3.1 Numeration Systems

Numbers, Numerals, and Digits

What exactly is a number? As elementary as it sounds, the question merits a pause for consideration. Depending on who you ask, a number is a mathematical object used to count things. People have counted things for thousands of years, but not everyone wrote numbers the same way. Different cultures used different symbols and different grouping rules. In order to further our understanding of what numbers are and how they behave, we're going to look at different numeration systems.

We're going to work our way up to the numbering system that you know, but we're not going to define it just yet. However, I still need to reference it to explain these other examples so you need to know its name. The symbols you're used to -

$$0,1,2,3,4,5,6,7,8,9$$

are called the Hindu-Arabic numerals. So when I use this word, don't be confused. I am referencing the numeration system that you're used to.

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Historical Numeration Systems

Egyptian Numerals

Egyptian numerals are additive. You count copies of each symbol and add everything. This is a link to an photo album containing images of real egyptian numerals, but for the sake of ease we'll use the unicode counterparts here.

Hindu-Arabic	$$1 000 000$$	$$100 000$$	$$10 000$$	$$1 000$$	$$100$$	$$10$$	$$1$$
Egyptian	𓁨	𓆏	𓂭	𓆼	𓍢	𓎆	𓏺

Very similar to our own counting system, the egyptians counted in groups of 10. To represent larger numbers, they just added more symbols: 1, 2, 3, 4,...9 becomes 𓏺, 𓏺𓏺, 𓏺𓏺𓏺, 𓏺𓏺𓏺𓏺, ... 𓏺𓏺𓏺𓏺𓏺𓏺𓏺𓏺𓏺. But when a group of 10 is met, we would moved to the next position in our positional based numerbering system and write 10, where the egyptians would use their symbol 𓎆.

MyOpenMath: Numeration Systems and Base Conversion

MyOpenMath: Numeration Systems and Base Conversion

Babylonian Numerals

Babylonian notation used a base-60 idea, this is where our notions of grouping will start to be challenged. Think of each new place as "groups of 60" instead of "groups of 10." When we count, we count up to $9$, but then we run out of symbols so we move to a new position and increment that value. $9$ goes to $10$, $1$ group of $10$ plus $0$ groups of $1$.

The Babylonians had few symbols, they used a type of cuneiform, but the numerals were repeated in groups of 60. For example, one would count all the way up to $59$, and then move to a new postion, or "roll over", and increment that position $1$ and reset the previous position to $0$. Using Hindu-Arabic numerals to mimic the base-60 counting, it would look something like this:

$$1,2,3,...,57,58,59,(1, 0),(1,1), (1,2),(1,3),...,(1,58),(1,59),(2,0),...$$

I'm using $(a,b)$ here to denote $a$ groups of $60$ and $b$ groups of $1$. $(3,15)$ would mean I have three groups of $60$ and fifteen groups of $1$, or $3(60) + 15(1) = 195$.

Even still, I'm hiding what a Babylonia numeral would really look like, because their writing was done by chiseling symbols into clay tablets. There wasn't a lot of articulation to be had with a chisel and clay, so the symbols are simple.

Using unicode glyphs, the symbol for $1$ is 𒐕, the symbol for $10$ is 𒐞, and the digit $23$ is written as 𒐞𒐞𒐕𒐕𒐕.

Once the count gets to $59$, we would roll over into a new position and increment there. $58,59,60,61,62$ becomes

𒐞𒐞𒐞𒐞𒐞𒐕𒐕𒐕𒐕𒐕𒐕𒐕𒐕

𒐞𒐞𒐞𒐞𒐞𒐕𒐕𒐕𒐕𒐕𒐕𒐕𒐕𒐕

𒐕

𒐕 𒐕

𒐕 𒐕𒐕

You see the problem here? Babylonians didn't have a numeral for zero and what was written was ambiguous. The context of the writing would give an idea of what the number should have been. For the sake of simplicity, I'm going to add a comma (,) between positions so it is easier to distinguish when places and to make it clear when there was a space left for a zero.

𒐕𒐕 , 𒐞𒐕

should clearly be $2(60)+11(1)=131$, and

𒐕𒐕 , , 𒐞𒐕

should clearly be $2(3600) + 0(60) + 11(1) = 7211$. A small note here, when I moved into the third position of the numeral, that represented $60$ groups of $60$ which is $60\cdot 60=3600$. The third position represent groups of $60^2=3600$. The fourth would represent $60^3=216000$ and so on.

MyOpenMath: Numeration Systems and Base Conversion

The other direction is a bit harder, but it is an important skill to build and will become very useful to us later in this course.

MyOpenMath: Numeration Systems and Base Conversion

Mayan Numerals

Across the pond in the Americas, Mayan's were doing their own counting. They even had a numeral for zero, 𝋠! In that way, mathematically speaking, the Mayans were more advanced.

Numeral	Value
𝋠	0
𝋡	1
𝋥	5

Mayan Numerals 1–30

They used dots 𝋡 and lines 𝋥 for 1's and 5's, but they counted in base 20. Here is what counting from 1 to 30 would look like.

1	2	3	4	5
𝋡	𝋢	𝋣	𝋤	𝋥
𝋦	𝋧	𝋨	𝋩	𝋪
𝋫	𝋬	𝋭	𝋮	𝋯
𝋰	𝋱	𝋲	𝋳	𝋡 𝋠
𝋡 𝋡	𝋡 𝋢	𝋡 𝋣	𝋡 𝋤	𝋡 𝋥
𝋡 𝋦	𝋡 𝋧	𝋡 𝋨	𝋡 𝋩	𝋡 𝋪

Mayan numeral positions extended vertically, with the largest positional value on top. For simplicity I will write the numbers horizontal with the largest positional value to the left to be more familiar. 𝋡 𝋫 𝋱 means 1 (𝋡) group of 400, 11 (𝋫) groups of 20, and 17 (𝋱) groups of 1: $$1(400)+11(20)+17(1)=637$$ Notice I used $400$ for the third position because 20 groups of 20 is 400.

