The Mysteries of Math

The History of Chinese Mathematic

2009-05-23T16:55:00.004+08:00

Although most people nowadays using mathematic system of Arabic,
which is the decimal system of 0-9,
but the old Chinese Mathematic still affect much to us in modern day.
included the invention of calculator and computer.

About Ancient Mathematic
Mathematic is an important system that is very important to our daily life.
This is also the same in ancient world.
A lot of archaeologist found and proved that,
even in ancient world that without any words,
they will also having some simple drawing to represent numbers.

China Mathematic in Old Dynasty
The oldest book about mathematic that can found in China is the Yi Jing,
which is actually an old book about fortune read,
but also contain a lot of Mathematic.
The book is written in Shang Dynasty,
which the dynasty found on 1600 BC to 1050 BC.

On Shang dynasty,
China already developed some expert decimal system,
included arithmetic, algebra, equations, and negative numbers.
China believe to be the first country that developed negative numbers.

Another impressive book of old China dynasty is the Zhou Bi Suan Jing,
which believe it's written in around 1200BC to 1000BC.
The book contain a theorem called Gougu Theorem,
which is an in-depth Pythagorean Theorem that focused in astronomical calculation.
Although the date that the book written still on debate,
most people accepted the date mention above due to some proof within the book.

During Qin Dynasty,
due to the burning of books and burying of scholars,
most book about China old mathematic have been burned.

The Abacus, Calculator and Computer
Most of history reader must know that in old China,
almost every person used Abacus, or called Suanpan,
to calculate simple arithmetic calculation.

In some dynasty,
there is some trading between old China and Europe,
which Chinese Abacus have been traded to Europe.
After some long improvement,
they finally invent another better calculation tools,
which is the calculator.
The calculator is actually a complicated form of Abacus,
which is actually invent by the idea of Abacus.

In which we know,
Computer is invent by idea of Calculator,
which in conclude,
Computer was invent because of existing of Chinese Abacus.

Conclude
Although Chinese Mathematic words,
or calculations tools not very widely-use in modern days,
but many technology product are actually invent by the idea of such tools.

The influence of Chinese mathematic might be bigger than you think.
Never underestimate anything or anyone.

Unable to Online

2009-04-17T01:59:00.002+08:00

I'm sorry to inform all of our visitors that due to local internet connection problem,
I'm not able to online (or experience many problem even I can online) within short time.
With this problem,
I'm currently not able to update this blog as the scheduled (1 post per week).
Please stay tune on our blog for new post once the connection is back.

System's Multiplication (1)

2009-03-30T14:46:00.003+08:00

Here's come a tricky part of the math system.
Multiplication is sometime confuse to many people when it come to system larger than Binary.

Binary Multiplication
For Binary multiplication,
it's actually easier than you think.
This is the basic theory:
0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1

Use the rules and multiply by using rules of decimal.
You can get the answer without problem.
Here's the example:
For 10011₂ * 11₂

   10011 *     11 --------    10011 + 10011 --------   111001 --------

Check:
10011₂ = 19
11₂ = 3
19 * 3 = 57
111001₂ = 57

Example:
1010111₂ * 1011₂

     1010111 *       1011 ------------      1010111     1010111    0000000   + 1010111    ------------   1110111101 ------------

Quaternary Multiplication
For queternary multiplication,
it is trickier than Binary multiplication.

The step is as below:
1) Multiply the number by decimal ways.
2) If the answer you get contain any number that is larger than 3, divide the number with 4 and add the answer to upper digits. Write down the remainder at same digits.
3) Repeat the process until there is no number larger than 3.

Example:
313₄ * 3₄

  313 *   3 -----   939   951 (The 9 from previous became 21) 2211 (The 5 from previous became 11, which make the upper digits 9 become 10, which again become 22 ) -----

The answer will be 2211₄

Example:
312₄ * 32₄

     312 *     32 --------     624    1230    936   2202 --------    1230 + 2202 --------   23250 = 23310 --------

Check:
312₄ = 54
32₄ = 14
54 * 14 = 756

23310₄ = 756

The system multiplication is too long to complete within single post.
To be continues......

System's Addition & Subtraction

2009-03-18T15:34:00.002+08:00

After learning about System Convert,
it's time for another 'hard' chapter to some people,
the operation of system.
Today,
we will cover additional and subtraction.

Addition
To learn additional,
first, you will need to understand the "maximum digit" of specific system.
For Binary, its 1,
for Octal, its 7,
for Hex, its F (15).

The additional work similar to decimal system,
the only different is,
you have to remember the maximum digit,
and the number over the digits should become 10.

