Binary Numbers
What is the difference between a number and a numeral?
We say that a numeral is a symbol for a number.
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 
Advantages of a positional numerical system has made it popular.
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?
We don’t have a symbol for ten, so we just use a 1 in the tens place.
We will just have to have a fives place in our system.
Number  Fives  Ones  Numeral 

Five  †  ☉  †☉ 
Six  †  †  †† 
Seven  †  ♒  †♒ 
Eight  †  △  †△ 
Nine  †  ✦  †✦ 
The number seven is \(one \times five + two \times one\).
Writing Eleven
What about representing ten? Since ten is really two × five, it is still easy:
Number  Fives  Ones  Numeral 

Ten  ♒  ☉  ♒☉ 
Eleven  ♒  †  ♒† 
Twelve  ♒  ♒  ♒♒ 
Thirteen  ♒  △  ♒△ 
Fourteen  ♒  ✦  ♒✦ 
So thirteen is \(two \times five + three \times one\).
Writing Twentyfive
This works until we get to twentyfour.
Which is ✦✦ … \(four \times five + four \times one\)
How would we represent twentyfive?
One hundred is a new positional place in our system because it is ten × ten.
Since ours is based on five, we would have a new position for it:
Twentyfives  Fives  Ones 

†  ☉  ☉ 
In other words: one ✕ twentyfive + zero ✕ five + zero ✕ one
So twentyfive is †☉☉, and twentysix is? Yup: †☉†
What about writing eightynine?
This is \(75 + 10 + 4\):
Twentyfives  Fives  Ones 

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

Zero  ○ 
One  ● 
Tedious, but simple enough now:
Number  Eight  Four  Two  Ones  Number 

Zero  ○  ○  
One  ●  ●  
Two  ●  ○  ●○  
Three  ●  ●  ●●  
Four  ●  ○  ○  ●○  
Five  ●  ○  ●  ●○●  
Six  ●  ●  ○  ●●○  
Seven  ●  ●  ●  ●●●  
Eight  ●  ○  ○  ○  ●○○○ 
Nine  ●  ○  ○  ●  ●○○● 
Ten  ●  ○  ●  ○  ●○●○ 
The number ten is \(1 \times 8 + 1 \times 2\).
Why is this important? 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.
However, since we don’t have an empty and filled in circles on most
keyboards, we just reuse 0
and 1
:
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 
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
Writing numbers in binary is pretty tedious, but decimal notation (a number system with ten symbols that we use all the time), isn’t good for computers.
Why not?
Need a number system for both computers and people. This is why you may run across hexadecimal which uses sixteen symbols.
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 

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).