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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 the difference between a number and a numeral?

We say that a numeral is a symbol for a number.

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

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 Twenty-five

This works until we get to twenty-four.

Which is ✦✦ … \(four \times five + four \times one\)

How would we represent twenty-five?

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:

Twenty-fives Fives Ones

In other words: onetwenty-five + zerofive + zeroone

So twenty-five is †☉☉, and twenty-six is? Yup: †☉†

What about writing eighty-nine?

This is \(75 + 10 + 4\):

Twenty-fives 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 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).