10 + 10 = 32

Most people will read the title of this post and say to themselves; “What!?  Is he off his rocker or something?”  Perhaps, but that’s not the question here.  How did I arrive at such a ridiculous answer?  Let me show you, read along with me for a bit and hopefully feel more educated for the experience.

Most of us learn how to count from our parents before we even get to school.  Our parents teach us first to count to ten.  We will sit with our toddlers and tell them, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10!  Just about everyone does this, but did we ever stop to wonder why we do it?

hands

That’s right!  Because we have 10 fingers.  Once you fill up your hands, you must increment to the next set numbers until you fill up your hands again.  Thus defined by our common physical shape a base 10 number or decimal number system was born.

Take a number like 394, base ten refers to the position, the 4 is in the one’s place, the 9 is in the ten’s place and the 3 is in the hundred’s place. Each number is 10 times the value to the right of it, hence the term base ten.

Hang on a titch, we are getting slightly ahead of ourselves.  What is the first number in a base system and most people would say 1, but it’s not, it’s 0.  We fill up the first set of numbers in the 1’s place and then we must increment the number in the 10’s place to keep going.  Let me demonstrate.  The column on the left has the numbers starting from zero and counting to twelve hiding the leading digit in the ten’s column if that digit is a zero.  The second column shows the preceding zero even if it is a zero.  This enables us to see what is going on more clearly.

0

00

1

01

2

02

3

03

4

04

5

05

6

06

7

07

8

08

9

09

10

10

11

11

12

12

Imagine if you would that we only had four fingers on each hand instead of 5.  It is most likely that the base number system that humans would use would have a base of 8.  That chart above would look something like this.  This numeric base is called octal.

0

00

1

01

2

02

3

03

4

04

5

05

6

06

7

07

10

10

11

11

12

12

Here is a list of the most commonly used number systems and their bases.

Base

Number System

2

binary

3

ternary

4

quaternary

5

quinary

6

senary

7

septenary

8

octal

9

nonary

10

decimal

11

undenary

12

duodecimal

16

hexadecimal

20

vigesimal

60

sexagesimal

Lovely, now I understand numeric bases and how they operate, but how does 10 + 10  = 32?  If I add two numbers from one base system and show you the answer in another, then I could get to such a result.  For my title I used a base 16 system more commonly known as Hexadecimal.

Here is a list of all of the hex numbers that you need to know.

0

1

2

3

4

5

6

7

8

9

0

A

B

C

D

E

F

Let’s figure out what 10 hexadecimal is equal to in our common decimal format.  Using the number chart above and what we have learned so far see if you can follow along.

7 + 8 = F

So far, we have added the number up to, but not exceeded the 16 numbers of 1’s column.  If we then add the two numbers shown below, we must increment the base’s column.

8 + 8 = 10

We called this the 10’s column earlier when dealing with a base 10 number system.  Now that we are dealing with a base 16, it is the 16’s column.

Still wondering how I got the answer of 32 for 10 + 10?

Willy Wonka: The suspense is terrible… I hope it’ll last.

If you understand that the number 10 in hexadecimal is equivalent to the number 16 in decimal, then the following will make sense.

10 + 10 = 20 = 32 = 40

This however would be very confusing if I never told you the base in which the numbers were in, in the first place.

10 hex + 10 hex = 20 hex = 32 decimal = 40 octal

This was fun to write, numeric bases were always a favorite of mine and made math SO much easier once I understood it.  I hope that I was clear enough that you understood it, if not drop me a line and let me know!

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