Number

Calculus

Imagine living twenty-eight thousand years ago. Among your hobbies, you enjoy rock collecting. It isn't long before you start to notice patterns in various arrangements of your rocks.

You don't yet have a number system to talk about these patterns, but you have a sense of quantity. You notice that only certain quantities of rocks can form this nice pattern:

Pattern 1

\(\dots\)

\(\dots\)

\(\dots\)

and only certain quantities of rocks can form this other nice pattern:

Pattern 2

\(\dots\)

\(\dots\)

\(\dots\)

Moreover, you notice that you can combine adjacent arangements from Pattern 1 to get arrangements from Pattern 2. Pretty neat! Good thing cell phones don't exist...

Then your friend Chelsea, also from twenty-eight thousand years ago, notices another connection: sometimes she can combine two arrangements from Pattern 2 to get a bigger arrangement also in Pattern 2, but it doesn't always work. She finds this one:

   with    

   makes    

Pretty soon your whole village is obsessed with these amazing patterns. They find some nice rocks that stack into towers and begin to make arrangements of towers in patterns like this one:

Pattern 3

\(\dots\) \(\dots\)

Chelsea, who is rightly proud of her discovery about combinations in Pattern 2, searches for similar combinations in Pattern 3. She wants to find two piles in Pattern 3 that can combine together to get another pile in Pattern 3, but this last format is challenging and it will be quite a wiles before someone figures it out.

Factors

In modern times, rocks still exist. You can even find your own in nature. Let's build some numbers with rows of rocks:

One

Two

     

Three

     

Four

     

     

Five

     

Six

     

     

     

These are all the ways that you can construct the numbers one through six with regular rows (rows of the same size so they line up). In each construction, a row represents a factor or divisor of the number. Four example, \(4\) is a factor of \(12\) because we can construct 12 with rows of size \(4\). Another way to think about this is that \(12\) can be dividied into rows of size \(4\).

Since \(4\) "goes into" \(12\), we may write \(4 \mid 12\). We can also write \(5 \nmid 12\) to say that \(5\) does not divide \(12\).

We haven't yet talked about negative rocks, which could be viewed as factors, but when we say "\(d\) divides \(n\)", we mean for positive integer values of \(d\).

Algorithm

Once we have stacks of rocks, we begin to move those stacks around. Like Chelsea, we start to find patterns not only in the arrangements, but also in relationships between arrangements. An algorithm is a sequence of steps to perform a calculation or solve a problem. The algorithms we will work with will take some inputs and return some output.

As we play with rock constructions, a question arrises:

The algorithm below can be used to find the greatest common divisor of Left and Right piles of rocks.

In this algorithm, we start with two piles and repeatedly take the smaller amount away from the bigger amount until the piles are equal. Let's run through it with an example of \(55\) and \(25\).

We are done, and we find that \(\gcd(55,25) = 5\).

Decimal

Our standard number system is base-10, which uses ten symbols: \[0, 1, 2, 3, 4, 5, 6, 7, 8, 9\] Every place value is a power of 10. The number \[4703\] is 4 thousands (\(10^{3}\)), 7 hundreds (\(10^{2}\)), 0 tens (\(10^{1}\)), and 3 ones (\(10^{0}\)). We could make a table like this: \[ \begin{array}{c|c|c|c|c} \text{Place-Value} & 10^3 & 10^2 & 10^1 & 10^0 \\ \hline \text{Digit} & 4 & 7 & 0 & 3 \\ \end{array} \] or represent it like this:

Counting

Given a set of symbols like \(0,1,2,3,4,5,6,7,8,9\), we start counting with the symbol \(0\) reprsenting zero, then move to the next symbol to represent the next number, and so on: \[ \begin{array}{c|c} \textbf{Number} & \textbf{Decimal} \\ \hline \text{zero} & 0 \\ \text{one}& 1 \\ \text{two}& 2 \\ \text{three} & 3 \\ \text{four} & 4 \\ \text{five} & 5 \\ \text{six}& 6 \\ \text{seven} & 7 \\ \text{eight} & 8 \\ \text{nine} & 9 \\ \end{array} \] At this point, we have run out of symbols! With every number system:

In each place, we start with our first symbol, \(0\), and count up to our last symbol \(9\). Every time we run out of symbols, we start over at \(0\) and add one to the left.

