Binary Numbers
What is a Numeral?
What is the difference between a number and a numeral?
Number  IndoArabic  Chinese  Thai  Hebrew  Roman 

Zero  0  零  ๐  
One  1  一  ๑  א  I 
Two  2  二  ๒  ב  II 
Three  3  三  ๓  ג  III 
Four  4  四  ๔  ד  IV 
Five  5  五  ๕  ה  V 
Why do we have 10 symbols? Could we use others?
Our New System
Let’s invent one:
Number  Numeral 

Zero  ☉ 
One  † 
Two  ≈ 
Three  △ 
Four  ✦ 
What if this is all the symbols we had?
Writing Five
How would we represent five?
No ten symbol: we use a 1 in the tens place.
We use a fives place in our system.
Number  Fives  Ones  Numeral 

Five  †  ☉  †☉ 
Six  †  †  †† 
Seven  †  ≈  †≈ 
Eight  †  △  †△ 
Nine  †  ✦  †✦ 
The number seven is one × five + two × one.
Writing Eleven
What about representing ten? Ten is two × five:
Number  Fives  Ones  Numeral 

Ten  ≈  ☉  ≈☉ 
Eleven  ≈  †  ≈† 
Twelve  ≈  ≈  ≈≈ 
Thirteen  ≈  △  ≈△ 
Fourteen  ≈  ✦  ≈✦ 
Thirteen is two × five + three × one.
Writing Twentyfive
Twentyfour is ✦✦ … four × five + four × one
What is twentyfive?
Like the hundreds place, we a new position for it:
Twentyfives  Fives  Ones 

†  ☉  ☉ 
Twentyfive is †☉☉, and twentysix is? Yup: †☉†
What about writing eightynine?
Writing EightyNine
Eightynine is 75 + 10 + 4:
Twentyfives  Fives  Ones 

△ (3)  ≈ (2)  ✦ (4) 
Written: △≈m✦
Binary Numerals
What if we only had two numerals!? Crazy, huh?
Number  Numeral 

Zero  ○ 
One  ● 
Binary Table
Number  Eight  Four  Two  Ones  Number 

Zero  ○  ○  
One  ●  ●  
Two  ●  ○  ●○  
Three  ●  ●  ●●  
Four  ●  ○  ○  ●○  
Five  ●  ○  ●  ●○●  
Six  ●  ●  ○  ●●○  
Seven  ●  ●  ●  ●●●  
Eight  ●  ○  ○  ○  ●○○○ 
Nine  ●  ○  ○  ●  ●○○● 
Ten  ●  ○  ●  ○  ●○●○ 
Why Binary?
Computers only have two digits … an electrical current that is on, and a missing a current that is off.
We call numbers represented with only two symbols, binary.
Binary Table … Again
Number  Eight  Four  Two  Ones  Number 

Zero  0 
0 

One  1 
1 

Two  1 
0 
10 

Three  1 
1 
11 

Four  1 
0 
0 
10 

Five  1 
0 
1 
101 

Six  1 
1 
0 
110 

Seven  1 
1 
1 
111 

Eight  1 
0 
0 
0 
1000 
Nine  1 
0 
0 
1 
1001 
Ten  1 
0 
1 
0 
1010 
Binary Joke
Telling the computer that the number 1000
is eight instead of one
thousand is another story.
Now, you will be able to understand the popular joke:
There are only
10
types of people in the world, Those who understand binary and those that who don’t.
Hexadecimal
Binary is tedious, decimal notation isn’t good for computers.
Number  Sixteen  Ones  Number 

Zero  0 
0 

One  1 
1 

Two  2 
2 

Three  3 
3 

Four  4 
4 

Five  5 
5 

Six  6 
6 

Seven  7 
7 

Eight  8 
8 

Nine  9 
9 
Hexadecimal … The Rest
Ten  A 
A 

Eleven  B 
B 

Twelve  C 
C 

Thirteen  D 
D 

Fourteen  E 
E 

Fifteen  F 
F 

Sixteen  1 
0 
10 
Seventeen  1 
1 
11 
Eighteen  1 
2 
12 
Nineteen  1 
3 
13 
Twenty  1 
4 
14 
Hundred  6 
4 
64 
Two Hundred  C 
8 
C8 
Two Hundred and Fiftyfive  F 
F 
FF 
Hundred is 64
because six ✕ sixteen is ninetysix plus four is one hundred.
Two hundred is C8
because C
is twelve and twelve ✕ sixteen is
onehundred and ninetytwo (just add eight more to get twohundred).