the smallest distances visible (but only barely) with microscopes that use ordinary light. Important laser technologies have been developed in the femtosecond domain, such as the frequency comb laser. Its spectrum consists of a tremendously large number of extremely thin frequency lines at a set distance from each other, resembling a comb. This technique is used for precision measurements, for example in atomic clocks, but also to determine extremely small and short movements. John Hall and Theodor Hänsch received the Nobel Prize in Physics in 2005 for their contributions to the development of the optical frequency comb technique.

10⁻¹³ seconds = 0.000,000,000,000,1 seconds = 0.1 picoseconds

In 100 femtoseconds, light travels a distance of 0.03 millimeters. This is less than half the width of a human hair.

Middle infrared light has wavelengths between 0.01 and 0.03 millimeters. Heat radiation with these wavelengths is emitted by objects at a very low temperature, around 100 kelvins, or −173 degrees Celsius. Radiation at this temperature has the strongest intensity at frequencies around 10 THz (terahertz).

The picosecond, referring to the Italian piccolo (very small), is abbreviated to 1ps. It is 1/1,000,000,000,000th, or 1/1,000,000th of 1/1,000,000th of a second. Light travels a distance of 0.3 millimeters in this time, about the width of the dot at the end of this sentence. With picosecond laser pulses, researchers can document and photograph ultra-short movements of molecules like the oscillations in vibrating molecules in hydrogen bridges in water, which vibrate with a period of about one picosecond.

In 0.000,000,000,01 seconds, or 10 picoseconds, light or a radio signal travels a distance of three millimeters. That is the distance between the two arrows, below:

The fastest oscilloscopes are able to observe electric vibrations at timescales of just dozens of picoseconds. Many other technologies function at these speeds, such as picosecond lasers, optic fibers and chip technology.

100 picoseconds is a tenth of a nanosecond. The space between the two arrows below marks the distance light or a radio signal travels within this time…

Nanos is the Greek word for ‘dwarf’ — nanus in Latin. A nanosecond lasts 0.000,000,001 seconds. Light travels 30 centimeters in this time. If you hold this book about 30 centimeters away, it takes a nanosecond for our words to reach your eyes. The nanosecond is also the timescale in which reading and writing processes occur in the memory of a computer with a RAM, or random access memory. That is much faster than reading or writing on a hard disk, with timespans of milliseconds, where you must wait for the reader-head to get to the right position first.

The following sections is included:

• 10⁻⁸ seconds = 3 meters of light

10⁻7 seconds is 0.1 microseconds. In this time period, light travels 30 meters (in a vacuum). Sound waves travel only 34 micrometers (or 34 × 10⁻⁶ meters) in the same time; that is about a million times shorter. The speed of sound in air depends on the temperature, but we usually assume that we are talking about room temperature. At higher temperatures, the speed of sound is faster.

The timespan of 10⁻⁶ seconds is called a microsecond, or 1 μs, with the Greek letter mu as prefix. It is one-millionth of a second, or 0.000,001 seconds. Light travels 100 meters in only 0.33 × 10⁻⁶ seconds (a third of a microsecond). By contrast, the fastest human being in history set the world record for running 100 meters at just below 10 seconds.

In 10 microseconds, or 1/100,000th of a second, sound travels a distance of 3.4 millimeters. Light, on the other hand, is able to travel almost three kilometers in the same time. The wavelength of radio waves with a frequency of 100 kilohertz is, therefore, three kilometers. The wave period is 10⁻⁵ seconds.

10⁻⁴ seconds is a tenth of a millisecond. Light can travel about 30 kilometers in 0.0001 seconds. The protons in the 27-kilometer long tunnel of the Large Hadron Collider at CERN that are accelerated almost to the speed of light take 0.9 × 10⁻⁴ seconds to complete one round. In comparison, sound moving through air travels only 34 millimeters in 0.0001 seconds. The wavelength of sound at 10 kilohertz is thus 34 millimeters, and the sound waves with that frequency vibrate once in 10⁻⁴ seconds. For the elderly among us, sounds with this frequency cannot be picked up anymore. Young children are able to hear pitches of up to 16–20 kilohertz.

10⁻³ seconds, also called one millisecond, is the time that light and radio waves need to cross a distance of about 300 kilometers. Sound travels only 34 centimeters in this time. It is also the wavelength of the high B (the high ‘si’ in music, see the figure on the right) in air at room temperature. The fact that sound is like a wave means it can easily bend itself around obstacles. But a wall with much larger dimensions than the wavelength may well be soundproof, especially for higher tones, so barriers are often used to absorb traffic noise alongside highways.

We have arrived at 1/100th of a second, or 10 milliseconds. These are timescales that are important in speech technologies and speech recognition. It is the difference in time between the pronunciation of ‘sometime’ and ‘some time’. Another example is the shutter speed of cameras, which last around 1/100th of a second, depending on the amount of available light, the chosen aperture opening (the f-stop value) and the sensitivity of the film or sensor. As an aside, light travels almost 3,000 kilometers in 0.01 seconds.

Our reaction time is at least 0.1 seconds. This can be measured with an array of experiments, for example, having someone push a button in response to a signal. This is called a simple reaction. The response time with a simple reaction depends on various factors: we tend to react faster to sounds (on average within 0.16 seconds) than to visuals (on average 0.19 seconds); and the response time is age dependent as young people tend to have faster reflexes than the elderly…

In the final chapter, we come full circle back to where we started: the second. A second is the time period that we relate to the most — a heartbeat lasts a second when we are in a state of rest. It is also roughly the time you need to say the word ‘Amsterdam’. Of course, the second is the official scientific unit of time.

The following sections are included:

• Epilogue
• Glossary
• Literature and Websites
• Illustrations Credit
• Index