Binary Numbers
What is a Numeral?
What is the difference between a number and a numeral?
| Number | Indo-Arabic | 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 Twenty-five
Twenty-four is ✦✦ … four × five + four × one
What is twenty-five?
Like the hundreds place, we a new position for it:
| Twenty-fives | Fives | Ones |
|---|---|---|
| † | ☉ | ☉ |
Twenty-five is †☉☉, and twenty-six is? Yup: †☉†
What about writing eighty-nine?
Writing Eighty-Nine
Eighty-nine is 75 + 10 + 4:
| Twenty-fives | 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
10types 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 Fifty-five | F |
F |
FF |
Hundred is 64 because six ✕ sixteen is ninety-six plus four is one hundred.
Two hundred is C8 because C is twelve and twelve ✕ sixteen is
one-hundred and ninety-two (just add eight more to get two-hundred).