Scientific Notation: _{Working with orders of magnitude}

Numbers and numeration systems

A number is an abstract concept of an amount or measure, for example, two or three. A numeral, however, is a symbol representing a number, such as the symbol “2” or “3.” The set of symbols and rules that govern how numbers are represented is called a numeration system. Many early numeration systems were based upon the values that symbols represented, and to make larger numbers, symbols were written in order and their values were added together. For example, in the Egyptian numeration system, the symbol | represents the number one, and || represents “1 + 1,” or two. ||| represents three, and so on (see Figure 1 for an example). The problem with this system is that numbers became extremely long. This was partially addressed by giving new symbols to each new power of ten, for example, ten was denoted as ∩. But this didn’t completely solve the problem as 99 looks like this in Egyptian symbols:

In contrast to the Egyptian system, our modern numeration system is called a place value system. Ten unique symbols are used to indicate amounts between zero and nine, and the position of a particular symbol represents the value of a particular power of ten. For example, the numeral “1” represents different values based on its position within the numeral. The numeral “11” represents the number eleven, not the number two as it would in the Egyptian system, because there is a “1” in the ten's place and a “1” in the one's place. So ten plus one equals eleven. One characteristic our system has in common with the Egyptian system is that we still add the values in each position to get our final number. Let’s take a look at the numeral 2,576. If we expand this numeral, we can show the value of each digit based on its position and how we would determine its value. The numeral 2,576 can be written in expanded notation as:

Notice that each of the numerals is multiplied by a power of ten. For example, five is multiplied by the number 100. Since 100 is a multiple of ten, it can be written as 10 x 10 or 10². 1000 is also a multiple of ten and can be written as 10 x 10 x 10 or 10³. Using these powers of ten, we can rewrite the expanded notation to highlight that feature:

Writing it like this, you can see that the exponent denotes the position of the numeral. And the position of each numeral is important as it tells us the value, or power of ten, to be included in the total sum for the number. Being able to express a number using powers of ten is important to understanding order of magnitude, for example, take a look at the table below.

History of our numeration system

Where did our numeration system originate? Our base-ten numeration system began with symbols originating in India. These symbols traveled with Indian traders to the Arabic and Islamic people, and finally migrated to Europe. The symbols evolved over time as they were used by various civilizations. Abu’l Hasan Ahmad ibu Ibrahim al-Uqlidisi was the very first mathematician to use this type of system in a document dated 953. In this document, he discussed and wrote numbers in the base-ten place value form similar to what we use today. Because our system was influenced by developments of the Indians and Arabs, it is called the Hindu-Arabic system. (See Figure 2 for an example of the evolution of numerals.)

Scientific notation

While our modern numeration system makes it easier to write large numbers than the Egyptian system, writing very large and very small numbers can still be cumbersome and so scientists use scientific notation as a way to write very large or very small numbers in a much more concise way that makes working with these numbers easier. For example, the Earth has a surface area of 169,900,000 miles squared and the world’s population is approximately 7,403,000,000. What if you needed to divide these two numbers to find out how many people per square mile were on the earth? These giant numbers can be cumbersome to divide. Instead, we could express them in scientific notation as 1.699 x 10⁸ and 7.403 x 10⁹. How did we convert these numbers and why is it easier to work with them in scientific notation? Let’s look at where scientific notation came from towards understanding its use.

Scientific notation follows naturally from our base-ten system as a shorthand way to write very large or very small numbers. Scientific notation has two parts. Looking at an example, 1.6 x 10⁸, we can see that the first part is a decimal number greater than or equal to one, but less than ten (in this case, 1.6). The second part is a multiple of ten expressed using an exponent (here, 10⁸).

The notation we use today to denote an exponent was first used by Scottish mathematician, James Hume in 1636. However, he used Roman numerals for the exponents. Using Roman numerals as exponents became problematic since many of the exponents became very large so Hume’s notation didn’t last long. A year later in 1637, Rene Descartes became the first mathematician to use the Hindu-Arabic numerals of today as exponents. The exponent is used as a shorthand way to state how many times a number should be multiplied by itself, so 10³ is equal to 10 x 10 x 10, and 2⁴ is equal to 2 x 2 x 2 x 2.

