Section 1.4

Find the binary representation of each of the following positive integers by working through the algorithm by hand. You can check your answer using the sage cell above.

(d) 100

What positive integers have the following binary representations?

(d) 10011110000

Section 1.5

Calculate the following series:

(d) Σn k=1 (2k −1) for n=4

For any positive integer k, let Ak = {x ∈ Q : k − 1 < x ≤ k} and Bk = {x ∈ Q : −k < x < k}. What are the following sets?

(a) 5 ∪ i=1 A_i

(b) 5 ∪ i=1 B_i

(c) 5 ∩ i=1 A_i

d) 5 ∩ i=1 B_i

Show all work and steps

Let's tackle each part of your question step by step, showing all work and detailed explanations for better clarity.

### Section 1.4 (Binary Representations)
#### (d) Convert the integer 100 to binary:
To find the binary representation of 100, we will repeatedly divide by 2 and keep track of the remainders. Here's how:

1. \( 100 \div 2 = 50 \) remainder **0**
2. \( 50 \div 2 = 25 \) remainder **0**
3. \( 25 \div 2 = 12 \) remainder **1**
4. \( 12 \div 2 = 6 \) remainder **0**
5. \( 6 \div 2 = 3 \) remainder **0**
6. \( 3 \div 2 = 1 \) remainder **1**
7. \( 1 \div 2 = 0 \) remainder **1**

Now, writing the remainders from bottom to top, we get:
**Binary of 100** = \( 1100100_2 \)

#### (d) Convert the binary 10011110000 to decimal:
Now, we convert the binary number \( 10011110000_2 \) to its decimal equivalent.

The binary number \( 10011110000_2 \) represents:

\[
1 \times 2^{10} + 0 \times 2^9 + 0 \times 2^8 + 1 \times 2^7 + 1 \times 2^6 + 1 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0
\]

Now calculate the powers of 2 and sum the results:

\[
1024 + 128 + 64 + 32 + 16 = 1264
\]

So, \( 10011110000_2 \) in decimal is **1264**.

---

### Section 1.5 (Series Calculation)
#### (d) Calculate the series \( \sum_{k=1}^{n} (2k-1) \) for \( n = 4 \)

The given series is:
\[
\sum_{k=1}^{4} (2k - 1)
\]

We will calculate each term of the series:
1. For \( k = 1 \), \( 2(1) - 1 = 1 \)
2. For \( k = 2 \), \( 2(2) - 1 = 3 \)
3. For \( k = 3 \), \( 2(3) - 1 = 5 \)
4. For \( k = 4 \), \( 2(4) - 1 = 7 \)

Now sum the values:
\[
1 + 3 + 5 + 7 = 16
\]

So, \( \sum_{k=1}^{4} (2k - 1) = 16 \).

---

### Section 1.6 (Set Operations)
Given the sets:

- \( A_k = \{x \in \mathbb{Q} : k - 1 < x \leq k\} \)
- \( B_k = \{x \in \mathbb{Q} : -k < x < k\} \)

Let's evaluate the union and intersection of these sets for \( i = 1 \) to \( i = 5 \).

#### (a) \( \bigcup_{i=1}^{5} A_i \)
Here, each \( A_i \) represents an interval:
- \( A_1 = (0, 1] \)
- \( A_2 = (1, 2] \)
- \( A_3 = (2, 3] \)
- \( A_4 = (3, 4] \)
- \( A_5 = (4, 5] \)

The union of these intervals is:
\[
\bigcup_{i=1}^{5} A_i = (0, 5]
\]

#### (b) \( \bigcup_{i=1}^{5} B_i \)
Here, each \( B_i \) represents an interval:
- \( B_1 = (-1, 1) \)
- \( B_2 = (-2, 2) \)
- \( B_3 = (-3, 3) \)
- \( B_4 = (-4, 4) \)
- \( B_5 = (-

Section 1.4
Find the binary representation of each of the following positive integers by working through the algorithm by hand. You can check your answer using the sage cell above.

(d) 100

What positive integers have the following binary representations?

(d) 10011110000

Section 1.5
Calculate the following series:

(d) Σn k=1 (2k −1) for n=4

For any positive integer k, let Ak = {x ∈ Q : k − 1 < x ≤ k} and Bk = {x ∈ Q : −k < x < k}. What are the following sets?

(a) 5 ∪ i=1 A_i

(b) 5 ∪ i=1 B_i

(c) 5 ∩ i=1 A_i

(d) 5 ∩ i=1 B_i

Show all work and steps

Explore binary conversion (100 and 10011110000), series calculation Σn k=1 (2k − 1) for n=4, and set operations (union and intersection) for Ak and Bk sets. Learn the algorithms and steps involved in solving these mathematical problems, suitable for Grades 9-12 students.

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