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Base 10 and other Bases in Mathematics

Updated on May 9, 2010

A short introduction to Bases in Math

What's base 10? What's base 2? What's the difference between bases in math? What's this all about, anyway? And Who's on first?

What are "bases" in math?

Why do we use mathematical bases?

Maybe you know the answer to those questions, but if you are like most people, there has been a time in your life when you didn't, and found the whole "base" idea frustrating.

If you have children, or if you are a teacher, your children or students are definitely going to go through a period of frustration about bases if they haven't already.

The mission of this lens is to make that period as short and painless as possible for your child, your students and yourself.

First let's try to define bases in normal English. A base is a way to express numbers using place value (that means like using columns). The typical system that we use, and that you are familiar with, is called the base 10 system. In base 10, each column is worth 10 times the amount of the column in the place to the right of it.

The column furthest to the right, is always the ones column. (I must point out that we are talking about whole numbers here, not decimals or fractions, or negative numbers. The whole numbers are the numbers 0,1,2,3...)

So the column to the left of that would be the 10s column because 10 is 10 x 1.

The next highest column to the left would be 10 times the 10s column, which of course would make it the hundreds column.

I imagine you already know what comes after the hundreds column. It's the thousands column of course because 100 x 10 = 1,000. There is no end to how high you can count when you use the base system.

This seems all matter-of-fact, until you realize that humanity didn't start using base systems until very late in its development. Think about Roman numerals - they consisted of letters like I, V, X, L, C, M., and, um, what came after that? See, with other systems you typically run out of letters or symbols, because each symbol stands for a different amount of numbers. And if you had large amounts of numbers, you'd have to memorize lots and lots of symbols.

Some ancient "programmer" must've figured out that there was a better way. To that nameless programmer, we owe a great debt of gratitude. I know some of you may not feel that way, because some people just "hate math." But imagine how much more someone might hate math if instead of learning how to multiply, say 14 x 8, they'd have to multiply XIV x VIII !

Are there other bases besides base 10, and if so, why?

Why can't we just use base ten?

There are plenty of other bases. there can be a base of just about any number you like, like base two, base three, base four, etc. There are many reasons you might want to use a base other than 10. One major reason will be explained in the next section.

A typical reason to use other bases, is to solve problems concerning specific amounts. For example, some items are sold in dozens, and grosses. A dozen is 12 x 1. A gross is 12 x 12. I'll bet you can guess what base we are dealing with there. Computer programs that are required to inventory merchandise that is sold in dozens can be written to solve problems using base 12, which makes them more streamlined and efficient, than calculating in base 10.

Great Math Books for Every Home - Personally Recommended by Professor Homunculus

The Number Devil: A Mathematical Adventure
The Number Devil: A Mathematical Adventure

From the amazon.com review: "...introduce basic concepts of numeracy, from interesting number sequences to exponents to matrices. Author Hans Magnus Enzensberger's dry humor and sense of wonder will keep you and your kids entranced while you learn (shhh!) mathematical principles."
Professor Homunculus sez: "If you are helping a child learn math, this book is a MUST!"

 
Playing with Infinity: Mathematical Explorations and Excursions
Playing with Infinity: Mathematical Explorations and Excursions

From the a reader's review on amazon.com: "This book explains the why's behind math from principles as basic as counting to as complicated as series, geometry, and even some calculus principles. It is written in a conversational tone with lots of pictures (yes, and numbers). Each chapter builds upon the last, and it is easy to follow (though sometimes dense). It was my first "fun" math book and is still by far my favorite."
Professor Homunculus sez: "I couldn't agree more!"

 
Mathematics for the Million: How to Master the Magic of Numbers
Mathematics for the Million: How to Master the Magic of Numbers

This book is a classic. It was the book that launched the intellectual self-improvement movement in America. It is perhaps the best book on general math ever written. Hogben takes you from the birth of mathematics to calculus in a lucid, human way. Even if you use only this one book, you will not only understand everything you will need for all of high-school math, and most of college math, but you also have a good basis for understanding the spirit of mathematics. H.G. Wells called it, "A great book, a book of first-class importance."

