binary packet 2

8/12/2019 Binary Packet 2

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Binary Math!We will be looking at two kinds of binary number operations arithmetic (addition)

operations and logical operations. We're going to learn how to add numbers and

compare numbers. First, we talk about adding numbers.

Addition in decimal (Base 10)

04

+09

9+4 is 13

04

+09

3

1carry the 1

Write down 3, then

04

+09

13

1

09+04 is 13, all done!

Base 10

single digitaddition

0 + 0 = 00 + 1 = 10 + 2 = 20 + 3 = 30 + 4 = 40 + 5 = 50 + 6 = 60 + 7 = 7

0 + 8 = 80 + 9 = 9

1 + 1 = 21 + 2 = 31 + 3 = 41 + 4 = 51 + 5 = 61 + 6 = 71 + 7 = 81 + 8 = 9

1 + 9 = 10

2 + 2 = 42 + 3 = 52 + 4 = 62 + 5 = 72 + 6 = 82 + 7 = 92 + 8 = 102 + 9 = 11

3 + 3 = 63 + 4 = 73 + 5 = 8

...

9 + 9 = 18


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Addition in binary (Base 2)

01+01

1 + 1 = 10

01+01

0

1carry the 1

Write down 0, then

01

+0110

1


Base 10single digit

addition

0 + 0 = 0

0 + 1 = 11 + 1 = 10

Remember,10 in binary = 2 in decimal

What's1 + 1 + 1?

1 + 0 + 0 = 1


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0011

+0101


Binary addition with larger numbers

It's great to be able to add numbers like 1 and 1, but how about

adding larger numbers like 3 + 5?

First, convert the numbers to binary.

3 in decimal converts to 11 in binary.

5 in decimal converts to 101 in binary.

0 1 1

124

= 3

1 0 1

124

= 5

1 + 1 = 10

1carry the 1

Write down 0, then

0011

+01010

0

8

0

8


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1

0011

+0101

0

1 + 1 = 10

1carry the 1

Write down 0, then

0011

+010100

1

0011+0101

00

1 + 1 = 10


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1carry the 1

Write down 0, then

0011

+0101

000

1

0011

+0101000

1 + 1 + 0 = 1

1

0011

+0101

10001000 in binary = 8 in decimal

0 0 0

124

1

8

0 1 1

124

0

8

1 0 10

0 0 01

+

3

5

8

All done!

+


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0 1 11

1 0 11

0

0+

0 1 011

Check your work

11

13+

24 0 1 0

124

1

8

1

1624 in decimal

is equal to

11010 in binary,

so our math is right!

0 1 10

0 0 11

0

0+

0 1 11

1 1 10

0

0+

0 0 11

1 0 11

0

0+

1 1 11

1 0 11

0

0+

1 1 11

0 0 01

0

0+

0 0 01

1 1 10

0

0+

0 1 10

0 0 11

0

0+

0 0 11

0 0 11

0

0+


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1 1 11

0 0 01

0

0+

0 1 11

1 1 10

0

0+

1 1 10

1 0 11

0

0+

1 0 01

1 0 01

0

0+

0 0 11

1 0 11

0

0+

0 0 10

1 1 11

0

0+

0 0 10

0 1 11

0

0+

1 1 00

1 1 00

0

0+

0 1 11

1 0 11

0

0+

0 0 11

1 1 11

0

0+

0 1 10

1 0 10

0

0+

1 1 11

1 0 10

0

0+

0 0 01

1 0 01

0

0+

0 1 11

1 0 10

0

0+

1 0 10

0 1 01

1

0+


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Logical operations with binary numbersAdding is a really helpful operation to have, but are there other ways

to compare binary numbers?

Of course there is! Why else would we have an entire sheet on otheroperations if they didn't exist?

So besides adding binary numbers, you can perform so called logical

operations on those numbers.

We will be learning about these operations:

- AND

- OR

- NOT- XOR

A logical operation compares binary numbers and gives a result

based on whatever operation it's implementing.

Let's start with AND. It's helpful to think of the logical operation as a

"gate". This AND gate has two inputstwo places where you can

input binary numbers and a single output. Logic gates ALWAYS have

one single output though they may have many inputs.

AND

Input A

Input B

Output


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Logic gates make decisions. The AND logic gate will output a 1 if

both of its inputs are 1. It outputs 1 if both Input A AND Input B are 1.

