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# Big and Small Numbers

What's the biggest number you can think of offhand? 1,000? 1 million? Believe it or not, the answer may depend on where you live. The number 1,000,000 is called 1 million all over the world, but there's some international confusion if you go above that. For example, Americans see ${10}^{9}$ ${10}^{9}$ (or 1,000,000,000) and call it one billion, but the British refer to ${10}^{9}$ it as "one thousand million." The British do have a number called a billion, but it's ${10}^{12}$ instead of ${10}^{9}$ . Americans call ${10}^{12}$ one trillion, creating some confusion when the U. S. and U. K. collaborate on math projects.

The British are starting to embrace the American names to avoid any issues, but the two countries aren't the only ones that disagree when it comes to big and small numbers. In India, commas are placed after every two zeros once you get to 1,000, leading to numbers such as the lakh (1,00,000, equivalent to the American 100,000) and crore (1,00,00,000, equivalent to the American ten million).

The rest of this article will explore the standard American and scientific nomenclature for both very large and very small numbers. You may want to review the properties of exponents, scientific notation, and exponent tables to deepen your understanding of these numbers. You may also recognize some of the prefixes from your study of the metric system. For example, 1,000 grams is called a kilogram while 1,000,000,000 watts make a gigawatt. Let's get started!

## The big numbers

 Number Name Metric Prefix ${10}^{1}=10$ Ten Deka- ${10}^{2}=100$ Hundred Hecto- ${10}^{3}=1000$ Thousand Kilo- ${10}^{4}=10000$ Ten Thousand Myria- ${10}^{6}=1000000$ One Million Mega- ${10}^{9}=1000000000$ One Billion Giga- ${10}^{12}$ One Trillion Tera- ${10}^{15}$ One Quadrillion Peta- ${10}^{18}$ One Quintillion Exa- ${10}^{21}$ One Sextillion Zetta- ${10}^{24}$ One Septillion Yotta- ${10}^{27}$ One Octillion ${10}^{30}$ One Nonillion ${10}^{33}$ One Decillion ${10}^{36}$ One Undecillion ${10}^{39}$ One Duodecillion ${10}^{42}$ One Tredecillion ${10}^{45}$ One Quattuordecillion ${10}^{48}$ One Quindecillion ${10}^{51}$ One Sexdecillion ${10}^{54}$ One Septendecillion ${10}^{57}$ One Octodecillion ${10}^{60}$ One Novemdecillion ${10}^{63}$ One Vigintillion ${10}^{100}$ One Googol ${10}^{303}$ One Centillion ${10}^{{10}^{100}}$ One Googolplex

Some of the prefixes above may surprise you. For example, you might associate "octo-" with the number eight and point to examples like "octopus" to justify your answer. Yet an octillion is represented by ${10}^{27}$ , a number that seemingly has nothing to do with eight. The secret is simple. If you write out ${10}^{27}$ , you get the following:

1,000,000,000,000,000,000,000,000,000

That's a one followed by nine groups of three zeros, each separated by a comma. Since we're writing the exponent in base 10 and not one, we subtract one from the nine and get eight: the exact number you would expect from an octillion.

Fun fact: the terms "googol" and "googolplex" were coined by a nine-year-old kid named Milton Sirotta in 1938, with the latter defined as writing a 1 "followed by zeros until your arm gets tired." The original definition was replaced with the standardized exponent above when people realized that different people would experience arm fatigue at different times. The googolplex is considered the largest power of 10 with a name, though obviously larger numbers exist, such as Grams number, which is so large we don't even know its first digit, but it ends in a 7.

You may have also noticed that the popular website Google sounds like a googol, and you would be right! Google's founders wanted to name their company after the googol but accidentally spelled it incorrectly. Oops! Similarly, Google's headquarters is called the Googleplex to reference the googolplex.

## The small numbers

Very small numbers are small numbers that most people fail to take fully into account. Although the difference between 49.9% and 49.99% may seem insignificant, it can make a difference. Take a look at the difference just nine-hundredth of a point makes:

$0.499×259380359=129430799$

$0.4999×259380359=129664241$

That's a difference of over 200,000! Small numbers help us make more precise calculations. When solving problems, be sure to take advantage of all the small numbers that you are provided to come to as accurate an answer as possible.

 Number Name Metric Prefix ${10}^{-1}=0.1$ One Tenth Deci- ${10}^{-2}=0.01$ One Hundredth Centi- ${10}^{-3}=0.001$ One Thousandth Milli- ${10}^{-6}=0.000001$ One Millionth Micro- ${10}^{-9}=0.000000001$ One Billionth Nano- ${10}^{-12}$ One Trillionth Pico- ${10}^{-15}$ One Quadrillionth Femto- ${10}^{-18}$ One Quintillionth Atto- ${10}^{-21}$ One Sextillionth Zepto- ${10}^{-24}$ One Septillionth Yocto-

Your study of the metric system likely familiarized you with many of the prefixes in this set such as the centimeter and milligram. Similarly, the nanosecond and microscope are examples of these prefixes at work outside of the metric system. You may have noticed popular numbers like a zillion absent from both tables as well. That's because there is no specific number called a zillion, it's just a word used as an undefined large number.

## Big and small numbers practice questions

a. What number is represented by ${10}^{-18}$ ?

One quintillionth

b. What number is represented by ${10}^{303}$ ?

One centillion

c. How would you express a googol in scientific notation?

${10}^{100}$

d. How would you express one sextillionth in scientific notation?

${10}^{-21}$

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Your student might not be directly tested on extremely big and small numbers very often, but learning them can help young learners develop strategies for rote memorization and practical problem-solving skills that can be applied to a multitude of classes. Big and small numbers can also be a fascinating way for students to practice working with scientific notation or associate math class with any interests in the sciences. Working with a tutor is an excellent way for students to get a better grasp on both big and small numbers. If you would like to learn more about the benefits of 1-on-1 instruction, Varsity Tutors has Educational Directors standing by to field any questions you may have!

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