 
Math Doesn't Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail
Math Doesn't Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail

When have you heard of a Hollywood actress doing something for the minds of her fans? Think back, back...

I can't really think of anyone. Until now. Danica McKellar, actress and mathematician (summa cum laude from UCLA) has done just that. She's written a book, "Math Doesn't Suck."

It's about the best book I can think of for middle-school girls who think that math isn't cool.

 

Why do we have to learn this stuff?

Well of course we don't have to. But we don't want to walk around and be ignorant of things that can be useful, do we? You may not have a use for them now, just like you don't have any use for a car right now, but if you never learn to drive, you're limiting yourself for no reason.

Learning bases is easy - you already use base 10 quite well, and believe it or not, you already use base 2, whether you know it or not. As a matter of fact, you're using it right now.

That's right. Are you aware that almost all computers and circuits operate on the base 2 system? What do you think all those zeros and ones are about? You've seen a number like 100010011, haven't you? Numbers like that, only much longer, he turn up in lots of science-fiction movies like The Matrix all the time. I'm sure you're aware that a number like that usually doesn't mean, "10 billion - is something," it's actually a number in base two.

The reason computers and circuits use base 2, is because based to only consists of ones and zeros. That conveniently can represent two states of a circuit - on, or off. There's more to it than I can explain in this simple lesson, but that's the basic deal.

So in a nutshell, no base 2, no computer games for you!

Anyone who cannot cope with mathematics is not

fully human."

- Robert A. Heinlein

Some of My Other Squidoo Lenses about Basic Math

Have no fear, there's no algebra here... (or not much, anyway).

Where can you learn more about bases?

This is a very basic explanation of bases, and I'm pretty sure you followed all of it.

To learn a little more about bases, and how you can start using them and manipulating them, please check out my series of posts about base 10 and other bases at my website at:

Bases - What are They?

That series of lessons will help you easily learn and understand things like:

  • How can we change a number from base 10 to base 2?
  • What bases are commonly used for ?
  • Which bases are commonly used?
  • How different bases are written.
  • Do bases have anything to do with exponents (powers)?
  • Operations ( addition, subtraction, division, application, etc.) with other bases besides base 10.
  • Can there be bases higher than base 10?

There will also be some trivia about bases, and much much more.

Got a question about bases? Got a weird math story? - Or just want to give me some encouragement? (I could use it!)

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    • profile image

      Albert_Tesla 4 years ago

      Being bored in math. Programming my calculator to translate from base 10 to base 2 or base 3. Aren't nerds amazing?

    • Homunculus profile image
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      Homunculus 4 years ago

      @anonymous: I'm not sure what you are asking. What do you mean by "the main number systems?"

    • profile image

      anonymous 4 years ago

      so would you be able to give examples of the bases for the main number systems?

    • Homunculus profile image
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      Homunculus 5 years ago

      @anonymous: Sounds like you're asking me to do your homework for you.

    • profile image

      anonymous 5 years ago

      My question: a # is divisible by 6 if and only if the # is divisible by both 2 and 3. so i started with the baseline from 0-5, 10-15, 20-25, and so on to 100. From my results i have to have an example that works with the divisibility of 6 that is divisible by both 2 and 3 in the base 6 system and i also need to have a non-example that is not able to be divisible by 2 and 3 in the base 6 system.

    • Homunculus profile image
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      Homunculus 5 years ago

      @anonymous: Michelle,

      I wrote a long post just to answer your question on my blog. You can check out the detailed answer at http://www.mathmojo.com/chronicles/2012/09/02/base...

    • Homunculus profile image
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      Homunculus 5 years ago

      @anonymous: Joct,

      Good try, but I think you forgot the 1s column. There would be seven columns, and the seventh column in base 2 is 64. So 1,000,000 (base 2) is 64 (base 10).

    • profile image

      anonymous 5 years ago

      @anonymous: 128

    • profile image

      anonymous 5 years ago

      If someone has $1,000,000 in base 2, how much money does she have in base 10?

    • profile image

      anonymous 5 years ago

      @Homunculus: Are there other bases besides base 10, and if so, why?