Place Value

Each place we write a decimial has a value associated with it. That value is a power of \(10\), increasing from right to left.

If this is the only number system you know, then it might be suprising to learn that we don't have to represent numbers this way.

Binary

Binary is a base-2 number system that uses the symbols \(0\) and \(1\). When we start counting in binary, we run out of symbols quickly: \[0\] \[1\] We reset the one's place and add \(1\) to the left of it: \[10\] That looks like a ten, but it is a two written in binary. Here are several numbers written in binary: \[ \begin{array}{c|r} \textbf{Number} & \textbf{Binary} \\ \hline \text{zero} & 0 \\ \text{one} & 1 \\ \text{two} & 10 \\ \text{three} & 11 \\ \text{four} & 100 \\ \text{five} & 101 \\ \text{six} & 110 \\ \text{seven} & 111 \\ \text{eight} & 1000 \\ \text{nine} & 1001 \\ \end{array} \] Some very nice patterns emerge. In fact, once we notice patterns counting in base-2, we might go back and look for similar patterns in base-10 that were there all along.

Place Value

In binary, we have numbers that look like: \[10111\] which has place-values in base-2: \[ \begin{array}{c|c|c|c|c|c} \text{Place Value} &2^4 & 2^3 & 2^2 & 2^1 & 2^0 \\ \hline \text{Bit} & 1 & 0 & 1 & 1 & 1 \\ \end{array} \] We can convert \(10111\) to decimal by calculting: \[ 1(2^4) + 0 (2^3) + 1 (2^2) + 1 (2^1) + 1(2^0) \] \[ = 16 + 0 + 4 + 2 + 1 \] \[ = 23 \]

Try creating some binary numbers by clicking on the cells below:

Decimal to Binary

Let's convert \(57\) to binary. We look at the place values and fill in the largest possible place until we have a value that sums to our target of \(57\).

\[ \begin{array}{c|c|c|c|c|c|c|c r } \text{Place Value} & 2^6 & 2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0 & \quad \quad \text{Target: }57 \\ \hline \text{Bit} & & & & & & & & \quad \quad \text{Sum: } 0 \\ \end{array} \] \[ \begin{array}{c|c|c|c|c|c|c|c r } \text{Place Value} & 2^6 & 2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0 & \quad \quad \text{Target: }25 \\ \hline \text{Bit} & & 1 & & & & & & \quad \quad \text{Sum: } 32 \\ \end{array} \] \[ \begin{array}{c|c|c|c|c|c|c|c r } \text{Place Value} & 2^6 & 2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0 & \quad \quad \text{Target: }9 \\ \hline \text{Bit} & & 1 & 1 & & & & & \quad \quad \text{Sum: 48 } \\ \end{array} \] \[ \begin{array}{c|c|c|c|c|c|c|c r } \text{Place Value} & 2^6 & 2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0 & \quad \quad \text{Target: } 1 \\ \hline \text{Bit} & & 1 & 1 & 1 & & & & \quad \quad \text{Sum: 56 } \\ \end{array} \] \[ \begin{array}{c|c|c|c|c|c|c|c r } \text{Place Value} & 2^6 & 2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0 & \quad \quad \text{Target: } 0 \\ \hline \text{Bit} & & 1 & 1 & 1 & 0 & 0 & 1 & \quad \quad \text{Sum: 57 } \\ \end{array} \] We find that \(57\) in binary is \(111001\).

Recommended Resources

• Joseph, G. G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton University Press. http://www.jstor.org/stable/j.ctt7sdsb
• Eglash, R. (2009). African Origins of Maths. https://www.youtube.com/watch?v=7n36qV4Lk94.