1. what is the sum of the numbers and convert the result to the decimal system: a) 1011101 + 110111; b) 10H101 + 1010; c) 101 101 + 1010; d) 11101 + 1 101; e) 10111 + 101; f) 101 101 + 11; g) 101 + 111001; 2. Perform subtraction and convert the result to the decimal system: a) 101 101 - 10111; b) 101 101 - 1010; c) 101101 - 101011; d) 101 - 101; e) 101111 - 11; f) 10111 - 101; g) 101 101 - 1; 3) 101 - 1001; 3. Perform multiplication and convert the result to the decimal system: a) 1011 multiplied by 101; b) 101 multiplied by 1010; c) 10101 multiplied by 10101; d) 101 divided by 101.
Валерия
1. a) 1011101 + 110111
To add these binary numbers, align them vertically and add each column from right to left, just like adding in the decimal system:
\[
\begin{array}{ccccccc}
& 1 & 0 & 1 & 1 & 1 & 0 & 1 \\
+ & 0 & 1 & 1 & 0 & 1 & 1 & 1 \\
\hline
& 1 & 0 & 1 & 1 & 0 & 0 & 0 \\
\end{array}
\]
The sum of 1011101 and 110111 in the binary system is 1011000.
To convert this sum to the decimal system, we can use the positional notation of the system. We multiply each digit of the binary number by the corresponding power of 2 and sum the results:
\[
\begin{align*}
1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \\
= 64 + 0 + 16 + 8 + 0 + 0 + 0 \\
= 88
\end{align*}
\]
Therefore, the decimal representation of 1011101 + 110111 is 88.
b) 10H101 + 1010
It seems like "H" is not a valid digit in the binary system. Please make sure all the digits are either 0 or 1.
c) 101 101 + 1010
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{ccccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
+ & & & 1 & 0 & 1 & 0 & 1 \\
\hline
& & 1 & 1 & 0 & 0 & 1 & 0 \\
\end{array}
\]
The sum of 101101 and 1010 in the binary system is 110010.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 16 + 0 + 0 + 2 + 0 \\
= 50
\end{align*}
\]
Therefore, the decimal representation of 101101 + 1010 is 50.
d) 11101 + 1 101
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{cccccc}
& 1 & 1 & 1 & 0 & 1 \\
+ & & 1 & 1 & 0 & 1 \\
\hline
& 1 & 1 & 0 & 1 & 0 \\
\end{array}
\]
The sum of 11101 and 1101 in the binary system is 11010.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 16 + 8 + 0 + 2 + 0 \\
= 26
\end{align*}
\]
Therefore, the decimal representation of 11101 + 1101 is 26.
e) 10111 + 101
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{cccc}
& 1 & 0 & 1 & 1 \\
+ & & & 1 & 0 & 1 \\
\hline
& 1 & 1 & 0 & 0 \\
\end{array}
\]
The sum of 10111 and 101 in the binary system is 1100.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \\
= 8 + 4 + 0 + 0 \\
= 12
\end{align*}
\]
Therefore, the decimal representation of 10111 + 101 is 12.
f) 101 101 + 11
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{cccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
+ & & & & & 1 & 1 \\
\hline
& & 1 & 0 & 1 & 1 & 1 & 0 \\
\end{array}
\]
The sum of 101101 and 11 in the binary system is 101110.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 0 + 8 + 4 + 2 + 0 \\
= 46
\end{align*}
\]
Therefore, the decimal representation of 101101 + 11 is 46.
g) 101 + 111001
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{cccccc}
& & & 1 & 0 & 1 \\
+ & & 1 & 1 & 1 & 0 & 0 & 1 \\
\hline
& & 1 & 1 & 1 & 1 & 1 & 0 \\
\end{array}
\]
The sum of 101 and 111001 in the binary system is 111110.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 16 + 8 + 4 + 2 + 0 \\
= 62
\end{align*}
\]
Therefore, the decimal representation of 101 + 111001 is 62.