      Why can't we just use base ten?

      There are plenty of other bases. there can be a base of just about any number you like, like base two, base three, base four, etc. There are many reasons you might want to use a base other than 10. One major reason will be explained in the next section.

      A typical reason to use other bases, is to solve problems concerning specific amounts. For example, some items are sold in dozens, and grosses. A dozen is 12 x 1. A gross is 12 x 12. I'll bet you can guess what base we are dealing with there. Computer programs that are required to inventory merchandise that is sold in dozens can be written to solve problems using base 12, which makes them more streamlined and efficient, than calculating in base 10

    • Homunculus profile image
      Author

      Homunculus 5 years ago

      @anonymous: OK, let's take it slow. What's the greatest 3 digit number in base 10 (or normal base)? It's 999 of course, because the greatest digit we can use in base 10 is 9.

      What's the greatest digit we can use in base 5? 4 of course, so the greatest three-digit number we can use in base 5 is 444.

      I'll let you figure out the greatest three-digit number we can use in base 3 for yourself, now that you have some hints.

    • profile image

      anonymous 7 years ago

      i cannot understand a single thing about maths

    • Homunculus profile image
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      Homunculus 5 years ago

      @anonymous: Yeah, if you'd check out Bases - What are They? Part 1, you would learn a lot about how they work.

    • Homunculus profile image
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      Homunculus 5 years ago

      @anonymous: Heinlein didn't say anyone "isn't human" if they don't understand bases - he said they "aren't fully human if they can't cope with math".

      I think what he means is that we aren't living up to our full human potential if we can't deal with math. That is not an unreasonable statement.

      It is also proven by your question. Math partially about details. You can learn details and logic from math. Your question was pretty shaky on the details and the logic.

      It doesn't mean you aren't human. It means that if you could deal better with logic and details, you would get more out of yourself.

      We all could be more fully human if we could deal better with those things.

    • Homunculus profile image
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      Homunculus 5 years ago

      @anonymous: Did you read the series at : Bases - What are They? Part 1

    • Homunculus profile image
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      Homunculus 5 years ago

      @anonymous: It seems to be the best system for everyday use because we have ten fingers, and learn to conceptualize ten things fairly easily. Less digits would be too limiting, and more digits might be too complicated.

      Basically, (excuse the pun) it's for convenience.

    • Homunculus profile image
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      Homunculus 5 years ago

      @anonymous: You needn't shout. You also need to lighten up.

    • kpp2385 profile image

      kpp2385 5 years ago

      extremely useful, specifically to students. great lense. If you get chance have a look at my vedic mathematics related blog. If you like, comment and squidlike if more than welcome!

    • profile image

      anonymous 5 years ago

      Robert A. Heinlein

    • profile image

      anonymous 5 years ago

      To anyone who's wondering, 0 is considered a whole number.

      Integers only include whole numbers, either positive or negative (eg. 1, 2, 8, -13, -19999 you get the idea...) and 0.

    • profile image

      anonymous 5 years ago

      This was kinda helpful

    • profile image

      anonymous 5 years ago

      Hi im 28 i never learnt this way of counting at school and my 7 year old daughter is bringing home homework for me to help her with which she needs extra help with maths anyway and i can't help her with something i don't no or understand myself.... please help me in a simple way possible, would be very greatfull to you. Ive read about base 10 on your site but still don't understand about these collums and hoe do you get an answer from it for eg 14 - 9 = using this base 10 method??

    • profile image

      anonymous 5 years ago

      Then, for say, would base 3 be represented as 3X1=3. then 3X30=90. then 3X300=900. ext....???

    • profile image

      anonymous 5 years ago

      I need to make a poster about base ten and I don't get it!!!!!!!

    • profile image

      anonymous 5 years ago

      @Homunculus: It's The base we americans use to count in

    • profile image

      anonymous 5 years ago

      Why do we use base 10

    • profile image

      anonymous 5 years ago

      @anonymous: because people have ten fingers

    • profile image

      anonymous 7 years ago

      I need help understanding how to find the greatest three-digit number in base three & base five. I'm lost!!!!

