Hindu-Arabic numeral system - The Hindu-Arabic System (800 BC) Today the most universally used system of numeration is the Hindu-Arabic system, also known as the decimal system or base ten system. The system was named for the Indian scholars who invented it at least as early as 800 BC and for the Arabs who transmitted it to the western world. Since the base of the system is ten, it requires special symbols for the numbers zero through nine. The following list the features of this system:

The decimal Hindu-Arabic numeral system was invented in India around 500 CE.^[7][8] The system was revolutionary in that it included a zero andpositional notation. It is considered an important milestone in the development of mathematics. One may distinguish between this positionalsystem, which is identical throughout the family, and the precise glyphsused to write the numerals, which vary regionally. The glyphs most commonly used in conjunction with the Latin alphabet since early modern times are 0 1 2 3 4 5 6 7 8 9.

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The Hindu–Arabic numeration system, which the United States and many other countries presently use, is a place-value or positional-value system. Place-value systems require a base. We use the base 10 or decimal number system. The symbols in this system (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) are called digits. The base of a system is most evident when a numeral is written in expanded form.

The number 6309 is written in base 10. If we wanted to emphasize this, we could write a 10 as a subscript to the right of the number: 630910.

Let's write this same number in expanded form:

(6 x 1000) + (3 x 100) + (0 x 10) + (9 x 1)

Another method for writing the number in expanded form uses the powers of 10:

(6 x 103) + (3 x 102) + (0 x 101) + (9 x 100)

(As discussed in the next module, any number to the zero power is 1; thus, the ones place in powers of ten is ten to the zero power.)

In this course, you are required to convert base 10 numerals to base 2 and base 2 numerals to base 10 (see examples 2 and 3 below). You need not be concerned with any of the other number bases. Base 2 is also called the binary system and is used in computer technology and codes like postal bar codes.

Example 2—Converting from base 10 to base 2

Convert 5710 to base 2.

Solution       To convert base 10 numbers to base 2, we first make a list of base 10 numbers associated with different powers of 2 (i.e., the positional values in the base 2 system). This is not as obvious as with base 10 powers in our earlier example. The list would look like this:

26 = 2 x 2 x 2 x 2 x 2 x 2 = 64

25 = 2 x 2 x 2 x 2 x 2 = 32

24 = 2 x 2 x 2 x 2 = 16

23 = 2 x 2 x 2 = 8

22 = 2 x 2 = 4

21 = 2

20 = 1

The place values in the base 2 system, therefore, are . . ., 26, 25, 24, 23, 22, 2, 1, or 64, 32, 16,

8, 4, 2, 1.

Since changing a base 10 number to a number in a different base involves division, proceed as follows:

1. Divide the base 10 number by the highest power of 2 that is less than or equal to the given number. Obviously, 64 (26) is too large, but we can divide by 32 (25).

57 ÷ 32 = 1 with remainder 25

2. Repeat this procedure for each remainder you obtain, always asking whether the remainder can be divided by the next power of 2 and how many times. If not (i.e., if the place value is too large to be included in the remainder), record a 0 and try the next power:

25 ÷ 16 = 1 with remainder 9

9 ÷ 8 = 1 with remainder 1

1 ÷ 4 = 0 with remainder 1

1 ÷ 2 = 0 with remainder 1

1 ÷ 1 = 1 with remainder 0

3. Tally your findings. The number 57 can be represented as one group of 32, one group of 16, one group of 8, no groups of 4, no groups of 2, and one group of 1:

57 = (1 x 32) + (1 x 16) + (1 x 8) + (0 x 4) + (0 x 2) + (1 x 1)

   = (1 x 25) + (1 x 24) + (1 x 23) + (0 x 22) + (0 x 21) + (1 x 20)

   = 1110012

The base 2 equivalent of 5710 is 111001.

Base 2 numbers only have the numerals 0 and 1 as place values, so no power can occur twice. If 2 x 32 is included in a number, then 64 (26) is included once. Also, note that all place values must be indicated even if they are zeros. This is true for any base system, really, since we couldn't skip the ten place even though it was zero in our earlier base 10 number, 6309.

Converting from base 2 to base 10 reverses the process. Instead of division, multiplication is involved.

Example 3—Converting from base 2 to base 10

Convert 10112 to base 10.

Solution

10112 = (1 x 23) + (0 x 22) + (1 x 21) + (1 x 20)

        = (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1)

        =     8     +    0     +    2     +     1

        = 11

The base 10 equivalent of 10112 is 11.

You all know how to perform arithmetic operations with base 10 numbers. Adding binary numbers is very similar.

Example 4—Adding binary numbers

Add 1110012 and 10112.

Solution     The rules for adding binary numbers are as follows. Note: Since base 2 numbers only have the numerals 0 and 1 as place values, you should never have a 2 in the answer.

1 + 0 = 1

0 + 1 = 1

1 + 1 = 0 with remainder of 1

Thus, lining up the numbers and adding, we get:

 111001

     1011

1000100

You can check. 57 + 11 = 68. Is 10001002 = 68? Let's see.

10001002 = (1 x 26) + (0 x 25) + (0 x 24) + (0 x 23) + (1 x 22) + (0 x 21) + (0 x 20)

             = 64 + 0 + 0 + 0 + 4 + 0

             = 68

Other operations with base 2 or any operations with other bases are not required in this course.