> For the complete documentation index, see [llms.txt](https://wiki.zacheller.dev/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://wiki.zacheller.dev/hardware/nand2tetris/boolean-arithmetic-and-the-alu.md).

# Boolean Arithmetic and the ALU

**In a nutshell:** Using the chipset that we've built in the previous module, we will now proceed to build a family of *adders* -- chips designed to add numbers. We will then take a big step forward and build an *Arithmetic Logic Unit*. The ALU, which is designed to perform a whole set of arithmetic and logical operations, is the computer's calculating brain. Later in the course we will use this ALU as the centerpiece chip from which we will build the computer's *Central Processing Unit*, or CPU. Since all these chips operate on binary numbers (0's and 1's), we will start this module with a general overview of binary arithmetic, and only then delve into building the ALU.

**Key concepts:** Binary numbers, binary addition, the two's complement method, half-adders, full-adders, n-bit adders, counters, Arithmetic Logic Unit (ALU), combinational logic.

## Binary Numbers

![](/files/-MIMiMpxA2uaPVPrl-IV)

* What is the decimal value of 1101?
  * 1101(Binary) = 8+4+1(Decimal) = 13(Decimal)

## Binary Addition

![](/files/-MIMn1sXQ8xStods-9LG)

![](/files/-MIMnJf-mUt-v4wevuj_)

![](/files/-MIMnQRK83sFKx-1IqEk)

![](/files/-MIMnU8Gk61k2Tcuc3yf)

![](/files/-MIMncnZWbZMhXRNRd9l)

![](/files/-MIMnwxU_dpUucTFmE3u)

![](/files/-MIMnpPLDruFhpqULnRh)

## Negative numbers

![](/files/-MIMoZYpsW7wXObxyHow)

![](/files/-MIMpUCwc8NTu5KbqKNb)

![](/files/-MIMqBNwmKFDBGegojjH)

## Arithmetic Logic Unit

![](/files/-MIMrJpfHNGKMab_PZeP)

![](/files/-MIMrBELNWEa6a7O-1Ka)

![](/files/-MIMr5bAgJ-iL8jueRn1)

![](/files/-MIMsAKdVJZcc81yjM9P)

![](/files/-MIMt-NDNihI5327No9J)

## Project 2 Overview

![](/files/-MIMtLCQZaTnSq28LtSX)

![Sum: xor, Carry: and](/files/-MIMtSJ_FA2Aa7aTtYwf)

![Half adder is so named because it takes 2 to make a full adder](/files/-MIMu8mov2rI8IfTaVr9)

![](/files/-MIMwctL8IBSvVLrfVLI)

![](/files/-MIMySjAsaswmNH6V1xE)

![](/files/-MIMzW1EC16xPsBe3hEE)


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