Number

Computing and Data Science

Pattern 1

\(\dots\)
 
\(\dots\)


\(\dots\)



Pattern 2

\(\dots\)

\(\dots\)


\(\dots\)





   with    



   makes    





Pattern 3

\(\dots\) \(\dots\)
One
Two
     
Three
     
Four
     
     
Five
     

Exercise

Find all of the "rock constructions" up to \(15\).

Algorithm — Square Root

Given a number \(x\), calculate \(\sqrt{x}\) to desired precision.


  1. Make a guess, \(g\)
  2. Calculate new guess: \(\frac{1}{2}( g + \frac{x}{g} )\)
  3. Repeate step (2) as many times as you like, plugging in the new guess each time.

Algorithm — Square Root

Given a number \(x\), calculate \(\sqrt{x}\) to desired precision.


  1. Make a guess, \(g\)
  2. Calculate new guess: \(\frac{1}{2}( g + \frac{x}{g} )\)
  3. Repeate step (2) as many times as you like, plugging in the new guess each time.

Find \(\sqrt{2}\):
  • \(g=1\)
  • \(g = \frac{1}{2} (1+\frac{2}{1}) \)
  • \(g = \frac{1}{2} \left( \frac{1}{2} (1+\frac{2}{1})+ \frac{2}{\frac{1}{2} (1+\frac{2}{1})} \right) \)
  • \( g = \frac{1}{2} \left( \frac{1}{2} \left( \frac{1}{2} (1+\frac{2}{1})+ \frac{2}{\frac{1}{2} (1+\frac{2}{1})} \right) + \frac{2}{\frac{1}{2} \left( \frac{1}{2} (1+\frac{2}{1})+ \frac{2}{\frac{1}{2} (1+\frac{2}{1})} \right)} \right) \)

Counting

0
1
2
\(\vdots\)
8
9

Counting

0
1
2
\(\vdots\)
8
9
10

Counting in Binary

0

Counting in Binary

0
1

Counting in Binary

0
1
10

Counting in Binary

0
1
10
11

Counting in Binary

0
1
10
11
100

Counting in Binary

0
1
10
11
100
101
110
111
1000
1001
1010

Place Value


\(10^{4}\) \(10^{3}\) \(10^{2}\) \(10^{1}\) \(10^{0}\)
0 0 0 0 0 \(=\) 0


\(2^{6}\) \(2^{5}\) \(2^{4}\) \(2^{3}\) \(2^{2}\) \(2^{1}\) \(2^{0}\)
0 0 0 0 0 0 0 \(=\) 0

Place Value


10000 1000 100 10 1
0 0 0 0 0 \(=\) 0


64 32 16 8 4 2 1
0 0 0 0 0 0 0 \(=\) 0

©2025 Jedediyah Williams
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