# 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).

Date: 2014 Nov 01

Created: 2020-12-23 Wed 10:14