2. a) 101 101 - 10111
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{ccccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
- & & & & 1 & 0 & 1 & 1 \\
\hline
& & 1 & 0 & 0 & 0 & 1 & 0 \\
\end{array}
\]
The difference of 101101 and 10111 in the binary system is 100010.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 0 + 0 + 0 + 2 + 0 \\
= 34
\end{align*}
\]
Therefore, the decimal representation of 101101 - 10111 is 34.
b) 101 101 - 1010
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{ccccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
- & & & & & 1 & 0 & 1 & 0 \\
\hline
& & 1 & 0 & 0 & 0 & 1 & 1 \\
\end{array}
\]
The difference of 101101 and 1010 in the binary system is 100011.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \\
= 32 + 0 + 0 + 0 + 2 + 1 \\
= 35
\end{align*}
\]
Therefore, the decimal representation of 101101 - 1010 is 35.
c) 101101 - 101011
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{ccccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
- & & & 1 & 0 & 1 & 0 & 1 \\
\hline
& & & & & & 0 & 0 \\
\end{array}
\]
The difference of 101101 and 101011 in binary is 0.
To convert this difference to the decimal system, we have 0.
Therefore, the decimal representation of 101101 - 101011 is 0.
d) 101 - 101
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{cccc}
& & 1 & 0 & 1 \\
- & & & 1 & 0 & 1 \\
\hline
& & & 1 & 1 \\
\end{array}
\]
The difference of 101 and 101 in the binary system is 11.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^1 + 1 \times 2^0 \\
= 2 + 1 \\
= 3
\end{align*}
\]
Therefore, the decimal representation of 101 - 101 is 3.
e) 101111 - 11
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{cccccc}
& & 1 & 0 & 1 & 1 & 1 \\
- & & & & & 1 & 1 \\
\hline
& & 1 & 0 & 1 & 0 & 0 \\
\end{array}
\]
The difference of 101111 and 11 in the binary system is 10100.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \\
= 16 + 0 + 4 + 0 + 0 \\
= 20
\end{align*}
\]
Therefore, the decimal representation of 101111 - 11 is 20.
f) 10111 - 101
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{ccccccc}
& & & 1 & 0 & 1 & 1 \\
- & & & & & 1 & 0 & 1 \\
\hline
& & & & 0 & 1 & 0 \\
\end{array}
\]
The difference of 10111 and 101 in the binary system is 010.
To convert this difference to the decimal system:
\[
\begin{align*}
0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 0 + 2 + 0 \\
= 2
\end{align*}
\]
Therefore, the decimal representation of 10111 - 101 is 2.
g) 101 101 - 1
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{cccccc}
& & 1 & 0 & 1 & 1 \\
- & & & & & 0 & 1 \\
\hline
& & 1 & 0 & 1 & 0 \\
\end{array}
\]
The difference of 101101 and 1 in the binary system is 101010.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 0 + 8 + 0 + 2 + 0 \\
= 42
\end{align*}
\]
Therefore, the decimal representation of 101101 - 1 is 42.
3. a) 1011 умноженное на 101
To multiply these binary numbers, align them vertically and perform the multiplication as you would do in the decimal system:
\[
\begin{array}{ccccccc}
& & & & 1 & 0 & 1 & 1 \\
\times & & & 1 & 0 & 1 \\
\hline
& & 1 & 0 & 1 & 1 & 0 & 1 \\
& & 0 & 0 & 0 & 0 & 0 & \\
& 1 & 0 & 1 & 1 & \\
\hline
& 1 & 1 & 1 & 1 & 0 & 1 & 1 \\
\end{array}
\]
The product of 1011 and 101 in the binary system is 1111011.
To convert this product to the decimal system:
\[
\begin{align*}
1 \times 2^6 + 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \\
= 64 + 32 + 16 + 8 + 0 + 2 + 1 \\
= 123
\end{align*}
\]
Therefore, the decimal representation of 1011 multiplied by 101 is 123.
b) 101 умноженное на 1010
To multiply these binary numbers, align them vertically and perform the multiplication as you would do in the decimal system:
\[
\begin{array}{ccccccc}
& & & & 1 & 0 & 1 & 0 \\
\times & & & 1 & 0 & 1 & 0 \\
\hline
& & 1 & 1 & 0 & 1 & 0 & 0 \\
& 1 & 0 & 1 & 0 & 0 & \\
\hline
& 1 & 0 & 1 & 1 & 1 & 0 & 0 \\
\end{array}
\]
The product of 101 and 1010 in the binary system is 1011100.
