🚧 Setup for 2.5

chapter_2/exercises.md

@@ -3647,3 +3647,200 @@ $$ A \wedge B \equiv (A \downarrow A)\downarrow(B \downarrow B) $$
 So:
 
 $$ (((P \downarrow P)\downarrow Q) \downarrow ((P \downarrow P)\downarrow Q)\downarrow ((P \downarrow P)\downarrow Q)) \downarrow (((Q \downarrow Q)\downarrow P) \downarrow ((Q \downarrow Q)\downarrow P) \downarrow (Q \downarrow Q)) $$
+
+---
+
+**Exercise Set 2.5**
+
+Page 129
+
+Represent the decimal integers in 1-6 in binary notation.
+
+1. $19$
+
+2. $55$
+
+3. $287$
+
+4. $458$
+
+5. $1609$
+
+6. $1424$
+
+Represent the integers in 7-12 in decimal notation.
+
+7. $1110_2$
+
+8. $10111_2$
+
+9. $110110_2$
+
+10. $1100101_2$
+
+11. $1000111_2$
+
+12. $1011011_2$
+
+Perform the arithmetic in 13-20 using binary notation.
+
+13.
+
+$$
+1011_2 \\
+\underline{+ 101_2}
+$$
+
+14.
+
+$$
+1001_2 \\
+\underline{+ 1011_2}
+$$
+
+15.
+
+$$
+101101_2 \\
+\underline{+ 11101_2}
+$$
+
+16.
+
+$$
+110111011_2 \\
+\underline{+ 1001011010_2}
+$$
+
+17.
+
+$$
+10100_2 \\
+\underline{+ 1101_2}
+$$
+
+18.
+
+$$
+11010_2 \\
+\underline{- 1101_2}
+$$
+
+19.
+
+$$
+101101_2 \\
+\underline{- 10011_2}
+$$
+
+20.
+
+$$
+1010100_2 \\
+\underline{- 10111_2}
+$$
+
+21. Give the output signals $S$ and $T$ for the circuit shown below if the input
+    signals $P$, $Q$, and $R$ are as specified. Note that this is _not_ the
+    circuit for a full-adder.
+
+a. $P = 1$, $Q = 1$, $R = 1$
+
+b. $P = 0$, $Q = 1$, $R = 0$
+
+c. $P = 1$, $Q = 0$, $R = 1$
+
+(See Page 130)
+
+22. Add $11111111_2 + 1_2$ and convert the result to decimal notation, to verify
+    that $11111111_2 = (2^8 - 1)_10$.
+
+Find the 8-bit two's complements for the integers in 23-26.
+
+23. $-23$
+
+24. $-67$
+
+25. $-4$
+
+26. $-115$
+
+Find the decimal representations for the integers with the 8-bit two's
+complements given in 27-30.
+
+27. $11010011$
+
+28. $10011001$
+
+29. $11110010$
+
+30. $10111010$
+
+Use 8-bit two's complements to compute the sums in 31-36.
+
+31. $57 + (-118)$
+
+32. $62 + (-18)$
+
+33. $(-6) + (-73)$
+
+34. $89 + (-55)$
+
+35. $(-15) + (-46)$
+
+36. $123 + (-94)$
+
+37.
+
+a. Show that when you apply the 8-bit two's complement procedure to the 8-bit
+two's complement for $-128$, you get the 8-bit two's complement for $-128$.
+
+b. Show that if $a$, $b$, and $a + b$ are integers in the range $1$ through
+$128$, then
+
+$$ (2^8 - a) + (2^8 - b) = (2^8 - (a + b)) + 2^8 \geq 2^8 + 2^7 $$
+
+Explain why it follows that if integers $a$, $b$, and $a + b$ are all in the
+range $1$ through $128$, then the 8-bit two's complement of $(-a) + (-b)$ is a
+negative number.
+
+Convert the integers in 38-40 from hexadecimal to decimal notation.
+
+38. $A2BC_{16}$
+
+39. $E0D_{16}$
+
+40. $39EB_{16}$
+
+Convert the integers in 41-43 from hexadecimal to binary notation.
+
+41. $1C0ABE_{16}$
+
+42. $B53DF8_{16}$
+
+43. $4ADF83_{16}$
+
+Convert the integers in 44-46 from binary to hexadecimal notation.
+
+44. $00101110_2$
+
+45. $1011011111000101_2$
+
+46. $11001001011100_2$
+
+47. **Octal Notation:** IN addition to binary and hexadecimal, computer
+    scientists also use _octal notation_ (base 8) to represent numbers. Octal
+    notation is based on the fact that any integer can be uniquely represented
+    as a sum of numbers of the form $d \cdot 8^n$, where each $n$ is a
+    nonnegative integer and each $d$ is one of the integers from $0$ to $7$.
+    Thus, for example,
+    $5073_8 = 5 \cdot 8^3 + 0 \cdot 8^2 + 7 \cdot 8^1 + 3 \cdot 8^0 = 2619_{10}$.
+
+a. Convert $61502_8$ to decimal notation.
+
+b. Convert $20763_8$ to decimal notation.
+
+c. Describe methods for converting integers from octal to binary notation and
+the reverse that are similar to the methods used in Examples 2.5.9 and 2.5.10
+for converting back and forth from hexadecimal to binary notation. Give examples
+showing that these methods result in correct answers.

