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Babblings of an aging geek in love with the Absurd, his family, and his own hubris.... oh, and Lisp.

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    
One 1 א I
Two 2 ב II
Three 3 ג III
Four 4 ד IV
Five 5 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 Š
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 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 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).