Table of Contents

Last modified on October 30th, 2023

Binary addition involves the addition of two binary numbers, 0 and 1. Each digit is called a bit.

- 0 denotes LOW or OFF
- 1 denotes HIGH or ON

The addition in binary is similar to decimal addition involving 2 numbers. It follows 4 rules:

- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 10 (0 carry 1)

We can summarize it as:

A + B | Sum | Carry |

0 + 0 | 0 | 0 |

0 + 1 | 1 | 0 |

1 + 0 | 1 | 0 |

1 + 1 | 0 | 1 |

The result of binary addition is the same as we obtain from decimal addition, with the difference in the place values of the digits.

Regrouping is done when the sum of 2 or more binary digits is greater than 1.

Let us add the binary numbers 1101_{2} and 101_{2}. Here, we show the overflow of the process.

**Step 1**: Aligning the numbers based on their place values

**Step 2**: Adding the digits in ones place using the binary addition rules.

1 + 1 = 10

Thus,

**Step 3**: Moving on to the next column.

1 + 0 + 0 = 1

**Step 4**: In the 3^{rd} column

1 + 1 + 10, so carry 1

**Step 5**: Finally,

1 + 1 = 10

Thus, 1101_{2} + 101_{2 }= 010010_{2}

When the 2 binary numbers being added are 0 and 1, regrouping is unnecessary.

Let us add the binary numbers 101_{2} and 10_{2}

**Step 1**: Aligning the numbers based on their place values

**Step 2**: Adding the digits in ones place using the binary addition rules.

1 + 0 = 1

**Step 3**: Moving on to the second column.

Similarly, 0 + 1 = 1

Here, also there is no carry

**Step 4**: Finally, in the last column

1 + 0 = 1

Thus, 101_{2} + 10_{2} = 0111_{2}

There are 2 types of complement of a binary number: 1’s complement and 2’ complement. In 1’s complement, we add two negative binary numbers, while 2’s complement of a binary number is always 1, added to the 1’s complement of the binary number.

- 0 represents a positive sign
- represents a negative sign

There are 2 possible situations:

**Case 1: When the Positive Number is Greater in Magnitude**

Here, we take the 1’s complement of the negative number, and the carried number is added to the sum of the numbers at the 1’s place. To find the 1’s complement of a number, we alter every 0 by 1 and every 1 by 0.

For example, 1’s complement of the binary number 100 is 011.

Let us add the numbers: 1010_{2} + (-1101_{2})

Rewriting, we get:

1110_{2} – 1101_{2}

+ 1110_{2 }→ 01110_{2}

-1101_{2 }→ 10010_{2} (Taking 1’ complement)

Thus, the required sum is + 0001.

**Case 2: When the Negative Number is Greater in Magnitude**

In this case, we take the 1’s complement of the negative number, and there is no end-around carrying. Finally, the result is obtained by taking the 1’s complement of the sum.

The result will always be a negative number.

Let us add the numbers: 0111_{2} + (-1000)_{2}

1’s complement of -1000 → 0111_{2}

Now, adding the 1’s complement to 0111_{2}

Finally, by taking the 1’s complement of the sum, we get

1 1 1 0 → 0 0 0 1

Here, we will add a negative sign before the number (as the negative number was greater) to get the final result.

0 0 0 1 → – 0 0 0 1

Here, we find the 1’s complement of both the negative numbers and then add both the complement numbers. Here, we will get the end-around carry to get the final result.

Let us add the numbers: -1010_{2} +(- 0011_{2})

Taking 1’s complement of both the numbers

1010_{2 }→ 10101_{2}

0011_{2 }→ 11100_{2}

Taking the 1’s complement of 10010_{2} to get the result10010_{2 }→ 11101_{2}

Last modified on October 30th, 2023