Number Systems- Part-2

                                                                                         

This article is a continuation of last week's article on Number Systems.
This week, we’ll discuss Octal and Hexadecimal Number Systems.

Octal System

‘octa’ means 8. The octal number system has a base of eight. It uses eight digits from 0 to 7, that is: 0, 1, 2, 3, 4, 5, 6, 7. The octal number system is also called the ‘base 8 number system’.

The octal number system is very important in digital computer field.

octal number system, octal system, number system, hexadecimal system


Similar to the decimal and binary number systems, the octal number system is also a positional number system. Just as the decimal system is based on powers of 10 and the binary system is based on powers of 2, the octal system is based on powers of 8. Place values in an octal system increase from right to left by powers of 8 since the base here is 8 (…64, 8, 1, …). That is, from right to left an octal system has a ones place, an 8’s place, a 64’s place etc.

Each digit in an octal number has a value dependent on its position in the number. A digit's value is the digit multiplied by a power of eight according to its position in the number.

For example, consider the octal number 562.

The digit 5 is in the ‘64’s place (8^2)’ and its value is 5 × 64 = 320.
The digit 6 is in the ‘8’s place (8^1)’ and its value is 6 × 8 = 48.
The digit 2 is in the ‘ones place (8^0)’ and its value is 2 × 1 = 2.

To convert an octal number to a decimal, find out the actual value represented by each digit and add them together. (Also recall from last week’s article that this same strategy worked for converting binary to decimal.)

So, the decimal equivalent of the octal number 562 is 320 + 48 + 2 = 370.

To convert from decimal to octal:

Step 1: Divide the octal number by 8.
Step 2: Record the remainder.
Step 3: Repeat Steps 1 and 2 with the quotient until the quotient becomes zero.

Let’s look at a simple example. Let’s find the octal equivalent of the decimal 253.

253/8 = 31 --- Remainder 5
31/8 = 3 --- Remainder 7
3/8 = 0 --- Remainder 3

Write the remainders in the reverse order. So, the octal equivalent of the decimal 253 is 375.

Let’s now talk about octal fractions.

An octal fraction is the same as a decimal fraction, but with the base of 8 instead of 10.

An octal point (called a decimal point in the decimal system) separates the integer part of an octal number from its fractional part. The octal point indicates the place where values change from positive to negative powers of 8. Each place to the left of the octal point represents a positive power of 8 (8, 64, 512, and so on) and each place to the right of the octal point represents a negative power of 8 [8^(-1), 8^(-2), 8^(-3), 8^(-4), and so on]. When a base is raised to a negative power, it just means the reciprocal of the base raised to the positive power. So:

8^ (-1) = 1/ (8^1) = 1/8
8^ (-2) = 1/ (8^2) = 1/64
8^ (-3) = 1/ (8^3) = 1/512 etc.

For example,

The octal number 351.024 represents 3 x 8^ (2) + 5 x 8^ (1) + 1 x 8^ (0) + 0 x 8^ (-1) + 2 x 8^ (-2) + 4 x 8^ (-3)

= 3 x 64 + 5 x 8 + 1 × 1 + 0 x 1/ (8^1) + 2 x 1/ (8^2) + 4 x 1/ (8^3)
= 3 x 64 + 5 x 8 + 1 × 1 + 0 x 1/8 + 2 x 1/64 + 4 x 1/512
= 192 + 40 + 1 + 0 + 1/32 + 1/128
= 233 + 5/128
= 233 + 0.0390625
= 233.0390625

Hexadecimal System

‘hex’ means 6, ‘deci’ means 10, ‘hexadecimal’ is 6 + 10 = 16. Just as the decimal system has a base of 10 (10 digits from 0 to 9), binary is base 2 (2 digits, 0 and 1), and octal is base 8 (8 digits from 0 to 7), the hexadecimal system has a base of 16. It uses 16 digits from 0 to 15.
Hexadecimal system uses the letters A-F to represent the digits 10 through 15. Here are the digits used in hexadecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. The hexadecimal system is also referred to as "base 16 system" and “hex system”.

