Binaries

Binary is the base-2 numbering system consisting solely of 0 s and 1 s.

Decimal and Binary Conversion

Binary to Decimal

Reverse the order of the binary code and multiply each binary number with its corresponding power of 2, and add the products.

To convert binary 1101 to decimal:

\[ \begin{align}\begin{aligned}1 * (2^0) = 1\\0 * (2^1) = 0\\1 * (2^2) = 4\\1 * (2^3) = 8\end{aligned}\end{align} \]

Binary 1101 in decimal: \(8 + 4 + 0 + 1 = 13\)

Octal to Decimal

Method A

Starting from the right, multiply the digits with the corresponding powers of 8 and add the results.

To convert octal number 15 to decimal:

\[ \begin{align}\begin{aligned}1|5 -> 1 * (8^1)|5 * (8^0)\\8|5 -> 8 + 5\\13\end{aligned}\end{align} \]

Octal number 15 in decimal: 13

Method B

Write the three-digit binary representation of each octal digit side-by-side.

\[ \begin{align}\begin{aligned}0 = 000\\1 = 001\\2 = 010\\3 = 011\\4 = 100\\5 = 101\\6 = 110\\7 = 111\end{aligned}\end{align} \]

To convert octal number 15 to binary:

\[ \begin{align}\begin{aligned}15 -> 1|5\\1|5 -> 001|101\\001101\end{aligned}\end{align} \]

Octal number 15 in binary: 001101

Convert binary to decimal.

Binary 001101 in decimal: 13

Hexadecimal to Decimal

Method A

Convert each digit to its corresponding decimal value. Multiply each digits by powers of 16, starting from the rightmost, and sum the products.

\[ \begin{align}\begin{aligned}0 = 0\\1 = 1\\2 = 2\\3 = 3\\4 = 4\\5 = 5\\6 = 6\\7 = 7\\8 = 8\\9 = 9\\10 = A\\11 = B\\12 = C\\13 = D\\14 = E\\15 = F\end{aligned}\end{align} \]

To convert hexadecimal 0xD to decimal:

\[|D -> |13 |13 -> |13 * (16 ^ 0) 13\]

Hexadecimal 0xD in decimal: 13

Method B

Convert each digit to its corresponding binary value.

\[ \begin{align}\begin{aligned}0 = 0000\\1 = 0001\\2 = 0010\\3 = 0011\\4 = 0100\\5 = 0101\\6 = 0110\\7 = 0111\\8 = 1000\\9 = 1001\\A = 1010\\B = 1011\\C = 1100\\D = 1101\\E = 1110\\F = 1111\end{aligned}\end{align} \]

Convert hexadecimal 0xD to binary:

\[D -> 1101\]

Hexadecimal 0xD in binary: 1101

Convert binary to decimal.

Binary 1101 in decimal: 13

Decimal to Binary

Return the modulo of the decimal until quotient becomes 0, and reverse the order of remainders. To convert decimal 13 to binary:

\[ \begin{align}\begin{aligned}13 / 2 = 6 r.1\\6 / 2 = 3 r.0\\3 / 2 = 1 r.1\\1 / 2 = 0 . 1\end{aligned}\end{align} \]

Decimal 13 in binary: 1101.

Decimal to Octal

First, convert decimal to binary.

Starting from the right, group the binary digits into 3. Per group, each digit is represented as 4-2-1—add these up and concatenate the sum of every group. To convert binary 1101 to octal:

\[ \begin{align}\begin{aligned}1101 -> 1|101\\1|(4 + 1) -> 1|5\\15\end{aligned}\end{align} \]

Binary 1101 in octal: 15

Decimal to Hexadecimal

First, convert decimal to binary.

Group the binary digits into sets of four, starting from the right, and then convert each group into its corresponding hexadecimal digit:

\[ \begin{align}\begin{aligned}0 = 0\\1 = 1\\2 = 2\\3 = 3\\4 = 4\\5 = 5\\6 = 6\\7 = 7\\8 = 8\\9 = 9\\10 = A\\11 = B\\12 = C\\13 = D\\14 = E\\15 = F\end{aligned}\end{align} \]

To convert binary 1101 to hexadecimal:

\[ \begin{align}\begin{aligned}1101 -> |1101\\|(8 + 4 + 0 + 1) -> |13\\13 -> D\end{aligned}\end{align} \]

Binary 1101 in hexadecimal: #D or 0xD

Binary Operations

Binary Addition

	Carry Over	Results
0 + 0	0	0
0 + 1	0	1
1 + 0	0	1
1 + 1	1	0
1 + 1 + 1	1	1

Binary Subtraction

	Result
1 - 0	1
1 - 1	0
0 - 0	0
0 - 1	1

Binary Multiplication

	Result
0 * 0	0
0 * 1	0
1 * 0	0
1 * 1	1

Binary Division

	Result
0 / 1	0
1 / 1	1