What is the largest binary number that can be expressed with 16 bits

a. What is the largest binary number that can be expressed with 16 bits? (Provide the answer in binary i.e., something like 1100111100011110)

b. What is the equivalent of the number in decimal? (Provide the value in decimal i.e., something like 9276)

c. What is the equivalent of the number in hexadecimal? (Provide the value in hexadecimal i.e., something like 9F2A)

The Correct Answer and Explanation is:

Let’s solve the problem step by step:

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a. What is the largest binary number that can be expressed with 16 bits?

The largest binary number with 16 bits is:

1111111111111111

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b. What is the equivalent of the number in decimal?

This binary number in decimal is:

65,535

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c. What is the equivalent of the number in hexadecimal?

This binary number in hexadecimal is:

FFFF

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✅ Detailed Explanation (300+ Words)

A bit is the smallest unit of data in computing and represents a binary digit: either 0 or 1. A 16-bit number means the number is represented using 16 binary digits (bits).

a. Largest 16-bit Binary Number:

When all 16 bits are set to 1, we get the largest possible binary number for that bit length. That is:

CopyEdit1111111111111111

This is equivalent to having all the binary “places” filled with the highest possible value (1), similar to writing 9999 in decimal for a 4-digit number.

b. Decimal Equivalent:

To convert this binary number to decimal, we use powers of 2. Each position in a binary number represents a power of 2, starting from the rightmost bit (2⁰) to the leftmost bit (2¹⁵ for 16 bits).

Mathematically:

CopyEdit= 2⁰ + 2¹ + 2² + ... + 2¹⁵
= 1 + 2 + 4 + 8 + 16 + ... + 32768
= 65,535

This is the maximum value you can store using unsigned 16-bit binary, meaning there is no sign bit (i.e., only positive values).

c. Hexadecimal Equivalent:

Hexadecimal is a base-16 number system, using digits 0–9 and letters A–F. Each group of 4 binary bits (a nibble) converts to one hexadecimal digit.

Break 16-bit binary into 4 groups:

yamlCopyEdit1111 1111 1111 1111

Each group of 4 bits equals:

• 1111 = F (15 in decimal)

So, the 16-bit binary number is:

scssCopyEditFFFF (hexadecimal)

✅ Summary:

• Binary: 1111111111111111
• Decimal: 65535
• Hexadecimal: FFFF

This concept is widely used in computing, particularly in defining memory addresses, data types, and setting register limits in microprocessors.