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Base 10 Numbering System

The Base 10 numbering system is also known as the Decimal numbering system. The Indian culture, known as the Mohenjo Daro culture, of the Hindu-valley was using the early form of the Decimal numbering system for more than 5000 years ago. Subsequent cultural growth in this area developed the ten-based numbering system into a long lasting decimal numbering system. The use of 10 digits for this numbering system may be seen to arise from counting on our 10 fingers.

The most important ingenuity in this Hindu-valley numbering system is the use of zero. More than 5000 years ago, it was not easy to prove the power of this zero-value digit. A zero is used in the decimal system to represent nothing of a particular place-value. For example, in the decimal number of 1230, the 1st place-value has the value of 0, while the 2nd , 3rd and 4th place-values have the values of 3, 2 and 1 respectively. The place-value idea enables us to write billions and trillions by using only ten digits. Unlike Hindu civilization, some other ancient cultures used additional special characters to represent 10s, 100s, 1000s and so on. For example, the number 36 is written as XXXVI in the Roman number system where the place-value idea does not exist. The ten digits used in the decimal system are from 0 to 9, not from 1 to 10.

Some other numbering systems are, Base 2 numbering system, known as the Binary numbering system; Base 8 numbering system, known as the Octal numbering system and Base 16 numbering system, known as the Hexadecimal numbering system. First few numbers in each system are given below.

Base 2
0
1
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
1111
Base 8
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
20
21
Base 10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Base 16
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
10
11

For instance, 10102 = 128 = 1010 = A16

Subscripts are used to identify the base of a number. If we do not specify the base, then it is assumed to be base 10.

Base 60:
The Base 60 numbering system is also known as Sexagesimal. The Sumerian Culture developed the Sexagesimal number system and the Babylonians got it from the Sumerian.



BCD

Binary Coded Decimal numbers are a special type of binary numbers. In this system, each Base-10 digit is represented in 4-digit binary form. There are only ten Binary Coded Decimals and they represent the numbers from 0 to 9.

Base 10
0
1
2
3
4
5
6
7
8
9
BCD
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001

The rest of the binary patterns 1010, 1011, 1100, 1101, 1110 and 1111 are not used since they do not represent any valid BCD numbers.

For example, in the BCD notation, the decimal 123 is represented as 0001 0010 0011 while in the normal binary notation, 123 is represented as 1110011.



BEDMAS

The order of mathematical operations for rational numbers follows the rule of BEDMAS (where B: brackets, E: exponents, D: division, M: multiplication, A: addition and S: subtraction). No one knows the origin of the order of operation denoted by BEDMAS. Note that even though D comes before M in BEDMAS, they have the same priority. Therefore it could also be called BEMDAS. Likewise, addition and subtraction also have the same priority. In general,

Step 1: Evaluate inside the brackets first. If there are brackets inside brackets, then the innermost values get evaluated first.

Step 2: Then evaluate all the exponents, if any

Step 3: Thirdly, execute all multiplication (s) and/or division (s) from left to right

Step 4: Finally, execute all the addition (s) and/or subtraction (s) from left to right.



Place-value and Zero

In the decimal system (base 10), the value of a particular digit depends on both the digit itself and its position within the number. Also, each place (digit) has a value of 10 times the place to its right. Zero is used as an empty place indicator. For example, the number 98765.432 has 9 ten thousands, 8 thousands, 7 hundreds, 6 tens, 5 ones, 4 tenths, 3 hundredths and 2 thousandths. The following expanded form shows the number can also be expanded into an addition statement.

98765.432 = 9x10000 + 8x1000 + 7x100 + 6x10 + 5 + 4/10 + 3/100 + 2/1000
98765.432 = 90,000 + 8,000 + 700 + 60 + 5 + .4 + .03 + .002

 
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