To convert this product to the decimal system:
\[
\begin{align*}
1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \\
= 64 + 0 + 16 + 8 + 4 + 0 + 0 \\
= 92
\end{align*}
\]
Therefore, the decimal representation of 101 multiplied by 1010 is 92.
**c) 10101 умн
To add these binary numbers, align them vertically and add each column from right to left, just like adding in the decimal system:
\[
\begin{array}{ccccccc}
& 1 & 0 & 1 & 1 & 1 & 0 & 1 \\
+ & 0 & 1 & 1 & 0 & 1 & 1 & 1 \\
\hline
& 1 & 0 & 1 & 1 & 0 & 0 & 0 \\
\end{array}
\]
The sum of 1011101 and 110111 in the binary system is 1011000.
To convert this sum to the decimal system, we can use the positional notation of the system. We multiply each digit of the binary number by the corresponding power of 2 and sum the results:
\[
\begin{align*}
1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \\
= 64 + 0 + 16 + 8 + 0 + 0 + 0 \\
= 88
\end{align*}
\]
Therefore, the decimal representation of 1011101 + 110111 is 88.
b) 10H101 + 1010
It seems like "H" is not a valid digit in the binary system. Please make sure all the digits are either 0 or 1.
c) 101 101 + 1010
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{ccccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
+ & & & 1 & 0 & 1 & 0 & 1 \\
\hline
& & 1 & 1 & 0 & 0 & 1 & 0 \\
\end{array}
\]
The sum of 101101 and 1010 in the binary system is 110010.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 16 + 0 + 0 + 2 + 0 \\
= 50
\end{align*}
\]
Therefore, the decimal representation of 101101 + 1010 is 50.
d) 11101 + 1 101
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{cccccc}
& 1 & 1 & 1 & 0 & 1 \\
+ & & 1 & 1 & 0 & 1 \\
\hline
& 1 & 1 & 0 & 1 & 0 \\
\end{array}
\]
The sum of 11101 and 1101 in the binary system is 11010.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 16 + 8 + 0 + 2 + 0 \\
= 26
\end{align*}
\]
Therefore, the decimal representation of 11101 + 1101 is 26.
e) 10111 + 101
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{cccc}
& 1 & 0 & 1 & 1 \\
+ & & & 1 & 0 & 1 \\
\hline
& 1 & 1 & 0 & 0 \\
\end{array}
\]
The sum of 10111 and 101 in the binary system is 1100.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \\
= 8 + 4 + 0 + 0 \\
= 12
\end{align*}
\]
Therefore, the decimal representation of 10111 + 101 is 12.
f) 101 101 + 11
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{cccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
+ & & & & & 1 & 1 \\
\hline
& & 1 & 0 & 1 & 1 & 1 & 0 \\
\end{array}
\]
The sum of 101101 and 11 in the binary system is 101110.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 0 + 8 + 4 + 2 + 0 \\
= 46
\end{align*}
\]
Therefore, the decimal representation of 101101 + 11 is 46.
g) 101 + 111001
To add these binary numbers, align them vertically and add each column from right to left:
\[
\begin{array}{cccccc}
& & & 1 & 0 & 1 \\
+ & & 1 & 1 & 1 & 0 & 0 & 1 \\
\hline
& & 1 & 1 & 1 & 1 & 1 & 0 \\
\end{array}
\]
The sum of 101 and 111001 in the binary system is 111110.
To convert this sum to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 16 + 8 + 4 + 2 + 0 \\
= 62
\end{align*}
\]
Therefore, the decimal representation of 101 + 111001 is 62.
2. a) 101 101 - 10111
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{ccccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
- & & & & 1 & 0 & 1 & 1 \\
\hline
& & 1 & 0 & 0 & 0 & 1 & 0 \\
\end{array}
\]
The difference of 101101 and 10111 in the binary system is 100010.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 0 + 0 + 0 + 2 + 0 \\
= 34
\end{align*}
\]
Therefore, the decimal representation of 101101 - 10111 is 34.
b) 101 101 - 1010
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{ccccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
- & & & & & 1 & 0 & 1 & 0 \\
\hline
& & 1 & 0 & 0 & 0 & 1 & 1 \\
\end{array}
\]
The difference of 101101 and 1010 in the binary system is 100011.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \\
= 32 + 0 + 0 + 0 + 2 + 1 \\
= 35
\end{align*}
\]
Therefore, the decimal representation of 101101 - 1010 is 35.
c) 101101 - 101011
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{ccccccc}
& & 1 & 0 & 1 & 1 & 0 & 1 \\
- & & & 1 & 0 & 1 & 0 & 1 \\
\hline
& & & & & & 0 & 0 \\
\end{array}
\]
The difference of 101101 and 101011 in binary is 0.
