Solve the following addition problems and express the following base ten numbers in base two.

The sum is $\boxed{\mathbf{77764}}$.

b) $5324+8765+6543$

The sum is $\boxed{\mathbf{20632}}$.

2) Express the following in base two.

a) $15_{ten}$

$15 \div 2 = 7$ remainder 1
$7 \div 2 = 3$ remainder 1
$3 \div 2 = 1$ remainder 1
$1 \div 2 = 0$ remainder 1

b) $16_{ten}$

$16 \div 2 = 8$ remainder 0
$8 \div 2 = 4$ remainder 0
$4 \div 2 = 2$ remainder 0
$2 \div 2 = 1$ remainder 0
$1 \div 2 = 0$ remainder 1

c) $18_{ten}$

$18 \div 2 = 9$ remainder 0
$9 \div 2 = 4$ remainder 1
$4 \div 2 = 2$ remainder 0
$2 \div 2 = 1$ remainder 0
$1 \div 2 = 0$ remainder 1

d) $20_{ten}$

$20 \div 2 = 10$ remainder 0
$10 \div 2 = 5$ remainder 0
$5 \div 2 = 2$ remainder 1
$2 \div 2 = 1$ remainder 0
$1 \div 2 = 0$ remainder 1

3) Express the following correct to two significant figures.

a) 3.4631

Since $6 \ge 5$, round up the second significant figure.

b) 54.621

Since $6 \ge 5$, round up the second significant figure.

c) 532.64

Since $2 < 5$, keep the second significant figure as it is and replace subsequent digits before the decimal point with zeros.

d) 43.5567

Since $5 \ge 5$, round up the second significant figure.

4) Binary arithmetic

i) $10011_{two} + 111_{two}$

$1+1=10_{two}$ (write 0, carry 1)
$1+1+1=11_{two}$ (write 1, carry 1)
$0+1=1_{two}$ (write 1)
$0+0=0_{two}$ (write 0)
$1+0=1_{two}$ (write 1)

The sum is $\boxed{\mathbf{101110_{two}}}$.

ii) $11111_{two} + 10011_{two}$

$1+1=10_{two}$ (write 0, carry 1)
$1+1+1=11_{two}$ (write 1, carry 1)
$1+0+1=10_{two}$ (write 0, carry 1)
$1+0+1=10_{two}$ (write 0, carry 1)
$1+1+1=11_{two}$ (write 1, carry 1)

The sum is $\boxed{\mathbf{110010_{two}}}$.

iii) $101_{two} \times 11_{two}$

$101_{two} \times 1_{two} = 101_{two}$

$101_{two} \times 1_{two} = 101_{two}$ (shifted becomes $1010_{two}$)

The product is $\boxed{\mathbf{1111_{two}}}$.

5) a. Define Binary numbers.
Binary numbers are a numeral system that uses only two symbols, typically 0 and 1, to represent numerical values. This base-2 system is fundamental to digital electronics and computer science, where 0 and 1 correspond to the "off" and "on" states of electrical switches.

b. Write five binary numbers you know.
Here are five binary numbers:

• $0_{two}$
• $1_{two}$
• $10_{two}$
• $11_{two}$
• $100_{two}$