chapter_2/notes.md

@@ -320,3 +320,70 @@ Page 112
 
 Two digital logic circuits are **equivalent** if, and only if, their
 input/output tables are identical.
+
+---
+
+Page 122
+
+**Definition**
+
+**The 8-bit two's complement** for an integer $a$ between -128 and 127 is the
+8-bit binary representation for
+
+$$
+\begin{cases}
+a & \text{if } a \geq 0 \\
+2^8 - |a| & \text{if } a < 0
+\end{cases}
+$$
+
+---
+
+Page 123
+
+**The 8-Bit Two's Complement for a Negative Integer**
+
+The 8-bit two's complement for a negative integer $a$ that is at least -128 can
+be obtained as follows:
+
+- Write the 8-bit binary representation for $|a|$.
+
+- Switch all the 1's to 0's and all the 0's to 1's. (This is called flipping, or
+  complementing, the bits.)
+
+- Add 1 in binary notation.
+
+---
+
+Page 124
+
+To find the decimal representation of the negative integer with a given 8-bit
+two's complement:
+
+- Apply the two's complement procedure to the given two's complement.
+
+- Write the decimal equivalent of the result.
+
+---
+
+Page 125
+
+To add two integers in the range -128 through 127 whose sum is also in the range
+-128 through 127:
+
+- Convert both integers to their 8-bit two's complement representations.
+
+- Add the resulting integers using ordinary binary addition, discarding any
+  carry bit of 1 that may occur in the 2<sup>8</sup>th position.
+
+- Convert the result back to decimal form.
+
+---
+
+Page 128
+
+To convert an integer from hexadecimal to binary notation:
+
+- Write each hexadecimal digit of the integer in 4-bit binary notation.
+
+- Juxtapose the results.

chapter_2/test_yourself.md

@@ -174,3 +174,69 @@ NOT; AND
    _______ gate.
 
 NOT; OR
+
+---
+
+**Test Yourself**
+
+Page 129
+
+1. To represent a nonnegative integer in binary notation means to write it as a
+   sum of products of the form _______, where _______.
+
+$d \cdot 2^n$ $d = 1$, and $n$ is a nonnegative integer.
+
+2. To add integers in binary notation, you use the facts that $1_2 + 1_2 = $
+   _______ and $1_2 + 1_2 + 1_2 = $ _______.
+
+$10_2$; $11_2$
+
+3. To subtract integers in binary notation, you use the facts that $10_2 - 1_2 =
+   $ _______ and $11_2 - 1_2 = $ _______.
+
+$1_2$; $10_2$
+
+4. A half-adder is a digital logic circuit that _______, and a full-adder is a
+   digital logic circuit that _______.
+
+outputs the sum of any two binary digits; outputs the sum of any three binary
+digits
+
+5. If $a$ is an integer with $-128 \leq a \leq 127$, the 8-bit two's complement
+   of $a$ is _______ if $a \geq 0$ and is _______ if $a < 0$.
+
+The 8-bit representation of $a$; The 8-bit representation of $2^8 - |a|$
+
+6. To find the 8-bit two's complement of a negative integer $a$ that is at least
+   -128, you _______, _______, and _______.
+
+- Write the 8-bit binary representation for $|a|$.
+
+- Switch all the 1's to 0's and all the 0's to 1's. (This is called flipping, or
+  complementing, the bits.)
+
+- Add 1 in binary notation.
+
+7. To add two integers in the range -128 through 127 whose sum is also in the
+   range -128 through 127, you _______, ______, _______, and _______.
+
+- Convert both integers to their 8-bit two's complement representations.
+
+- Add the resulting integers using ordinary binary addition,
+
+- Discarding any carry bit of 1 that may occur in the 2<sup>8</sup>th position.
+
+- Convert the result back to decimal form.
+
+8. To represent a nonnegative integer in hexadecimal notation means to write it
+   as a sum of products of the form _______, where _______.
+
+$d \cdot 16^n$; $d = 0, 1, 2, \dots 9, A, B, C, D, E, F$ and $n$ is a
+nonnegative integer.
+
+9. To convert a nonnegative integer from hexadecimal to binary notation, you
+   _______ and _______.
+
+- Write each hexadecimal digit of the integer in 4-bit binary notation.
+
+- Juxtapose the results.