In scientific notation we also use a decimal numeral. The Flemish mathematician Simon Stevin (Figure 3) first used a decimal point to represent a fraction with a denominator of ten in 1585. While decimals had been used by both the Arabs and Chinese long before this time, Stevin is credited with popularizing their use in Europe. An English translation of Stevin’s work was published in 1608 and titled Disme, The Arts of Tenths or Decimal Arithmetike, and it inspired President Thomas Jefferson to propose a decimal-based currency for the United States. (For example, one tenth of a dollar is called a dime.)

While we can trace the history of the components (decimals and exponents) of scientific notation, it is difficult to determine who actually first used the term scientific notation. In fact, it wasn’t until 1961 that the term could be found in a dictionary indicating that it was being widely used. Even though it is difficult to pinpoint the exact origins of the phrase, it is often thought to have begun with computer scientists. In the 1940’s, Konrad Zuse introduced a concept that he referred to as the "floating point." Zuse’s floating point was a method of representing any number as a decimal greater than or equal to one but less than ten, times a number raised to a power. This notation made it easier to represent and conduct calculations on large and small numbers in the binary code used by computers, even given their limited computing power at the time. Over the next two decades, the term scientific notation often referred to a number expressed as a decimal (described above) times any second number raised to a power. For example, 2.45 x 2³ would have been described as scientific notation in the 1960s or earlier. Today, we only use the term scientific notation when the second number is the numeral 10 raised to a power, such as in 2.45 x 10³.

How to write a number in scientific notation

Did you know that the Earth is 4,543,000,000 years old or that a carbon atom weighs just 0.0000000000000000000000199265 grams? Those are really large and small numbers! If you had to write them several times, or worse yet, use them to calculate something, you may have a difficult time keeping up with all of those zeros. To help simplify these numbers and make them easier to work with, we can express them using scientific notation.

Consider the age of the Earth, 4,543,000,000 years. To rewrite 4,543,000,000 years in scientific notation we must express it as a decimal between one and ten. To do that we divide it by 1,000,000,000 giving us 4.543 years. But the age of the earth is not 4.543 years. So to express this number correctly, we must show that it has to be multiplied by 1,000,000,000. So we can write this as 4.543 times 1,000,000,000. But this can be shortened further because 1,000,000,000 is equal to 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 which can be expressed as 10⁹. So we can write the age of the earth in scientific notation as 4.543 x 10⁹ years.

Table 3 shows how some small and large numbers are expressed in terms of powers of ten.

As you can see from the table, when the exponent on the number ten is a negative number, we can think of this as multiplying by 1/10. For example, 10^-4 is the same as multiplying by the following:

This number can be written in different ways, such as:

The takeaway is that negative exponents are used to express very small numbers in scientific notation.

Let’s think about the weight of our carbon atom, 0.0000000000000000000000199265 grams. To express the weight of the carbon atom as a decimal number greater than or equal to one and less than ten, we would have to multiply by 100,000,000,000,000,000,000,000, or 10²³. This would give us a decimal number of 1.99265, but we then have to add in the factor to show how the number is truly expressed. To reverse this process and return the atom to its original weight we would have to divide by 10²³. So the original weight of a carbon atom can be written as:

This representation is very close to scientific notation, but scientific notation is always written using multiplication, not division. So we can rewrite it as:

Finally, we express the power of ten with a negative exponent and place it in the numerator:

We now have the weight of an atom written correctly in scientific notation.

Scientific notation and the decimal place

A faster method for rewriting a number in scientific notation is to think of how many times the decimal point would have to be moved. For example, in our age of the Earth example, to express 4,543,000,000 as a number greater than one and less than ten we can simply move the decimal.

In this case, we moved our decimal nine places to the left to get the number 4.543. Each shift of the decimal represents division of the number by ten. To return the number back to its original form we must do the opposite, multiply by nine 10s. Therefore, to express the number in scientific notation, we would use a positive nine exponent, thus giving us the same answer as earlier: 4.543 x 10⁹. To convert a number with a positive exponent back, you would move the decimal to the right the same number of places as indicated by the exponent. For example, 3.79 x 10⁵:

We can use a similar but opposite process to write very small numbers in scientific notation. Let’s return to the carbon atom that weighed 0.0000000000000000000000199265 grams. To begin the process, we must move our decimal 23 times to the right.