AND

1

11

If you pass a 1 and a 0 as inputs to an AND gate, it will output 0. It

needs every input to be 1 before it will output a 1.

AND1

00 AND

0

10 AND

0

00

Meet the OR gate. How do you think the OR gate operates?

OR

Input A

Input B

Output

OR

What do you think about this? What's the output for an OR gate that

has a 1 and a 0 as inputs?

1

0

?


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OR

1

0

1

If you guessed 1, you're right!

An OR gate outputs a 1 if any of its inputs are 1. An OR gate outputs

a 1 if Input A OR Input B are 1.

OR0

1

1 OR1

1

1 OR0

0

0

This is the NOT gate. It's unique in that it can only receive a single

input. Like other logic gates, it has just one single output.

The NOT gate is easy to understand it just flips the binary number.

If you input a 1, a NOT gate outputs a 0. If you give it a 0, it outputs

1.

Input OutputNOT

0 1NOT 1 0NOT


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And finally, we have XOR. XOR stands for eXclusive OR. XOR is a

little more complicated than just a simple OR. XOR only outputs a 1 if

and only if one of its inputs is 1. If both inputs are 1, it outputs 0.

XOR

1

0

1

An OR gate outputs a 1 if any of its inputs are 1. An OR gate outputs

a 1 if Input A OR Input B are 1.

We will be using these logical operations/gates AND, OR, NOT,

and XOR because they can be used to create any kind of digital

circuitry magic that we can imagine. There are other logic gates like

NAND, NOR, and XNOR that you can explore on your own.

XOR0

11 XOR1

10 XOR0

00

One of the cool things about binary operations is that you can

perform them without using gates and you can apply them to

multibyte numbers.

For instance, say we wanted to run the logical operation AND on

10011 and 01110. How could we do that?


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0 1 10

1 1 01

1

0

It's easy to get started just write the numbers down like you are

going to add them.

But instead of writing a "+" down, write "AND"

0 1 10

1 1 01

1

0AND

Now just compare them, using AND as a logical operator, byte by

byte, writing down the answer to each paring underneath.

0 1 10

1 1 01

1

0AND

1 AND 0 outputs 0

1 AND 1 outputs 1

0 AND 1 outputs 0

0 AND 1 outputs 0

1 AND 0 outputs 0

So, 0 1 10

1 1 01

1

0

0 1 000

AND

Which you can write as "10011 AND 01110 = 00010"


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So what is 10011 OR 01110?

How about 10011 XOR 01110?

How about NOT 10011?

Or NOT 011110?

Try these.

0 1 10

0 0 10

1

1AND

1 1 00

1 0 01

1

1AND

1 0 10

1 1 10

1

0AND

1 0 10

1 1 01

1

0OR

0 0 10

0 1 00

1

0OR

0 1 11

1 1 11

1

1OR

0 1 10

0 1 11

1

0XOR

1 0 10

0 0 11

1

0XOR

1 1 11

1 0 01

1

1XOR

0 1 011NOT

0 1 100NOT

1 1 011NOT


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XOR

1

0

1 NOT

OR0

0AND

0

0

AND1

0

AND

0

1

AND1

1

OR1

0

OR1

1

0 NOT

XOR

0

1

XOR

1

1

XOR

0

0

Time to check your skills


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OR0

1

XOR

1

0

And here is where the real magic starts.

Logic gates can be chained together. Outputs can feed into inputs. By chaining logic gates,you can do anything a computer is capable of. Everything you do with a digital device happensbecause of chained logic gates performing arithmetic and logical operations. Comparing andadding numbersthat's all it is.

AND NOT

1

1

1 0

OR0

1

XOR

1

1

ANDNOT

0

1

0 1

Change the inputs and you can get a different output.


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ORXOR

1

1

AND

NOT

Solve these. What is the final output? The intermediate outputs?

1

0

Now draw your own.

A circuit with four inputs, two OR gates, one AND gate, and one NOT gate.

A circuit with six inputs, three XOR gates, two AND gates, one OR gate, and three NOTgates.

A circuit with eight inputs, one XOR gate, six NOT gates, one AND gate, and two OR gates.

Populate the starting inputs with values and see if your neighbor can work through the logic ofthe circuit.

AND

OR

XOR

NOT

1

1

1

1

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