# Base 10 and other Bases in Mathematics

## 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.

## Learn much more about bases - Click the link below for a free four-part series about bases

- Free four-part series about bases at The Math Mojo Chronicles

Someone wrote in, "What is a base?? I'm sorry but I'm in the sixth grade and never heard of a base and then all of the sudden it's in my homework. Will you please explain to me in easy fifth or fourth grade words what a base is? Pretend I'm stupid or

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

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!"

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!"

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."

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).

- How to learn and teach multiplication

you know that "tables" stuff they tortured you with in school? Well, not only are they not the only way to learn how to multiply, they are nowhere near the best, the easiest, the most efficient, the most effective, and they sure as poop aren't the mo - Using advanced thinking methods to "trick out" ways to learn math.

This isn't just a collection of silly tricks, like, "Take a number, multiply it by nine, add your age, divide by the number of socks in your sock drawer, subtract your grandmother's birthday, and I'll tell you some meaningless number that will bore y - Danica McKellar's Book "Math Doesn't Suck"

Read more about the actress/mathematician's .book that turns math from a drag to a dream for middle-school girls

## 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:

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!)

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

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

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.

@anonymous: 128

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

@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

i cannot understand a single thing about maths

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!

Robert A. Heinlein

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.

This was kinda helpful

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??

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

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

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

Why do we use base 10

@anonymous: because people have ten fingers

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

@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

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

hello!

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

thanks! (:

Why is the base 10 system supposedly the best system?

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!!!

oh i love this site

how can you teach number bases using discovery method

What is a Base 10 short?

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

Mathematics is the gymnastic of the brain.

How many bases are there?

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

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

Brian -

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

Keep up the GREAT work!

Jennifer

49