feat(num-rep): release

content/number_rep/index.qmd

@@ -3,4 +3,63 @@ title: "Number Representation"
 subtitle: "UC Berkeley, CS 61C"
 ---
 
-## TODO
+![](astronaut-alien.png){fig-alt="A astronaut talking to an alien with 2 fingers per hand and there are 4 rocks on the ground. Alien says 
+'There are 10 rocks.' Astronaut says 'Oh, you must be using base 4. See, I use base 10.' Alien says 'No. I use base 10. What is base 4?' Caption reads 'Every base is base 10'"}
+
+## Precheck Questions
+1. **True/False**: Depending on the context, the same sequence of bits may represent different things.    
+
+::: {.callout-note title="Answer" collapse="true"}
+**True.** The same bits can be interpreted in many different ways with the exact same bits! The bits
+can represent anything from an unsigned number to a signed number or even, as we will cover
+later, a program. It is all dependent on its agreed upon interpretation.
+:::
+
+2. **True/False**: If you interpret a $N$-bit Two's complement number as an unsigned number, negative numbers would be smaller than 
+positive numbers.
+
+::: {.callout-note title="Answer" collapse="true"}
+**False.** In Two’s Complement, the MSB is always 1 for a negative number. This means EVERY
+negative number in Two’s Complement, when converted to unsigned, will be larger than the
+positive numbers.
+:::
+
+3. **True/False**: We can represent fractions and decimals in our given number representation formats (unsigned, biased, and Two’s Complement).
+
+::: {.callout-note title="Answer" collapse="true"}
+**False.** Our current representation formats has a major limitation; we can only represent and do
+arithmetic with integers. To successfully represent fractional values as well as numbers with
+extremely high magnitude beyond our current boundaries, we need another representation
+format.
+:::
+
+4. How many numbers can be represented by an unsigned, base-4, $n$-digit number.    
+    **A.** 1    
+    **B.** $2^n - 1$    
+    **C.** $4^n$    
+    **D.** $4^{n-1}$    
+    **E.** $4^n - 1$
+
+::: {.callout-note title="Answer" collapse="true"}
+**C.**
+:::
+
+5. How many bits are needed to represent decimal number 116 in binary?
+
+::: {.callout-note title="Answer" collapse="true"}
+**7 bits**. $(116)_{10} =$ `0b111 0100` or $log{_2}{116} \approx 6.85$ which we round to 7 bits.
+:::
+
+
+## Useful Links
+[Fall 25 Slides](https://docs.google.com/presentation/d/1dmCk2fZz-P8VedzAXnVmJiYPKszVka5NKmTuLJ6hqZc/edit?slide=id.g3292f1ea280_3_0#slide=id.g3292f1ea280_3_0)    
+[Fall 25 Precheck](https://cs61c.org/fa25/pdfs/discussions/disc01/disc01-pre.pdf)    
+[Precheck Solutions](https://cs61c.org/fa25/pdfs/discussions/disc01/disc01-pre-sols.pdf)
+
+### Additional Reading
+**CS Illustrated Number Rep Handout**    
+
+* [Comparing Binary Integer Representations](https://csillustrated.berkeley.edu/PDFs/handouts/integer-representations-1-handout.pdf)    
+* [Negation and Zeroes](https://csillustrated.berkeley.edu/PDFs/handouts/integer-representations-2-comparing-handout.pdf)     
+* [Increments and Monotonicity](https://csillustrated.berkeley.edu/PDFs/handouts/integer-representations-3-comparing-handout.pdf)    
+* [The Thrilling Conclusion!](https://csillustrated.berkeley.edu/PDFs/handouts/integer-representations-4-comparing-handout.pdf)    

content/number_rep/signed.qmd

@@ -3,6 +3,82 @@ title: "Signed Representations"
 subtitle: "UC Berkeley, CS 61C"
 ---
 