Example:
For binary,
1 + 1 = 10
11 + 10 = 101

As you can see,
there is no 2 present.
The maximum digits is 2,
if you see 1 + 1, it will become 10.

For Octal,
7 + 1 = 10
7 + 7 = 16
17 + 2 = 21
71 + 10 = 101

So, the maximum digit is 7.
You cannot see any 8 within the whole thing.
For 7+1, it already reached it maximum,
thus will become 10.
For 7+7, it already reached maximum, and over it,
so it should become this: 7 + 1 + 6
7+1 give us 10, and 10+6 will give us 16.
In 17+2, it will become 17 + 1 + 1.
17 + 1 = 20, and 20+1 = 21.
That's it.
Practice make perfect.
Try to practice by yourself,
and you can later do it faster without ever mistake.

For Hex,
6 + 6 = C
F + 1 = 10
F + F = 1E
F5 + 16 = 10B

Hex is much complicated than any example above,
as it contain A to F that represented 10 to 15.
The maximum number will be F, which is represented 15.
For 6+6, it will be 12 in decimal,
but in Hex, we should find the represented symbol,
which is C.
For F+1, since F(15) is the maximum digits,
F+1 reached it's maximum,
thus will become 10 in Hex.
For F+F, it already over the maximum,
which will become F + 1 + E. equal 10 + E,
which is equal to 1E.

For other system,
the theory work similar,
just need to remember what is their maximum digit should work well.

Subtraction
If addition is hard for you,
subtraction will be even harder.
I prefer you to learn more about addition first become read this one.

Before you do the operator,
you also need to think about the maximum digits too.

For binary (Maximum : 1),
10 - 1 = 1
100 - 1 = 11
100 - 11 = 1

For 10 - 1,
since the first digit of the main is not enough to deduct (0 - 1),
we will have to use the second digits (1 of the 10).
Since the maximum digits is 1, the limits number will be 2.
Therefore, 10 will become 2,
and 10 - 1 = 2 - 1 = 1
For 100 - 1 = 20 - 1 = 12 - 1 = 11

For Octal (Maximum : 7),
10 - 1 = 7
100 - 5 = 73
100 - 15 = 63

Since the maximum digits is 7, therefore 10 will be (8).
For 10 - 1 will be 8-1 = 7

The theory work same with other system as well.
Try it yourself at home.

The next post will be about next two operator,
about Multiplication and Division.
Stay tuned,
or subscribe to our RSS Feed.

System Convert (3)

2009-03-06T09:02:00.002+08:00

This will be the last about system convert.
Here I will tell you about convert a number from a non-decimal system to other non-decimal system.

From the last post,
you might notice that the method to convert system mostly run with a similar theory.
Most of them have this at the start: a^x = b.
This is actually the basic requirement for system convert within non-decimal system.

From Small to Large
To convert a number from smaller base to larger base,
first, determine the power-relation: a^x = b,
which a is smaller than b, and x must be a natural number exclude 0(or positive integer).
Here is the step to convert it:

Define the x of equation a^x=b, where a is the base of your question and b is the base of the request system.

Convert the number into x digits per group started from the last number.

Convert number from each group into their decimal number.

The number you get will be the answer.

Example 1:
Find the base 9 number of 2212₃

3²=9. Therefore, x=2

| 22 | 12 |

| 8 | 5 |

Therefore, 85₉ is the base 9 number of 2212₃

Check:
2212₃
= (2 * 27) + (2 * 9) + (1 * 3) + (2 * 1)
= 77

85₉
= (8 * 9) + (5 * 1)
= 77

Example 2:
Convert 10001₃ into base 27 number.

3³=27. Therefore, x=3

| 10 | 001 |

| 3 | 1 |

Therefore 31₂₇ is the base 27 number of 10001₃

From Large to Small
To convert larger system to smaller system,
here is the required steps:
1) Define x of a=b^x, where a is given system and b is request system.
2) Separate the number given into each group.
3) Convert the number in each group to requested system.
4) If the number in each group is less than x digits, add 0s in front of them until they have x digits.
The number you get will be at requested system.

Example 1:
Convert 742₉ into base 3.
1) 9=3². Therefore, x = 2.
2) | 7 | 4 | 2 |
3) | 21 | 11 | 2 |
4) | 21 | 11 | 02 |
Therefore, 211102₃ = 742₉

Example 2:
Convert 962₂₇ into base 3.
1) 27=3³. x=3
2) | 9 | 6 | 2 |
3) | 100 | 20 | 2 |
4) | 100 | 020| 002 |
Therefore, 100020002₃ = 962₂₇

Conclusion
This should finish the whole part of system convert,
which might be useful for people who is interesting in Math.

