Question

G a key part to designing digital logic is manipulating binary numbers. you might have learned number representation while working on your dac since the dac used a binary representation of a number to generate an analog voltage that represented the number.

a.express the decimal digit "6", "129" and "600" in binary. how many binary bits are needed? how many decimal bits are needed? (6)
b.if you wish to add two 2-bit binary numbers together, construct a truth table to describe the correct outputs of the simple adder. first, express the logic equation in terms of inputs a = (a1a0), b = (b1b0), and output s = (s2s1s0). second, complete a truth table. (21)

Answer 1

`2^0=1_(10)=1_2`

`2^1=2_(10)=10_2`

`2^2=4_(10)=100_2`

`\vdots`

The pattern above tells you that the
`n`th power of 2 requires
`n+1` binary digits. So


`4=2^2<6<2^3=8\implies 6\text{ requires 3 binary digits}`

`6=4+2\implies6_(10)=110_2`


`128=2^7<129<2^8=256\implies129\text{ requires 8 binary digits}`

`129=128+1\implies129_(10)=10000001_2`


`512=2^9<600<2^(10)=1024\implies600\text{ requires 10 binary digits}`

`600=512+64+16+8\implies600_(10)=1001011000_2`

Meanwhile, you clearly need 1, 3, 3 decimal digits (respectively) to represent the numbers in base 10.