1.4: Adding Binary Whole Numbers

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Nov 1, 2024
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- 1.3: Character Representation
- 1.5: Integer Numbers (2's Complement)

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Before moving on to how integer values are stored and used in a computer for calculations, how to do addition of binary whole numbers needs to be covered.

When 2 one-bit binary numbers are added, the following results are possible: 0₂+0₂ = 0₂; 0₂+1₂ = 1₂; 1₂+0₂ = 1₂; and 1₂+1₂ = 10₂. This is just like decimal numbers. For example, 3+4=7, and the result is still one digit. A problem occurs, however, when adding two decimal numbers where the result is greater than the base of the number (for decimal, the base is 10). For example, 9+8. The result cannot be represented in one digit, so a carry digit is created. The result of 9+8 is 7 with a carry of 1. The carry of 1 is considered in the next digit, which is actually adding 3 digits (the two addends, and the carry). So 39 + 28 = 67, where the 10's digit (4) is the result of the two addends (3 and 2) and the carry (1).

The result of 1₂+1₂ = 10₂ in binary is analogous to the situation in base 10. The addition of 1₂+1₂ is 0₂ with a carry of 1₂, and there is a carry to the next digit of 1₂.

An illustration of binary addition is shown in the figure below.

Screen Shot 2020-06-27 at 6.27.54 PM.png

Here the first bit adds 1₂+1₂, which yields a 0₂ in this bit and a carry bit of 1₂. The next bit now has to add 1₂ +1₂ +1₂ (the extra one is the carry bit), which yields a 1₂ for this bit and a carry bit of 1₂. If you follow the arithmetic through, you have 0011₂ (3₁₀) + 0111₂ (7₁₀) = 1010₂ (10₁₀).

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