What will coming next?
I will then tell about operation of each system,
like plus, minus, times, divide or even square and root.
Visit our blog next week for more,
or subscribe to our RSS feed.

System Convert (2)

2009-02-25T16:58:00.002+08:00

Continues from the last post,
this post will write more about system convert.

The "2^x" System
To convert a number from one system to another system that both system aren't decimal,
there is some other ways to do than the old way,
(The old way: From first system to decimal, then from decimal to destination)
but there is some condition.

The first condition is, base "2^x".
This included Binary (Base 2), Quaternary (Base 4), Octal (Base 8),
Hex (Base 16) or any base that is with power 2.

Binary to Quaternary
Step 1: Arrange the binary number by two per group (from the last number).
Step 2: Convert the number on each group to it's decimal.
The number you get will be the Quaternary number.

Example 1:
10001₂ (17 in decimal)
Step 1: 1 | 00 | 01
Step 2: 1 | 0 | 1
The number we get is 101₄

Check:
101₄
= (1 * 4²) + (1 * 4⁰)
= 16 + 1
= 17

Example 2:
101101₂ (45 in decimal)
Step 1: 10 | 11 | 01
Step 2: 2 | 3 | 1
The number we get is 231₄

Binary to Octal
Step 1: Arrange the binary number by three per group (from the last number).
Step 2: Convert the number on each group to it's decimal.
The number you get will be it's Octal.

Example 1:
10001₂ (17 in decimal)
Step 1: 10 | 001
Step 2: 2 | 1
The Octal number will be 21₈

Example 2:
1010100₂ (84 in decimal)
Step 1: 1 | 010 | 100
Step 2: 1 | 2 | 4
The Octal number is 124₈

Binary to Other 2^x
From the above example,
we can get that to convert Binary to other 2^x system,
the step is just to rearrange the number by group.
Here is the concluded solution:
First, we will need to know the "x".
We have to do simple calculation to know that the base we want,
is equal to 2 to power x.
The number x is the number that should be in a group.

Now, rearrange the number by x number per group.
Convert the number within the group to it's decimal,
you will get the number of that system.

Example 1:
Convert 100101₂ to Hex system.
Hex = Base 16, and 16 = 2₄
4 is the x in this question.
So, rearrange the number with 4 number per group:
10 | 0101
We will get the answer as:
2 | 5
Therefore, 25₁₆ is the Hex number of 100101₂.

Example 2:
Let try something bigger.
Convert 100101000₂ into Base 32 system.
32 = 2₅
So, since the x is 5, we have to rearrange the number by 5 per group:
1001 | 01000
= 9 | 8
The answer will be 98₃₂

Check:
100101000₂ = 296 (In decimal).
98₃₂
= (9 * 32¹) + (8 * 32⁰)
= (9 * 32) + (8 * 1)
= 296

From Quaternary to Binary
When there is a method to convert Binary to any 2^x system,
there will also be a method to convert from any 2^x to Binary.

For Quaternary to Binary:
Step 1: Now, rearrange the number by 1 number per group
Step 2: Convert those number to binary number as those are decimal.
Step 3: Checking each group, if the binary you get within each group is less than 2 digits, add a 0 at front.
Make sure that all number within the group is two digits.
The number you get will be the Binary.

Example 1:
203₄

Step 1: 2 | 0 | 3
Step 2: 10 | 0 | 11
Step 3: 10 | 00 | 11
The answer will be 100011₂

Example 2:
31210₄

Step 1: 3 | 1 | 2 | 1 | 0
Step 2: 11 | 1 | 10 | 1 | 0
Step 3: 11 | 01 | 10 | 01 | 00
Therefore, 1101100100₂ = 31210₄ = 868 (in decimal)

From Octal to Binary
Step 1: Now, rearrange the number by 1 number per group
Step 2: Convert those number to binary number as those are decimal.
Step 3: Checking each group, if the binary you get within each group is less than 3 digits, add 0 at front until it have 3 digits.
Make sure that all number within the group is three digits.
The number you get will be the Binary.

Example 1:
27₈
Step 1: 2 | 7
Step 2: 10 | 111
Step 3: 010 | 111
Therefore, 10111₂ is equal to 27₈

Example 2:
7413₈
Step 1: 7 | 4 | 1 | 3
Step 2: 111 | 100 | 1 | 11
Step 3: 111 | 100 | 001 | 011
Therefore, 7413₈ = 111 100 001 011₂

From Other System to Binary
Observe the step we use above.
There is also many similarity.
What we can get is,
The digits in each small group is equal to "x" of 2^x
Therefore, we concluded the step as follow:
1) Get the "x" by 2^x
2) Arrange the number into 1 per group.
3) Convert those number within the group as decimal.
4) Add 0 if the number you get isn't fulfill the x digits.