    • profile image

      anonymous 7 years ago

      @anonymous: yes it is because it can challenge you at times but times you may understand so it is you just have to learn to keep with it

    • profile image

      anonymous 7 years ago

      I do not think that is true so if you don't understand bases that mean that you aren't human that is foolishness

    • profile image

      anonymous 7 years ago

      hello!

      can someone explian some examples of real life application of bases in detail?

      thanks! (:

    • Homunculus profile image
      Author

      Homunculus 7 years ago

      @anonymous: Wow, I haven't checked the site in awhile. Too late to help you I'm afraid. But then again, I never respond to last minute help. You probably had quite awhile to learn or ask before your test.

      I hope you finally learned it, though, even if it wasn't in time for the test.

    • Homunculus profile image
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      Homunculus 7 years ago

      @anonymous: Wow, I haven't checked the site in awhile. Too late to help you I'm afraid. But then again, I never respond to last minute help. You probably had quite awhile to learn or ask before your test.

      I hope you finally learned it, though, even if it wasn't in time for the test.

    • profile image

      anonymous 7 years ago

      Why is the base 10 system supposedly the best system?

    • profile image

      anonymous 8 years ago

      Hi, I have a packet due tomorrow on bases. This didn't really help me because I needed to know HOW to do them. Could you please comment on this ASAP? I need to know how!!!

    • profile image

      anonymous 8 years ago

      oh i love this site

    • Homunculus profile image
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      Homunculus 8 years ago

      [in reply to chinyere] Hi, Chinyere,

      I'm afraid I'm not sure what the "discovery method" is. I don't teach in the public schools, so I don't get some of the lingo they use. I assume there is some "official" way to discover things in the public schools.

      Like most things "official," I have as little contact as possible with them, so I don't have any info on it. As far as ways to "discover the bases for yourself," I have way too many thoughts on that for a lens like this. It'll have to wait for a future time.

      IN the meantime, check out the four-part series I wrote about bases that begins at:http://mathmojo.com/chronicles/bases-1/

      Let me know how you do.

      All the best,

      Brian (a.k.a. Professor Homunculus at MathMojo.com )

    • profile image

      anonymous 8 years ago

      how can you teach number bases using discovery method

    • Homunculus profile image
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      Homunculus 8 years ago

      Chella,

      I have no Idea what "Base 10 short" is. If you find out, would you please come and post it here?

      Anybody else know what it is?

    • profile image

      anonymous 8 years ago

      What is a Base 10 short?

    • Homunculus profile image
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      Homunculus 8 years ago

      [in reply to RAFAEL DEL ROSARIO]

      Rafael,

      I'm not sure what you mean. Can you give an example of how your lesson differs?

    • profile image

      anonymous 8 years ago

      NO MY LESSON IS DIFFERENT THAN YOURS I AM FINDING FOR BSE10 TO OTHER BASES AND BASE 10 TO OTHER BASES

    • profile image

      anonymous 8 years ago

      Mathematics is the gymnastic of the brain.

    • Homunculus profile image
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      Homunculus 8 years ago

      [in reply to Pathantel]

      There area an infinite amount of bases. You can use any natural number (if you don't count zero as a natural number) and there are an infinite amount of those. There isn't much use for base 1, though,

      Bases are just building blocks. They are sort of the amount you use are your basic "package" for numbers.

      We normally package our numbers in powers of 10 (ten's column, hundreds column, thousands column, etc.) But there's no reason we couldn't have the fifteens column (15^1), two hundred twenty-fives column (15^2), three thousand three hundred seventy-fives column (15^3), etc.

      You could have base 1,000,000 if you had enough things that that would make sense to do that.

      I hope that helped,

      - Professor Homunculus

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      anonymous 8 years ago

      How many bases are there?

    • profile image

      anonymous 8 years ago

      how do i convert 52 into base 12. HELP! im confused

    • profile image

      anonymous 8 years ago

      because we want to understand about bases when the teacher talking about

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      PotPieGirl 9 years ago

      Brian -

      Your lenses are so wonderful - makes me almost believe I like math! haha!

      Keep up the GREAT work!

      Jennifer