The following are the four important types of number systems.

1. Decimal

2. Binary

3. Octal

4. Hexadecimal

In this article, we’ll look at Decimal and Binary Number Systems. We’ll discuss the other two number systems in our next article.

Decimal System

‘Deci’ means ten. The decimal number system has ten as its base. It uses ten digits from 0 to 9. The decimal number system is also called Hindu Arabic, or Arabic, or base 10 system.

The decimal system is simple and versatile. It is the most commonly used number system. It has become the dominant number system in the world. Any number, no matter how large or small, can be written in the decimal system using only the 10 basic symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, the population of Asia is about 3,800,000,000. The diameter of an iron atom is about 0.000000025 cm.

It is believed that the decimal system is based on 10 digits because humans have 10 fingers and 10 toes. The word digit is derived from the Latin word ‘digitus’ which means finger or toe.

The decimal system is a positional number system. Each digit in a number has a value dependent on its position in the number. A digit's value is the digit multiplied by a power of ten according to its position in the number.

For example, consider the number 9,364.

The digit 9 is in the ‘thousands place’ and its value is 9 × 1,000 = 9,000.

The digit 3 is in the ‘hundreds place’ and its value is 3 × 100 = 300.

The digit 6 is in the ‘tens place’ and its value is 6 × 10 = 60.

The digit 4 is in the ‘ones place’ and its value is 4 × 1 = 4.

So:

9,364 = 9 × 1,000 + 3 × 100 + 6 × 10 + 4 × 1 = 9,000 + 300 + 60 + 4

In the decimal system, place values increase from right to left by powers of 10 (…1000, 100, 10, 1…). That is, from right to left a decimal system has a ones place, a tens place, a hundreds place, a thousands place, a ten thousands place etc.

For example,

In the number 22:

The 2 on the left is in the "tens place," and its value is 2 × 10 = 20.

The 2 on the right is in the "ones place," and its value is 2 × 1 = 2.

Observe that the value of the 2 on the left is 10 times as large as that of the 2 on the right.

Let’s now talk about decimal fractions.

A decimal fraction is a number with a decimal point in it, like 0.3, 0.75, 1.23, 7.049, 0.0058 etc.

In a decimal fraction (commonly called a decimal) a decimal point separates the whole numbers from fractions. The decimal point indicates the place where values change from positive to negative powers of 10. Each place to the left of the decimal point represents a positive power of 10 (10, 100, 1000, 10000, and so on) and each place to the right of the decimal point represents a negative power of 10 [10^ (-1), 10^ (-2), 10^ (-3), 10^ (-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:

10^ (-1) = 1/ (10^1) = 1/10 = 0.1

10^ (-2) = 1/ (10^2) = 1/100 = 0.01

10^ (-3) = 1/ (10^3) = 1/1000 = 0.001 etc.

So, the place values to the right of the decimal point are tenths, hundredths, thousandths, and so on. Negative powers of 10 are small numbers, less than 1 and greater than 0.

EXAMPLE

In the decimal 13.97:

1 is in the ‘tens place’.

3 is in the ‘ones place’.

9 is in the ‘tenths place’.

7 is in the ‘hundredths place’.

So:

13.97 = 1 × 10 + 3 × 1 + 9 × 0.1 + 7 × 0.01 = 10 + 3 + 0.9 + 0.07

Binary System

‘Bi’ means two. The binary number system works just like the decimal number system except that the binary number system uses base 2. It uses only two digits, 0 and 1. The binary number system is also referred to as "base 2 system".

The binary system plays an important role in technology and computer science. 0 and 1 can represent off and on, yes and no, etc. Computers, Calculators, Microwave, and other electronic equipments use binary. Binary number system is easy for machines, but, human beings find it very difficult, because it requires so many digits to represent a number in the binary system. For example, the number 213 takes only three digits (2, 1, and 3) to write in the decimal system, yet takes eight digits to write in the binary system (11010101).

All input to the computer and other electronic equipments is converted into binary numbers made up of 0’s and 1’s. But, decimal is what humans are accustomed to, so, the computer and other electronic equipments turn the binary into decimals and display information in a form that humans can easily understand.

The binary system is a positional number system. Each digit in a binary number has a value dependent on its position in the number. A digit's value is the digit multiplied by a power of two according to its position in the number.

For example,

Consider the binary number 10101 (should be read as "one zero one zero one" and not as "ten thousand one hundred one").

In the binary number 10101:

The first 1 on the left is in the sixteen’s place and its value is 1 × 16 = 16.

The first 0 on the left is in the eight’s place and its value is 0 × 8 = 0.

The second 1 is in the four’s place and its value is 1 × 4 = 4.

The second 0 is in the two’s place and its value is 0 × 2 = 0.

The last 1 on the rightmost is in the ones place and its value is 1 × (2 to the zero power) = 1 × (2^0) = 1 × (1) = 1.

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

For example, the decimal equivalent of the binary number 10101 (we just discussed) is 16 + 0 + 4 + 0 + 1 = 21.

To convert from decimal to binary:

Step 1: Divide the decimal number by 2.

Step 2: Record the remainder (0 or 1).

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 binary equivalent of 23.

23/2 = 11 --- Remainder 1

11/2 = 5 --- Remainder 1

5/2 = 2 --- Remainder 1

2/2 = 1 --- Remainder 0

1/2 = 0 --- Remainder 1

The sequence of remainders going up gives the answer. So, the binary equivalent of the decimal 23 is 10111.

Just as the place values in the decimal system increase by powers of 10, the place values in the binary system increase by powers of 2 from right to left (…32, 16, 8, 4, 2,…). That is, from right to left a binary system has a ones place, a two’s place, a four’s place, an eight’s place, a sixteen’s place etc.

For example, in the binary number 11,

The 1 on the left is in the two’s place and its value is 1 × 2 = 2.

The 1 on the right is in the ones place and its value is 1 × 2^0 = 1 × 1 = 1.

Observe that the value of the 1 on the left is twice as large as that of the 1 on the right.

Let’s now talk about binary fractions.

A binary fraction is the same as a decimal fraction, but with the base of 2 instead of 10.

In a binary fraction, a binary point separates the integer part of a binary number from its fractional part. The binary point indicates the place where values change from positive to negative powers of 2. Each place to the left of the binary point represents a positive power of 2 (2, 4, 8, 16, 32, and so on) and each place to the right of the binary point represents a negative power of 2 [2^(-1), 2^(-2), 2^(-3), 2^(-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:

2^ (-1) = 1/ (2^1) = 1/2

2^ (-2) = 1/ (2^2) = 1/4

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

The place values to the right of the binary point are one-half, one-fourth, one-eighth and so on. Negative powers of 2 are small numbers, less than 1 and greater than 0.

For example,

The binary number 11.011 represents 1 x 2^ (1) + 1 x 2^ (0) + 0 x 2^ (-1) + 1 x 2^ (-2) + 1 x 2^ (-3)

= 1 x 2 + 1 x 1 + 0 x 1/ (2^1) + 1 x 1/ (2^2) + 1 x 1/ (2^3)

= 1 x 2 + 1 x 1 + 0 x 1/2 + 1 x 1/4 + 1 x 1/8

= 2 + 1 + 0 + 1/4 + 1/8

= 3 + 3/8

= 3 + 0.375

= 3.375

Converting a decimal fraction to a binary fraction requires more steps.

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