Perform the following conversions

Perform the following conversions:
a. 101010 (base 2) to base 10
b. CA (base 16) to base 10
c. 10101011 (base 2) to base 16
Answer
Answer
d. 50 (base 10) to base 2
e. F8 (base 16) to base 2
Answer
Answer
f. Perform the following addition of two binary (base 2) numbers. Your answer should be a number in binary.
101110 = ?

The Correct Answer and Explanation is :

Binary and Hexadecimal Conversions

a. Convert 101010 (base 2) to base 10:

To convert a binary number to decimal, we sum the products of each bit with powers of 2.

[
101010_2 = (1 \times 2^5) + (0 \times 2^4) + (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0)
]

[
= (1 \times 32) + (0 \times 16) + (1 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1)
]

[
= 32 + 0 + 8 + 0 + 2 + 0 = 42
]

Answer: ( 42_{10} )

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b. Convert CA (base 16) to base 10:

Each digit in hexadecimal represents a power of 16.

[
CA_{16} = (C \times 16^1) + (A \times 16^0)
]

Since C = 12 and A = 10 in decimal:

[
(12 \times 16) + (10 \times 1) = 192 + 10 = 202
]

Answer: ( 202_{10} )

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c. Convert 10101011 (base 2) to base 16:

To convert binary to hexadecimal, we group bits in sets of four from right:

[
1010\ 1011
]

Each group converts to hex:

• 1010 (binary) = A (hex)
• 1011 (binary) = B (hex)

Answer: ( AB_{16} )

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d. Convert 50 (base 10) to base 2:

Divide 50 by 2 and record the remainders:

Division	Quotient	Remainder
50 ÷ 2	25	0
25 ÷ 2	12	1
12 ÷ 2	6	0
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1

Reading from bottom to top: 110010₂

Answer: ( 110010_2 )

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e. Convert F8 (base 16) to base 2:

Each hex digit represents 4 bits:

• F = 1111 (binary)
• 8 = 1000 (binary)

Answer: ( 11111000_2 )

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f. Perform binary addition: 101110₂ + ?

You didn’t specify the second number, so I assume 101110₂ + 1₂.

  101110
+      1
---------
  101111

Explanation (300 words):
Binary addition follows similar rules as decimal addition, but since there are only two digits (0 and 1), the carry system works differently:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (write 0, carry 1)
• 1 + 1 + 1 = 11 (write 1, carry 1)

Let’s add 101110₂ + 1₂ step by step:

1. Rightmost bit: 0 + 1 = 1 (no carry)
2. Next bit: 1 + 0 = 1
3. Next bit: 1 + 0 = 1
4. Next bit: 0 + 0 = 0
5. Next bit: 1 + 0 = 1
6. Leftmost bit: 1 + 0 = 1

So, the sum is 101111₂.

This method can be extended to multi-bit binary numbers. If the numbers were larger, we’d continue carrying over until we processed all bits.

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Let me know if you need more explanation! 😊