Since we are moving to the right we are multiplying 0.0000000000000000000000199265 by 10 twenty-three times. To return the number back to its original form we must reverse the process and divide by 10 twenty-three times. Since multiplying by a negative exponent is the same as dividing a number, the original weight of a carbon atom can be written as 1.99265 x 10^-23 grams.

Notice that when you moved the decimal to the left you made the number you started with smaller. To return it to its original size you had to multiply by a bunch of tens. This means that the exponent on the ten had to be positive. When you moved the decimal to the right, the opposite happened. You made the original number much bigger, so to return it to its original tiny size you had to divide by a bunch of tens, making the exponent on ten negative.

Orders of magnitude

Scientists describe the magnitude or size of numbers using something called the order of magnitude. We can think of order of magnitude to mean the power of ten closest to a given quantity. For example, we often hear there are a billion people in China. This is not the exact number of people in China. It is just an order of magnitude approximation. Since 1,000,000,000 can be written as 10⁹ we can say that the population of China has an order of magnitude of nine. We can find an order of magnitude by simply writing a number in scientific notation.

Dividing numbers in scientific notation

Scientists often use order of magnitude to compare the sizes or distances of items. Orders of magnitude give us a quick method for determining the relationship between two quantities. But to do this, we need to be able to divide numbers written in scientific notation. For example, the distance from the Earth to the Sun is 93,000,000 miles = 9.3 x 10⁷ miles. However, the distance to the next nearest star, Proxima Centauri, is about 2.522 x 10¹³ miles. How many times greater is the distance from Proxima Centauri to Earth than the distance from the Earth to the Sun?

To solve this problem, we need to divide the two distances.

Notice that when you divide powers of ten, the net result is that the exponents get subtracted from one another:

Similarly, when you multiply powers of ten the net result is to add the exponents together. Now back to our example. Since the exponent on ten is six, it is tempting to say that the two numbers differ by a magnitude of six, but this is not correct. The convention is to report the answer in scientific notation. The answer 0.2712 x 10⁶ is not in correct form, since 0.2712 is not expressed as a number at least one but less than ten. To convert this into scientific notation, you need to move the decimal one place to the right, which is the same as multiplying the number by ten. To compensate for this you would have to reduce the exponent by one digit, which is equivalent to dividing by ten, and arrive at the same answer 2.712 x 10⁵. We can now correctly report that the two distances differ by a magnitude of five. Since 1 x 10⁵ is equal to 10,000, we can say that Proxima Centauri is 10,000 times further from Earth than the Sun.

We see order of magnitude comparisons often arising in chemistry when discussing the concentrations of substances. For example, a scientist might say the concentration of arsenic in the hundreds of parts per million (ppm) range. This means, that for every one million “parts” of water you would find 0.01 parts of arsenic. For example, you could say that for every 1kg of water, you could find approximately 0.01mg of arsenic (1kg = 1,000,000mg). Let’s write the comparison as a ratio.

Now let’s express the numerator and denominator in scientific notation:

Dividing we get:

We can now say that the amount of arsenic is eight orders of magnitude less than the amount of water.

Example: All creatures great and small

Let’s return to the blue whale and plankton mentioned at the beginning of this module. The blue whale is the largest creature ever to inhabit the planet and yet it shares the ocean with one of the smallest, plankton, a creature that barely represents a tiny speck in the ocean. An average blue whale weighs approximately 190,000 kilograms while a single plankton weighs a mere 0.5 milligrams. To express these weights in the same units (if you need help with metric conversions, refer to our Metric System module) we can say that a Blue Whale weighs 190,000,000,000mg or 1.9 x 10¹¹mg and a Plankton weighs 0.5mg or 5 x 10^-1mg.

Comparing these two ocean dwellers we get:

Thus the whale is 10¹¹, or 100,000,000,000 times larger than the plankton!

Conclusion

Scientific notation and order of magnitude are fundamental concepts in all branches of science. Order of magnitude allows us to quickly estimate the size of something or the difference in measure between two things by expressing it as a power of ten. With practice, expressing numbers in scientific notation provides a quick and easy way to compare, multiply and divide numbers. These concepts are especially useful when comparing very large and very small measurements such as the weight of an atom, the distance to a star, or the length of a tiny, microscopic ocean dweller.