The hexadecimal number system is used widely in computer industry and digital electronics. Programmers use hexadecimal system as it is convenient and provides a much more compact way to represent binary numbers. It is very easy to convert from binary to hexadecimal and vice versa.

The hexadecimal system is also a positional number system. Just as the decimal system is based on powers of 10, the binary system is based on powers of 2, and the octal system is based on powers of 8, the hexadecimal system is based on powers of 16.
In the hexadecimal system, place values increase from right to left in powers of 16 since the base here is 16 (…256, 16, 1, …). That is, from right to left a hexadecimal system has a ones place, a 16s place, a 256s place etc.

Each digit in a hexadecimal number has a value dependent on its position in the number. A digit's value is the digit multiplied by a power of 16 according to its position in the number.

For example,

Consider the hexadecimal number A42.

A is in the 256s place (16^2) and its value is A × 256 = 10 × 256 = 2,560.
4 is in the 16s place (16^1) and its value is 4 × 16 = 64.
2 is in the ones place (16^0) and its value is 2 × 1 = 2.

To convert a hexadecimal number to a decimal, find out the actual value represented by each digit and add them together.

So, the decimal equivalent of the hexadecimal number A42 is 2,560 + 64 + 2 = 2,626.

To convert from decimal to hexadecimal:

Step 1: Divide the decimal number by 16.
Step 2: Record the remainder.
Step 3: Repeat Steps 1 and 2 with the quotient until the quotient becomes zero.

Let’s look at a simple example. Let’s determine the hexadecimal equivalent of 3746.

3746/16 = 234 --- Remainder 2
234/16 = 14 --- Remainder 10 = A
14/16 = 0 --- Remainder 14 = E

Write the remainders in the reverse order. So, the hexadecimal equivalent of the decimal 3746 is EA2.

Let’s now talk about hexadecimal fractions.

A hexadecimal fraction is the same as a decimal fraction, but with the base of 16 instead of 10.

A radix point (called a decimal point in the decimal system) separates the integer part of a hexadecimal number from its fractional part. The radix point indicates the place where values change from positive to negative powers of 16. Each place to the left of the radix point represents a positive power of 16 (16, 256, 4096, and so on) and each place to the right of the radix point represents a negative power of 16 [16^ (-1), 16^ (-2), 16^ (-3), 16^ (-4), and so on]. When a base is raised to a negative power, it just means the reciprocal of the base raised to the positive power. So:

16^ (-1) = 1/ (16^1) = 1/16
16^ (-2) = 1/ (16^2) = 1/256
16^ (-3) = 1/ (16^3) = 1/4096 etc.

For example,

The hexadecimal number C7.4D8 represents C x 16^ (1) + 7 x 16^ (0) + 4 x 16^ (-1) + D x 16^ (-2) + 8 x 16^ (-3)

= C x 16 + 7 x 1 + 4 x 1/ (16^1) + D x 1/ (16^2) + 8 x 1/ (16^3)
= 12 x 16 + 7 x 1 + 4 x 1/ (16^1) + 13 x 1/ (16^2) + 8 x 1/ (16^3)
= 12 x 16 + 7 x 1 + 4 x 1/16 + 13 x 1/256 + 8 x 1/4096
= 192 + 7 + 1/4 + 13/256 + 8/4096
= 192 + 7 + 1024/4096 + 208/4096 + 8/4096
= 199 + 1240/4096
= 199 + 0.302734375
= 199.302734375

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About the author:
I’m Chandrajeet, an in-house writer for iCoachMath. iCoachMath is an effective, convenient, easy-to-use online Math Program which has been used by thousands of students, teachers, and parents. iCoachMath strives to lead K-12 students to excellence in math by offering quality web-based educational solutions. iCoachMath’s instructional and lesson materials are aligned to State Curriculum Standards in all 50 states (USA).
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