# Why we need different number systems ?

• string1037
In summary, there are multiple number systems because they are more optimal for specific purposes. For example, time and date uses different bases to relate to the revolutions of the Earth and around the sun, and computers use octal and hexadecimal representations to make it easier to deal with collections of bits. The use of binary in present day computers also makes it easier to convert to octal or hexadecimal.
string1037
why there are so many number systems when we can only use anyone of them ?
What's the need of octal and hexadecimal ?

string1037 said:
why there are so many number systems when we can only use anyone of them ?
What's the need of octal and hexadecimal ?

They are easier to convert to binary.

string1037 said:
why there are so many number systems when we can only use anyone of them ?
What's the need of octal and hexadecimal ?

Because some are more optimal than others for a specific purpose.

Here are few examples:

1. Time and Date

Time and date uses multiple bases to express time and date in relation to the amount of time it takes the Earth to revolve around the sun, and also with respect to the time it takes for the Earth to make a complete revolution on its own rotation (around its axis).

In terms of seconds up to days, we define a second to be a standard measure (something to do with the cesium atom, or in terms of the speed of light, roughly the amount of time it takes light to travel from here to the moon).

Now we use 60 seconds for minute, 60 minutes for an hour and 24 hours for a day.

On top of this we use 365 days for a standard year or an approximation of using 7 days for a week and 52 weeks for a year.

Notice that for measuring a year in terms of seconds we have a 365:24:60:60 representation: a mixture of different basis that are optimized for a specific purpose of relating time and date to the revolutions of our Earth and around our sun.

If we used base 10, then we would probably get some weird number and it wouldn't make as much sense as our current format.

2. Computers

If you ever do a computer degree (Computer Science, Computer Engineering, Electrical Engineering etc), you'll end up learning about logical gates. You'll also learn that using these gates you can do everything from arithmetic computations to logical computations and so on. The atomic version of information is the bit, and based on this we use any kind of analysis in the context of computer science and engineering to refer to bits or collections of bits.

Octal and hexadecimal representations just make it easier to deal with collections of bits. Instead of reading a sequence of 32 1's and 0's, it's a lot easier to read 8 hexadecimal digits.

These are just two examples, and I'm sure that there are more out there.

Thank you sir. Its very helpful for me.

The crucial point is that present day computers are "binary". That arose from the fact that, rather than carefully controlling currents in computers to give 10 distinct values, it was far easier to have just "on" and "off" giving two values. "on" corresponds to 1 and "off" corresponds to 0.

Because of that numbers are stored in digital computers as "binary numbers". The number 27, for example, can be written as 1(16)+ 1(8)+ 0(4)+ 1(2)+ 1= 110112 and then stored on 5 "flip-flops". Because 8 and 16 are powers of two, it happens to be easy to convert binary to octal or hexidedimal- and humans find it easier to read them- especially hexidecimal.

8= 23= 10002 so we can separate 11011 as 11 011 and say 11= 3, 011= 3 and so 2710= 338. Similarly, 16= 24= 100002 so we can separate 11011 as 1 1011. 1011 is "11" or B in hexidecimal so 2710= 110112= 1B16.

"There are 10 kinds of people, those who understand binary and those who don't."

## 1. Why do we need different number systems?

The use of different number systems allows us to represent and manipulate different quantities and values. For example, the decimal system is useful for everyday counting and calculations, while binary is essential for computer programming and data storage.

## 2. How many number systems are there?

There are infinite number systems, but the most commonly used in mathematics and computer science are decimal, binary, octal, and hexadecimal.

## 3. What are the advantages of using different number systems?

Different number systems have different strengths and weaknesses. For example, binary is useful for representing and manipulating data in computers, while hexadecimal is used for compactly representing large numbers. Having multiple number systems allows us to choose the most efficient one for a specific task.

## 4. Can all numbers be represented in every number system?

No, not all numbers can be represented in every number system. For example, decimal numbers with repeating decimals cannot be accurately represented in binary. Similarly, binary numbers cannot represent fractions with denominators other than powers of 2.

## 5. How do we convert between different number systems?

To convert between number systems, we use the concept of place value. Each digit in a number represents a different power of the base, and we can use this to convert from one system to another. There are specific algorithms and techniques for converting between different number systems, such as the division method for converting from decimal to binary.

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