MyOpenMath: Numeration Systems and Base Conversion

Again, the other direction is a bit tougher but it gets easier with practice.

MyOpenMath: Numeration Systems and Base Conversion

Roman Numerals

While the Egyptians, Babylonians, and Mayans were most likely completely new to you, the Romans may be a bit more familiar. Their conquest of the world left a lasting impact on cultures around the world. To this day we still see them on historical buildings and other things. Notably, the most recent Super Bowl as of the typing of these notes was LXIII, or $63$.

Luckily for us, the romans mostly counted in base 10. Unluckily for us, they didn't use a positional system. They simply strung symbols together and added them up.

Symbol	I	V	X	L	C	D	M
Value	$$1$$	$$5$$	$$10$$	$$50$$	$$100$$	$$500$$	$$1000$$

So to count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 romans would write

I,II,III,IV,V,VI,VII,VIII,IX,X

To disect that Super Bowl numeral above, LXIII$=$L $+$ X $+$ III $$=50+10+3=63$$

Note an odd occurance here, for 4, I wrote IV instead of IIII. To help with repetition, the Romans would write a smaller valued numeral before a larger one to show that it was less. IV means one less than 5 as opposed to VI which means one more than 5. Here are the most common differences used when building a roman numeral.

Numeral	IV	IX	XL	XC	CD	CM
Value	$$4$$	$$9$$	$$40$$	$$90$$	$$400$$	$$900$$

This makes reading a roman numeral a little tough. In general, reading it left from right we are adding values up. That is unless we encounter a smaller value in front of a larger one, in that case we subtract the smaller from the larger and keep adding.

MyOpenMath: Numeration Systems and Base Conversion

MyOpenMath: Numeration Systems and Base Conversion

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Hindu-Arabic Place Value

Now we finally make it to the system we use now, the Hindu-Arabic numeral system. This numbering system started a little later than the other systems we've seen here, but it is still very old. It started in India somewhere between the 1st and 6th century. It was adopted by Arab and Persian scholars soon after and eventually made its way to Europe via Spain and North Africa. Famously, Fibonacci, our friend form a previous section, popularized it in his book Liber Abaci or Book of Calculation.

The reason why this numbering system has had the staying power and persistance that it has, is because it takes advantage of a positional system, (try subtracing MCMXCIV and DCCXCIX by hand), a relatively simple set of glyphs, and uses base 10 to match up with our 10 fingers making it easy to use on the fly.

The numerals are Hindu-Arabic numerals are

$${0,1,2,3,4,5,6,7,8,9}$$

When counting one starts in the first position and increments up to 9,

$$1,2,3,4,5,6,7,8,9$$

and then because we've run out of symbols we move to the next position and increment while resetting the pevious position.$10$ means we have $1$ group of $10$ and $0$ groups of $1$.

$$10=1(10)+0(1)$$

This may seem trivial now, because you're very used to the system, but it merits a second look. For example, $5123$ uses 4 positions, each a power of $10$.

$$5123 = 5(1000)+1(100)+2(10)+3(1)$$

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Counting in Other Bases

To better understand several of the algorithms and operations we'll be using later in the semester, we're going to limit out selection of digits to count in different bases. This lets us use numbers that we're familiar with but in a way that is unusual so we can experience the confusion first time learners might have with these algorithms again.

The intent here is to remove the familiarity with the systems at work so we can focus on the how instead of the what. Any calculator can add or subtract 5 digit numbers, what we care about is how its done, not what the answer is.

Here are the digits of some different base systems. Examples:

• Binary (Base-2) has two digits - $$D_2=\{0,1\}$$
• Base-5 has five digits - $$D_5=\{0,1,2,3,4\}$$
• Base-12 $$D_{12}=\{0,1,2,3,4,5,6,7,8,9,T,E\}$$
• Hexadecimal (Base-16) $$D_{16}=\{0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F\}$$

These might seem strange at first, but they're used the same way every positional system I've described in this section works. You start by counting in the first position until you run out of numerals, then you roll over into the next position and increment there resetting the first.

Lets start by counting in base five. Up to 4 there is no problem.

$$0,1,2,3,4$$

But we've run out of numerals. So we roll over into the next position, increment it by one, and begin again.

$$0,1,2,3,4,10,11,12,13,14$$

And now we're at $14$, which in base-5 means $1(5)+4(1)=9$. This is where it starts to get a little confusing so I'm going to add a subscript to numbers when they are in different bases.

$$14_{\text{five}}=1(5)+4(1)=9$$

When a number does not have a subscript, I mean for it to be in base 10.

For another quick example using Hexadecimal (Base-16), the number $$A3F1_{\text{hex}}=10(16^3)+3(16^2)+15(16^1)+1(16^0)=41969$$

This is one reason computer scientists uses hexadecimal. You can store larger numbers in smaller spaces.

MyOpenMath: Numeration Systems and Base Conversion

Backwards is always the harder direction, but we've had practice. If we wanted to convert a number from base 10 into any other base, we need only count how many groupings there are of each power of the base and remove them.

MyOpenMath: Numeration Systems and Base Conversion

After doing this conversion several times, one starts to notice a pattern. We get to learn our first real algorithm! Here are steps one can use to convert a number from base 10 into any other base.

MyOpenMath: Numeration Systems and Base Conversion

MyOpenMath: Numeration Systems and Base Conversion

Heres a closing graph on visualizing counting in different bases with blocks.