Introduction to very large numbers
Understanding how to write, say and compare very large numbers can help you in your everyday life. It is also a useful skill when you need to do calculations that involve adding, subtracting, or dividing the largest numbers.
To understand these concepts, we will first look at how to read and write very large numbers. Then we will compare two very large numbers by looking at their magnitudes.
One million is one thousand thousand.
One million is one thousand thousand.
One million is 1000000 (1 followed by 6 zeros).
One million is written as 1,000,000.
Here’s what that looks like 1,000,000
(Imagine a space for a comma after the second zero.)
That was easy!
Let’s look at some fun facts about one million.
If you start counting from 0 and count to one million, it’ll take you about 20 days. That’s a lot of counting!
If you had 1000000 seconds in front of you, they’d be gone before you could count to 11.5 days. That’s how long one million seconds would take up if you counted them all out!
If each new minute took up 1 minute of your time (so your watch ticked in real-time), it’d take almost 19 years to go through a million minutes. That’s a long time to what most people have been alive!
And finally: if every hour took up an hour of your time (again in real-time), it would take 114 years to go through a million hours.
One billion is one thousand million.
In the US, one billion is 1,000 million. One trillion is 1, followed by 12 zeros, one thousand billion. One million has six 0s in it (1,000,000), and one billion has nine 0s (1,000,000,000). These are incredibly large numbers and hard to wrap our heads around:
- A million seconds would take us back roughly 11 days.
- A billion seconds would take us back almost 32 years.
- A trillion seconds would take us back almost 32 millennia!
One trillion is one thousand billion.
Now that you’ve figured out how many zeros are in a trillion let’s look at it.
A trillion is just a bigger number than a billion and a million. A trillion is one thousand billions or one million million or one thousand thousand or one with 12 zeros after it.
Big Numbers Bigger Than a Trillion
Now that you’ve seen a few trillion let’s turn our attention to larger numbers.
We have names of numbers up to one thousand in the English language. After that, we add prefixes like ‘thousand’ or ‘million’ to the beginning of the number (like ‘one million) as the formula for the derivation of names of large numbers. This is called the decimal system. The decimal system also includes prefixes such as ‘billion’, which refers to a thousand millions (1,000,000 x 1,000,000), and ‘trillion’, which refers to a thousand billions (1,000 x 1,000 x 1,000 x 1,000).
This system works well if you are talking about small numbers. Unfortunately, people start to run out of words after a short time when discussing large values. To make matters worse, there is no official agreement about what comes after a trillion! Some people use “quadrillion” for 10^15, and some use “billion” for 10^12…
Grouping Zeros by Threes
Grouping zeros by threes is a method of writing and reading large numbers. In this method, you separate zeros into groups of three. For instance, consider the following number:
This two-digit number seems small. But when we look at its digits individually, we can see that it has four digits:
So how do we write this number instead? By putting together three zeros in a row, we get 1,000. But to go from four digits to just one digit would be too dramatic of a change for readers to adjust to quickly. To make the change easier for the reader, then, we group the first three digits:
Powers of 10 Shortcut
When you use powers of 10, you’re using a shorthand way to express multiplying a number by itself. For example, 106 can be expressed as “10 to the sixth power,” which means that 10 is multiplied six times. It’s useful for expressing very large numbers because it’s easier to say and write than a very long string of digits.
In science and math, powers of 10 are used often to express quantities that get too big or too small when expressed in standard notation—that is, notation with just one digit before the decimal point followed by lots of digits after it. Powers of 10 have been around for ages but have become more popular in recent years thanks to advances in technology and miniaturization. The smallest particle known today is 1.6 x 10-19 meters wide, and the largest star we’ve found is 2 x 1011 meters wide—both numbers that are easier to express with powers of 10 than with regular notation!
The Enormous Numbers: Googol and Googolplex
The largest number ever used in mathematics is the googolplex. It’s a 1 followed by a googol (10100) zeros. This is such a large number that it is impractical to write down. Imagine trying to fill an entire encyclopedia with zeros. That would take you up to about 1085, which is only one percent through writing out the googolplex!
Googol and googolplex are not practical numbers. There aren’t enough atoms in the universe for you to use them. But they help us think about how big numbers can be and why we need very large sizes of infinity.
Short and Long Scales of a Billion
The difference between the two scales is that a comma is added to separate the millions/billions/trillions after each number. The long scale adds three zeros after each comma, and the short scale adds six zeros. They both add another zero for every three zeros added.
For example, using the long scale: 1 million = 1,000,000
1 billion = 1,000,000,000
1 trillion = 1,000,000,000
Similarly: 1 million = 10^6
1 billion = 10^9
1 trillion = 10^12
Systems of Numeration
One thing that any mathematician will tell you is that numbers are infinitely fascinating. There are always new ways to use them, new questions about how they work, and new systems for how we organize them.
One of those systems is the scientific notation, which allows us to write really large numbers in a way that makes them easier to understand and use. A lot of us were taught to write extremely large numbers with an E, like this: 1E9. That’s how it would look in scientific notation, and it means one billion.
It’s not just in science that you’ll see those letters, though. Both the American system of numeration and the British system of numeration use E to indicate a number is so large it needs to be written in the scientific world, or the biggest number.
American scientists write large numbers in the form Nx10^n, where the number N is a number between 1 and 10, and number n is the exponent. For example, one hundred million would be written as 1×10^8 in American scientific notation. The British systems of numeration use the same formula but use a capital E instead of a lowercase x: 1E8 (one hundred million).
These systems are helpful for understanding very large numbers, but what about that next smallest number? Mathematician Stanley Skewes was wondering this very question in 1933. He began studying the properties of large values and found that no matter how you looked at it, there was no way all the whole numbers could fit below some certain value. The number he found was approximately 1.06×10^34, which is written as 1.6E34 in British scientific notation.
Final thoughts on Very Large Numbers
The key to understanding very large numbers is to use scientific notation.
When counting by powers of ten, you know how many places that particular number will be from the decimal point. This rule applies to any base number. For example, when counting in base 10, every power of 10 appears 9 places from the decimal point (e.g., 1,000 = 3; 1,000,000 = 6; etc.). The same can happen using other bases:
A googol is defined as one followed by 100 zeroes, and a googolplex is defined as one followed by googol zeroes. The actual value of these numbers is not important for comprehension purposes because these two numbers are extremely difficult to comprehend due to their size.