. Convert 2AF from hexadecimal to binary. Show your work.

To convert from hexadecimal base system to binary base system, first you can do an intermediate conversion from hexadecimal to decimal using this formula:

[tex]N = x_1 * 16^0 + x_2 * 16^1 + x_3 * 16^2 + x_4 * 16^3+ ... + x_n 16^n^-^1[/tex]

, where position of the x₁ is the rightmost digit of the number and:

A = 10.
B = 11.
C = 12.
D = 13.
E = 14.
F = 15.

2AF₁₆ = 2*16²+A*16¹+F*16⁰ = 512 + 160 + 15 = 687₁₀

Now, transform from decimal to binary the number 687. Divide the number repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to 0:

687 ÷ 2 = 343 + 1;

343 ÷ 2 = 171 + 1;

171 ÷ 2 = 85 + 1;

85 ÷ 2 = 42 + 1;

42 ÷ 2 = 21 + 0;

21 ÷ 2 = 10 + 1;

10 ÷ 2 = 5 + 0;

5 ÷ 2 = 2 + 1;

2 ÷ 2 = 1 + 0;

1 ÷ 2 = 0 + 1;

Now, construct the integer part base 2 representation, by taking the remainders starting from the bottom of the list:

687₁₀ = 1010101111₂