-## TODO
+## Precheck
+
+Unsigned binary numbers work for natural numbers, but many calculations use negative numbers as well. To deal with this, a number of different schemes have been used to represent signed numbers. Here are two common schemes:
+
+### Two’s Complement 
+
+(a) We can write the value of an $n$-digit two's complement number as 
+$$
+\sum_{i=0}^{n-2} 2^i d_i - 2^{n-1} d_{n-1}
+$$
+(b) Negative numbers will have a 1 as their most significant bit (MSB). Plugging in $d_{n-1} = 1$ to the formula above gets us 
+$$
+\sum_{i=0}^{n-2} 2^i d_i - 2^{n-1}
+$$
+(c) Meanwhile, positive numbers will have a 0 as their MSB. Plugging in $d_{n-1} = 0$ gets us
+$$
+\sum_{i=0}^{n-2} 2^i d_i
+$$
+which is very similar to unsigned numbers.
+
+(d) To negate a two's complement number: flip all the bits and add 1. 
+(e) Addition is exactly the same as with an unsigned number. 
+(f) Only one 0, and it’s located at `0b0`. 
+
+### Biased Representation
+(a) The number line is shifted so that the smallest number we want to be representable would be `0b0...0`. 
+(b) To find out what the represented number is, read the representation as if it was an unsigned number, then add the bias. 
+(c) We can shift to any arbitrary bias we want to suit our needs. To represent (nearly) as much negative numbers as positive, a commonly-used bias for $N$-bits is $-(2^{N-1} - 1)$.
+
+## Summary & Questions
+
+You should be able to do:
+
+* Conversion between binary, decimal, and hexademical
+* Conversion with signed numbers
+* Arithmetic with binary and hexadecimal numbers
+
+### Conceputal Review Questions
+1. What is a bit? How many bits are in a byte? Nibble?
+
+::: {.callout-note title="Answer" collapse="true"}
+A bit is the smallest unit of digital information and it can be either 0 of 1. There are 4 bits in a nibble and 8 bits in a byte.
+
+See: [Lecture 2 Slide 13](https://docs.google.com/presentation/d/1dmCk2fZz-P8VedzAXnVmJiYPKszVka5NKmTuLJ6hqZc/edit?slide=id.g2af3b38b3e2_1_154#slide=id.g2af3b38b3e2_1_154)
+:::
+
+2. What is overflow?
+
+::: {.callout-note title="Answer" collapse="true"}
+When the result of an arithmetic operation is outside the range of what is representable by given number of bits.
+
+See: [Lecture 2 Slide 26](https://docs.google.com/presentation/d/1dmCk2fZz-P8VedzAXnVmJiYPKszVka5NKmTuLJ6hqZc/edit?slide=id.g2af3b38b3e2_1_186#slide=id.g2af3b38b3e2_1_186)
+:::
+
+3. What is the range of numbers representable by $n$-bit unsigned, sign-magnitude, one's complement, two's complement, and biased notation?
+
+::: {.callout-note title="Answer" collapse="true"} 
+* **Unsigned**: $[0, 2^n-1]$
+* **Sign-Magnitude**: $[-(2^{n-1} - 1), 2^{n-1} - 1]$
+* **One's complement**: $[-(2^{n-1} - 1), 2^{n-1} - 1]$
+* **Two's complement**: $[-2^{n-1}, 2^{n-1} - 1]$
+* **Bias**: $[0+$bias$, 2^n-1+$bias$]$
+
+See:  [Lecture 2](https://docs.google.com/presentation/d/1dmCk2fZz-P8VedzAXnVmJiYPKszVka5NKmTuLJ6hqZc/edit?slide=id.g32e4dda2ba9_0_123#slide=id.g32e4dda2ba9_0_123)
+:::
+
+4. How many ways to represent zero do these representations have, $n$-bit unsigned, sign-magnitude, one's complement, two's complement, and biased notation?
+
+::: {.callout-note title="Answer" collapse="true"}
+* **Unsigned**: 1
+* **Sign-Magnitude**: 2
+* **One's complement**: 2
+* **Two's complement**: 1
+* **Bias**: 1 or 0 (depending on bias)
+
+See: [Lecture 2](https://docs.google.com/presentation/d/1dmCk2fZz-P8VedzAXnVmJiYPKszVka5NKmTuLJ6hqZc/edit?slide=id.g32e4dda2ba9_0_123#slide=id.g32e4dda2ba9_0_123)
+:::
 
 

content/number_rep/unsigned.qmd

@@ -3,7 +3,10 @@ title: "Unsigned Representations"
 subtitle: "UC Berkeley, CS 61C"
 ---
 
-## TODO
-
-
+## Precheck
+If we have an $n$-digit unsigned numeral $d_{n-1}$ $d_{n-2}$...$d_0$ in radix (or base) $r$, then the value of that numeral is:
+$$
+\sum_{i=0}^{n-1} r^i d_i
+$$
+which is just fancy notation to say that instead of a 10's or 100's place we have an $r$'s or $r^2$'s place. For the three radices binary, decimal, and hex, we just let $r$ be 2, 10, and 16, respectively.