Orders of Magnitude

The Order of Magnitude of a number is an approximation of its logarithm. Simply put, the order of magnitude tells us the scale of a number. Whether it is in the tens, hundreds, thousands, millions, and so on, or in the opposite direction (tenths, thousandths, millionths, etc.) which we can express with a single number.

In this way we can say that, for example, the order of a magnitude of a number is 6 (in the millions), 5 (in the hundreds of thousands), or -3 (in the thousandths) as you will see below. It also enables us to specify that one number is a number of orders of magnitude greater/less than another number. This is useful because, for one, it is an accurate way to group numbers based on their magnitude, which can then be compared relative to a reference value or relative to another number.

Contents

1. Logarithmic Scale of Orders of Magnitude
2. Orders of Magnitude In Population Growth
3. Computing The Order of Magnitude
4. Colloquial Use of the Term “Order of Magnitude”

Logarithmic Scale of Orders of Magnitude

Orders of magnitude are determined within a base 10 logarithmic scale, each power of 10 being used to denote an order of magnitude. As you will know, the powers of 10 go from 10⁰=1, to 10¹=10, 10²=100, 10³=1000, 10⁴=10,000, 10⁵=100,000 onwards literally towards infinity. We can also move in the opposite direction towards the infinitely small to fractional values between 1 and 0. From 10^-1=0.1, 10^-2=0.01, 10^-3=0.001, 10^-4=0.0001 and so on.

On such logarithmic scales there is a difference of 1 order of magnitude between 10 (10¹) and 100 (10²). You can see this as the difference between the powers. Between 10 and 1000 there is a difference of two orders of magnitude. Between 10 and 10,000 there is a difference of three orders of magnitude.

This becomes clear when we express these numbers in scientific notation as powers of 10 — which is fundamentally how an order of magnitude is defined. One million (1,000,000) is expressed in scientific notation as 10⁶, while one thousand (1,000) is expressed in scientific notation as 10³. There are three orders of magnitude difference between 1,000,000 and 1,000 because there is a difference of 3 between their exponents: 6-3=3.

Likewise, there is also a difference of 3 orders of magnitude between 1 and 1,000 because 1 is equivalent to 10⁰ and 1,000 is equivalent to 10³. The difference between their exponents is 3. On the other hand, there is only a 2 order of magnitude difference between 10 and 1,000 because 10 is equivalent to 10¹ and 1,000 is 10³, because there is a difference of 2 between their exponents.

Orders of Magnitude in Population Growth

Let us use an example of orders of magnitude derived from estimates of global population. These numbers are just rough estimates, and become even more uncertain the further we go back in time. In bold we have the general date, followed by the estimated global population.

• 10,000 BCE :: Estimated global population of 2,000,000 (2.0 × 10⁶) people.
• 5,000 BCE :: ~ 18,000,000 (18×10⁶ or 1.8×10⁷) people.
• 1,000 BCE :: ~ 115,000,000 people (1.15×10⁸).
• 600 CE :: ~ 213 million people (2.13×10⁸).
• 1800 CE :: ~ 1,000 million or 1 billion people (10⁹).
• 2020 CE :: ~7.4 billion people (7.4×10⁹, roughly equivalent to 10^9.87).

I chose each of the intervals above because the population estimates represent a change in the order of magnitude of global population (with the single exception being the population estimate at 600 CE). You will notice that most of the time I wrote the values in scientific notation, with the exception being the 2020 population (7.4 billion = 10^9.87). This last is the way we actually determine the order of magnitude, since scientific notation is just an estimate.

The earliest population estimate is that around 10,000 BCE the number of homo sapiens on Earth may have been around 2 million. If we take the log base 10 of 2 million we get the exponent 6.3. So the order of magnitude at 10,000 BCE was about 6.3 (since 2 million = 10^6.3). This means that from the time when the first homo sapiens evolved and was born in Africa around 300,000 BCE (representing the first order of magnitude of 0 since 10⁰=1 for the first homo sapiens) by 10,000 BCE the homo sapiens population has increased by about 6 orders of magnitude.

At 5,000 BCE we can see that the population increased from about 2 million to 18 million. Just by looking we see an increase of at least 1 order of magnitude during the period because the population increased from millions to the ten millions (10⁶ to 10⁷). More precisely, taking the logarithm base 10 of the argument 18 million, our exponent is 7.26. So the increase is from 10^6.3 to 10^7.26. An increase of a full order of magnitude.

Then at 1000 BCE we see the population has increased at least another order of magnitude from the ten thousands to the hundred thousands, an increase of 2 orders of magnitude from 10,000 BCE.

Now from 1,000 BCE to 600 CE (about 1,600 years) the population grows from about 115 million to 213 million, a jump of almost 100 million people. It is tempting for people who don’t yet fully understand orders of magnitude to look at this and say that this was an increase in an order of magnitude, but it is not!

The population increased by 100%, which is a way of saying that it doubled. But this is not an order of magnitude because both values are very close relative to one another, within the low hundred millions, even though objectively 100 million is a lot. Population would have had to move from the hundred millions to the ten millions to decrease by an order of magnitude (perhaps due to some cataclysmic event) or else move from the hundred millions to the high hundred millions or low billions to increase an order of magnitude. Specifically population went from an order of magnitude of 8.06 to 8.33 during this interval.

The increase of an order of magnitude didn’t occur for a further 1200 years until 1800 CE, around which time homo sapiens population reached 1 billion people. Here we have an increase from the population c.1000 BCE around 213 million (2.13×10⁸) up to a billion (10⁹) which is an increase in order of magnitude from 8.33 to 9.00.

This brings us to the present day when our population is about 7.4 billion (about 10^9.87) which is an order of magnitude of 9.87. In only 2 short centuries our global population is about to reach the next order of magnitude, speaking precisely. Though as orders of magnitude are often used as estimates, if we round 9.87 up to 10 and are already there. Meaning that we are, at present, about 10 orders of magnitude of population above the time when the first ever individual homo sapiens was born.

(As an aside, which is the subject of a long discussion in and of itself, you’ll notice that the time period between each order of magnitude increase is getting shorter and shorter. This illustrates unequivocally our rapidly growing population, even if the rate of population growth itself is not always increasing.)

Colloquial Use of the Term “Order of Magnitude”

As with most things, there is a difference between the casual way of using a word and its technical definition. In common conversation we use versions of the phrase “An order of magnitude greater” to essential mean the same thing as “On another level”.

In essence, this is a perfect use of the term, because that is exactly what the orders of magnitude are — quantitative levels that we can group numbers by. We can break down population, annual income for an individual or business, and the price levels for various products accurately into orders of magnitude. (Mr. Beast’s $1 versus $1,000,000 house, car, hotel, or meal type video is a perfect example of this.)

The only difference is that the term is used far more loosely — and less accurately — when used colloquially. There is nothing wrong with this, because the meaning is clear, which is what is truly important. It should be noted however, that from a technical definition, those casual orders of magnitudes don’t always equate to true “order of magnitude” level differences.

In a qualitative sense, this term also has value. We can say, for example that Steph Curry, Michael Jordan, Jon Jones, Cristiano Renaldo, or Lionel Messi (who are among the few truly greats in their sport) are at a higher order of magnitude relative to other pros. This is probably accurate. We can also define the orders of magnitude casually in this sport-specific sense as the difference between a professional, amateur, university level and high school level player. Although these distinctions are very casual, and impossible to numerically prove, in my opinion they capture the essential meaning, and are an example of how the term can be applied accurately in a casual sense.