Binary Numbers

Let’s invent one:

Number	Numeral
Zero	☉
One	†
Two	♒
Three	△
Four	✦

What if this is all the symbols we had?

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 \(one \times five + two \times one\).

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 \(two \times five + three \times one\).

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: one ✕ twenty-five + zero ✕ five + zero ✕ one

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: △♒✦

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.

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).