03. Arithmetic

This web page is associated with a book called called The Animated Computer

The book can be bought at: https://play.google.com/store/books/details/Dr_Jerome_Heath_THE_EBOOK_ON_COMPUTER_DESIGN?id=WxOTBQAAQBAJ

The book gives excellent explanations for the animated images on this web site. This web page shows the animation of the computer designs in action. The book gives explanations of what the animations are about. Combining the books explanations with the animation provides a well rounded understanding of the animated computer.

The next context is circuits for arithmetic. This context allows us to add and subtract numbers. The addition and subtraction is done by combining logical gates in ways that produce a truth table that is equivalent to addition or subtraction. In the arithmetic context we use the truth tables of NAND and XOR. NAND is the inversion of AND and XOR is the exclusive OR.

Decimal, Octal, and Digital Numbers

(and showing how digital numbers are represented)

Come Count with Me!

This is digital counting.

This arithmetic is based on numbers being integers.

Banks need to keep track of the pennies, and scientist need to calculate real numbers. We don't cover that in this book but the numbers are added using the functions available from the instruction set. There is the number value itself called the mantissa (the number as a fraction), and the exponent. You calculate how the number do in arithmetic (as though they were integers) and then place the decimal point by rules that determine the ultimate size of the number. This is much like slide rule calculations of old. Setting the decimal point was always the hard part of slide rule calculation.

Calculating the Truth Table for a One Bit Adder.

The ALU: Add & Subtract

https://www.youtube.com/watch?v=6L9hDBu3pyE

Listen and figure out what he is saying. Then try to determine if his design of an adder and subtractor works. That is one way to learn how this works. The subtract bit is to decide whether this is subtraction or adding. He says where that goes but I did not catch it. There is probably literature on a standard adder subtractor for just one or two bits to compare what someone else says.

The Abacus

Another interesting numbering base is abacus arithmetic. Note the abacus does arithmetic with the 5 numbering base; but the answer is kept in decimal as the results are put into bead motions. Why use fives when we all know how to do tens? Check out the answer to the puzzle:

Figure 3. Guessing Numbers

With the first set most people have to count to see how many. With the second, most people can tell without counting that there are five. Handling five is intuitive. That is the basis of abacus arithmetic. But by putting the result in beads on the abacus we see the final answer in decimal.

0.0 1.0 2.0 3.0 4.0 5.0

6.0 7.0 8.0 9.0 10.0 11.0. V

Figure 4. Counting On the Abacus

Adding and subtracting uses arithmetic complements so 6 + 7 = 13 becomes 6 - 5 + 2 + 10 = 13:

+ =

6 + 7 = 13

5 + 1 + 5 + 2 = 13 V

Figure 5. Abacus Addition

The 5 and 1, and the 5 and 2 are figured in the head. The five bead disappears and the ten bead appears. Then the 1 and 2 are added in by moving three beads. The beads show the answer. The abacus may require more manipulations than straight decimal arithmetic, but doing arithmetic using the abacus can be much faster and easier than on paper. Another side to this is that addition and subtraction become processes rather than the result of memorizing tables. That means that understanding arithmetic as a process is possible the abacus way. We will show later how complements are used in computer subtraction.

In the computer the only way to store data is in binary. Doing arithmetic is also more efficient for the computer in binary as the abacus method is more efficient for humans. In the computer any other way than digital (or binary) is too costly in computer memory and processing time. The only time that data needs to be in decimal form is when the user looks at the data on the screen or in a report. Then we need to represent data in decimal for the sake of the consumer.

DECIMAL DIGITAL

DECIMAL

DIGITAL

Dr. Jerome Heath