To convert this difference to the decimal system, we have 0.
Therefore, the decimal representation of 101101 - 101011 is 0.
d) 101 - 101
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{cccc}
& & 1 & 0 & 1 \\
- & & & 1 & 0 & 1 \\
\hline
& & & 1 & 1 \\
\end{array}
\]
The difference of 101 and 101 in the binary system is 11.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^1 + 1 \times 2^0 \\
= 2 + 1 \\
= 3
\end{align*}
\]
Therefore, the decimal representation of 101 - 101 is 3.
e) 101111 - 11
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{cccccc}
& & 1 & 0 & 1 & 1 & 1 \\
- & & & & & 1 & 1 \\
\hline
& & 1 & 0 & 1 & 0 & 0 \\
\end{array}
\]
The difference of 101111 and 11 in the binary system is 10100.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \\
= 16 + 0 + 4 + 0 + 0 \\
= 20
\end{align*}
\]
Therefore, the decimal representation of 101111 - 11 is 20.
f) 10111 - 101
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{ccccccc}
& & & 1 & 0 & 1 & 1 \\
- & & & & & 1 & 0 & 1 \\
\hline
& & & & 0 & 1 & 0 \\
\end{array}
\]
The difference of 10111 and 101 in the binary system is 010.
To convert this difference to the decimal system:
\[
\begin{align*}
0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 0 + 2 + 0 \\
= 2
\end{align*}
\]
Therefore, the decimal representation of 10111 - 101 is 2.
g) 101 101 - 1
To subtract these binary numbers, align them vertically and subtract each column from right to left:
\[
\begin{array}{cccccc}
& & 1 & 0 & 1 & 1 \\
- & & & & & 0 & 1 \\
\hline
& & 1 & 0 & 1 & 0 \\
\end{array}
\]
The difference of 101101 and 1 in the binary system is 101010.
To convert this difference to the decimal system:
\[
\begin{align*}
1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\
= 32 + 0 + 8 + 0 + 2 + 0 \\
= 42
\end{align*}
\]
Therefore, the decimal representation of 101101 - 1 is 42.
3. a) 1011 умноженное на 101
To multiply these binary numbers, align them vertically and perform the multiplication as you would do in the decimal system:
\[
\begin{array}{ccccccc}
& & & & 1 & 0 & 1 & 1 \\
\times & & & 1 & 0 & 1 \\
\hline
& & 1 & 0 & 1 & 1 & 0 & 1 \\
& & 0 & 0 & 0 & 0 & 0 & \\
& 1 & 0 & 1 & 1 & \\
\hline
& 1 & 1 & 1 & 1 & 0 & 1 & 1 \\
\end{array}
\]
The product of 1011 and 101 in the binary system is 1111011.
To convert this product to the decimal system:
\[
\begin{align*}
1 \times 2^6 + 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \\
= 64 + 32 + 16 + 8 + 0 + 2 + 1 \\
= 123
\end{align*}
\]
Therefore, the decimal representation of 1011 multiplied by 101 is 123.
b) 101 умноженное на 1010
To multiply these binary numbers, align them vertically and perform the multiplication as you would do in the decimal system:
\[
\begin{array}{ccccccc}
& & & & 1 & 0 & 1 & 0 \\
\times & & & 1 & 0 & 1 & 0 \\
\hline
& & 1 & 1 & 0 & 1 & 0 & 0 \\
& 1 & 0 & 1 & 0 & 0 & \\
\hline
& 1 & 0 & 1 & 1 & 1 & 0 & 0 \\
\end{array}
\]
The product of 101 and 1010 in the binary system is 1011100.
To convert this product to the decimal system:
\[
\begin{align*}
1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \\
= 64 + 0 + 16 + 8 + 4 + 0 + 0 \\
= 92
\end{align*}
\]
Therefore, the decimal representation of 101 multiplied by 1010 is 92.
**c) 10101 умн
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