Example 1:
C52₁₆ (C = 12, Check the first post of Binary)
16 = 2⁴
Therefore, there will be 4 digits per group.
So, the step will be:
C | 5 | 2
1100 | 101 | 10
1100 | 0101 | 0010
Therefore, C52₁₆ = 1100 0101 0010₂

Example 2:
633₃₂
32 = 2⁵
Therefore, there will be 5 digits per group:
6 | 3 | 3
110 | 11 | 11
00110 | 00011 | 00011
Therefore, 633₃₂ = 110 00011 00011₂

It already near the end of System.
The next one will concluded the last method of system convert,
and a conclusion of the whole System.

System Convert (1)

2009-02-18T15:34:00.004+08:00

Before start reading this,
please make sure that you know what is Math number system.
Read the previous post to learn more about Math System

From Decimal to Binary
There is many ways to convert number from decimal to binary.
Before start, let me remind you that decimal is base 10 number system and binary is base 2.
From the previous post,
there is already mention a way to convert Binary number to Decimal number.
Here how we convert Decimal number to Binary:
1) First, divide your number by 2. If the number can divide full by 2, you record down "0". If the number can't divide by 2 (remain 1), then record down "1".

2) Repeat step one with the balance you get with step one. Repeat again until you left a number smaller than 2 (in this case, it's 1).

3) The last remaining number will be the first binary number, and rearrange the number with descending order on it's back and you will get the binary number.

Check the example for clearer understand:
Let's try with number "9"

Divide 9 with two, we get 4 remain 1. (Record down 1)
Now, divide the balance (4) with 2, we get 2 with no remain. (Record down 0)
Again, divide the remain 2 with 2, we will get 1 with no remain. (Record down 0).
After getting the last number (1),
we have to write the last number (1) as initial number,
and arrange recorded number with descending order,
we will get 1001.

Thus, 1001₂ is the binary number of 9.

Let's try with number "12":

Divide 12 by 2, get 6 with no remain ( 0 ),
divide 6 with 2, get 3 with no remain ( 0 ),
divide 3 with 2, get 1 with remain 1 (Record 1)
Now, with the last number (1) as initial,
after rearrange, we will get 1100

Thus, 1100₂ is the binary number of 12.

Remember that you must put the last remaining number (1) at the start of your binary number.

From Decimal to any other system
To convert Decimal number to any other system have similar step with the binary case.
It is actually base on same method.

1) First, divide your number by the base number. If the number can be fully divide by the base number, record down 0. If the number cannot fully divided by base number, record down the remain number

2) Repeat step one with the balance you get with step one. Repeat again until you left a number smaller than the base number.

3) The last remaining number will be the first number, and rearrange the number with descending order on it's back and you will get the number of that system.

Again we will have some example:

Let's try to convert 153 into Octal system (Base 8):

Divide 153 with base number (8), we will get 19 with remain of 1.
Divide 19 with 8, we will get 2 with remain of 3.
Since the last balance number, 2, is smaller than 8.
So we rearrange the number and will get 231

Thus, 231₈ is the Octal number of 153.

Try to test it,
231₈
= (2 * 8²) + (3 * 8¹) + (1 * 8 ¹)
= (2 * 64) + (3 * 8) + (1 * 1)
=153

Let's try something special, convert 137 into base 6 system, the Senary:

Divide 137 by 6, get 22 with remain of 5.
Divide 22 by 6, get 3 with remain of 4.
3 is the last balance since it is smaller than 6.
Rearrange and we will get 345

Thus, 345₆ is the Senary (Base 6) number of 137.
345₆ can be read as "three four five base six"

Test it again,
345₆
= (3 * 6²) + (4 * 6¹) + (5 * 6⁰)
= (3 * 36) + (4 * 6) + (5 * 1)
= 137

Last example,
for system larger than decimal (base 10),
we will have their symbol as below:
10 = A
11 = B
12 = C
...
For the Hexadecimal system,
it will have A to F as 10 to 15.

Try to convert 548 into Hexadecimal:

Divide 558 by 16, get 34 with remain of 14, which is E.
Divide 34 with 16, get 2 with remain of 2.
The last remain 2 is smaller than 16,
thus we will get 22E

Thus, 22E₁₆ is the Hexadecimal number of 548.

