Convert from binary to decimal to hexadecimal (PLEASE SHOW ALL WORK) a. 1010 1011 – decimal: – hexadecimal b. 0011 0001 – decimal: – hexadecimal: c. 1110 0111 – decimal: – hexadecimal: d. 1111 1111 – decimal: – hexadecimal:
The Correct Answer and Explanation is:
. 1010 1011
Decimal Conversion:
To convert a binary number to decimal, we multiply each bit by 2 raised to the power of its position (starting from 0 at the far right).
So, for 1010 1011: 10101011=(1×27)+(0×26)+(1×25)+(0×24)+(1×23)+(0×22)+(1×21)+(1×20)1010 1011 = (1 \times 2^7) + (0 \times 2^6) + (1 \times 2^5) + (0 \times 2^4) + (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0)10101011=(1×27)+(0×26)+(1×25)+(0×24)+(1×23)+(0×22)+(1×21)+(1×20) =128+0+32+0+8+0+2+1= 128 + 0 + 32 + 0 + 8 + 0 + 2 + 1=128+0+32+0+8+0+2+1 =171= 171=171
Hexadecimal Conversion:
To convert a binary number to hexadecimal, group the binary digits in sets of four (starting from the right), and then convert each set to its hexadecimal equivalent. 10101011→1010=Aand1011=B1010 1011 \rightarrow 1010 = A \quad \text{and} \quad 1011 = B10101011→1010=Aand1011=B
So, the hexadecimal representation is AB.
b. 0011 0001
Decimal Conversion:
For 0011 0001: 00110001=(0×27)+(0×26)+(1×25)+(1×24)+(0×23)+(0×22)+(0×21)+(1×20)0011 0001 = (0 \times 2^7) + (0 \times 2^6) + (1 \times 2^5) + (1 \times 2^4) + (0 \times 2^3) + (0 \times 2^2) + (0 \times 2^1) + (1 \times 2^0)00110001=(0×27)+(0×26)+(1×25)+(1×24)+(0×23)+(0×22)+(0×21)+(1×20) =0+0+32+16+0+0+0+1= 0 + 0 + 32 + 16 + 0 + 0 + 0 + 1=0+0+32+16+0+0+0+1 =49= 49=49
Hexadecimal Conversion: 00110001→0011=3and0001=10011 0001 \rightarrow 0011 = 3 \quad \text{and} \quad 0001 = 100110001→0011=3and0001=1
So, the hexadecimal representation is 31.
c. 1110 0111
Decimal Conversion:
For 1110 0111: 11100111=(1×27)+(1×26)+(1×25)+(0×24)+(0×23)+(1×22)+(1×21)+(1×20)1110 0111 = (1 \times 2^7) + (1 \times 2^6) + (1 \times 2^5) + (0 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0)11100111=(1×27)+(1×26)+(1×25)+(0×24)+(0×23)+(1×22)+(1×21)+(1×20) =128+64+32+0+0+4+2+1= 128 + 64 + 32 + 0 + 0 + 4 + 2 + 1=128+64+32+0+0+4+2+1 =231= 231=231
Hexadecimal Conversion: 11100111→1110=Eand0111=71110 0111 \rightarrow 1110 = E \quad \text{and} \quad 0111 = 711100111→1110=Eand0111=7
So, the hexadecimal representation is E7.
d. 1111 1111
Decimal Conversion:
For 1111 1111: 11111111=(1×27)+(1×26)+(1×25)+(1×24)+(1×23)+(1×22)+(1×21)+(1×20)1111 1111 = (1 \times 2^7) + (1 \times 2^6) + (1 \times 2^5) + (1 \times 2^4) + (1 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0)11111111=(1×27)+(1×26)+(1×25)+(1×24)+(1×23)+(1×22)+(1×21)+(1×20) =128+64+32+16+8+4+2+1= 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1=128+64+32+16+8+4+2+1 =255= 255=255
Hexadecimal Conversion: 11111111→1111=Fand1111=F1111 1111 \rightarrow 1111 = F \quad \text{and} \quad 1111 = F11111111→1111=Fand1111=F
So, the hexadecimal representation is FF.
Summary of results:
- a. 1010 1011 → Decimal: 171 → Hexadecimal: AB
- b. 0011 0001 → Decimal: 49 → Hexadecimal: 31
- c. 1110 0111 → Decimal: 231 → Hexadecimal: E7
- d. 1111 1111 → Decimal: 255 → Hexadecimal: FF
I hope this breakdown helps you understand the conversion process! Let me know if you’d like to explore more examples.
