Question

"The binary pattern 01000001 represents the number 65." Write a brief response explaining whether you believe this statement is always true. Explain your reasoning.

Answer 1

Answer:

Yes

Explanation:

Yes this statement is always true.

Starting from right the first digit represents 1 and second digit represents 2 and so forth.

Starting from right the sequence becomes

1,2,4,8,16,32,62,128 for 8 binary digits

If 1 is written then we count it else for 0 we don't count it

for example 01 means 1 and 10 means 2 and 11 means 3 as 2+1=3

for 01000001 only place that is filled is for 1 and 64 therefore complete decimal digit shall be 64+1=65 and it would always remain same.

In subscript of 2 we can write  $2^{n}$, where n=0,1,2,3,4,...

For first digit from right $2^{n}$ will be $2^{0}$ which is 1. For second digit from right $2^{1}$ which will be 2 so on and so forth.

For this question we have $2^{0}$ and $2^{6}$ which will be 1+64=65

Answer 2

Binary patterns are numbers in base 2, and they are called binary numbers.

The statement that the binary pattern 01000001 represents number 65 is true

The binary number is given as:

$\mathbf{01000001_2}$

Convert the number to base 10, as follows:

$\mathbf{01000001_2 = 0 \times 2^7 + 1 \times 2^6 + 0 + \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0}$

Evaluate the exponents

$\mathbf{01000001_2 = 0 \times 128 + 1 \times 64 + 0 + \times 32 + 0 \times 16 + 0 \times 8+ 0 \times 4 + 0 \times 2 + 1 \times 1}$

Evaluate all products

$\mathbf{01000001_2 = 0 + 64 + 0 + 0+ 0 + 0 + 0 + 0 + 1}$

Add all numbers

$\mathbf{01000001_2 = 65}$

Rewrite as:

$\mathbf{01000001_2 = 65_{10}}$

Hence, the statement is true

Read more about binary numbers at:

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