Another test:
22E₁₆
= (2 * 16²) + (2 * 16¹) + ([E = 14] * 16⁰)
= (2 * 256) + (2 * 16) + (14 * 1)
=558

Today Summary
We have so far learn about what the system it, and convert number from Decimal to other system.
Stay tuned, to learn how to convert number from a non-decimal system to another non-decimal system,
like convert Binary to Octal directly without passing through Decimal.
You can also subscribe to this blog to stay inform of new post.

The Number & System

2009-02-15T18:38:00.005+08:00

The number is the basic and the "root" of Math.
Our daily common use number is called Arabic Numerals,
and the system of the Arabic Numerals is Decimal System,
which Deci- mean ten and decimal mean base ten.
The decimal also mean that,
the Arabic numbers contain of at least 10 digits,
which is (0,1,2,3,4,5,6,7,8,9)

What is The System?
System also called position system or position notation,
is a numeral system in which each position is related to the next by a constant multiplier,
a common ratio, called the base or radix of that numeral system.
Each position can be represented by unique symbols or limited sets of symbols.
Although the explanation seem a bit confuse,
but look at the example which might help you to understand a bit more.

Decimal System
As I mention above, Decimal system is used by Arabic numerals as our common daily use,
and it's contain of ten base.
Which mean, the numbers is contain of 10 digits,
which may be represent by 10 unique symbols.
The "symbols" we used is 0,1,2,3,4,5,6,7,8,9
The smallest numbers of the system is 0 (zero),
while the largest of the system is 9 (nine).
If the number "9" increase by "1",
it will over limit of the system,
which we have to use 10 (one-zero, read as ten) to represent it.
For the similar case, if 19 (one-nine) is increase by one,
the "1" of the example will increase by one and 9 will fall back to 0,
which give us 20 (two-zero).

The decimal is actually based on power of 10,
which the first is 10⁰, the second on is 10¹, continues on 10², 10³, 10⁴ ...
From the previous example,
20 contain of following element:
(2 * 10¹) + (0 * 10⁰)
= (2 * 10) + (0 * 1)

Other example:
346
= (3 * 10²) + (4 * 10¹) + (6 * 10⁰)
= (3 * 100) + (4 * 10) + (6 * 1)

9587
= (9 * 10³) + (5 * 10²) + (8 * 10¹) + (7 * 10⁰)
= (9 * 1000) + (5 * 100) + (8 * 10) + (7 * 1)

Binary System
The binary system is a number system with base of 2,
which contain only 2 unique symbol.
The theory of all number system is similar,
only the symbols used to represent is limited.
Let say the binary system contain of 0 and 1,
the system will worked as follow:
The smallest number is 0, the largest number is 1.
If "1" is increase by 1,
it will become 10 (one-zero, this cannot read as ten, must be read as one-zero binary system)
If "11" is increase by 1,
it will become 100.
(The first digits reach it's max, and give second digits an increment of 1. The second digits will also reach it's max, and thus increase the third digits form default 0 to 1.)
The correct way the write one-zero-zero binary system is 100₂
Without the "₂", the 100 is confused and hence is incorrect.

In decimal, the system use 10 as power.
In binary, the system will use power of 2.
From the previous example,
the 100₂ contain the following element:
(1 * 2²) + (0 * 2¹) + (0 * 2⁰)
= (1 * 4) + (0 * 2) + (0 * 1)
= 4 (in decimal system)

Other example:
101₂
= (1 * 2²) + (0 * 2¹) + (1 * 2⁰)
= (1 * 4) + (0 * 2) + (1 * 1)
= 5 (in decimal system)

10110₂
= (1 * 2⁴) + (0 * 2³) + (1 * 2²) + (1 * 2¹) + (0 * 2⁰)
= (1 * 16) + (0 * 8) + (1 * 4) + (1 * 2) + (0 * 1)
= 22 (in decimal system)

The above ways not only worked as root of binary system,
but can also used to convert binary numbers to decimal numbers.
The Binary System is also famous because it is also the roof of computer system.

Other System
Beside than decimal and binary system,
there is a lot of other system.
Common use system will be Decimal, Binary, Quaternary, Octal and Hex.
They are base on 10, 2, 4, 8 and 16 simultaneously.
There is also system with base 3, 5, 7, 9 ... without limit (limitless).
All of the system worked similar,
hence I will stop it here.

How to convert decimals number into other system?
How to convert number from any system to other system?
Visit our blog later for new post,
or subscribe to us to read every updates.

Welcome

2009-02-14T17:27:00.000+08:00

Welcome to my new blog,
The Mysteries of Math,
a blog about Math from basic to deep.