Gain insights into essential number systems for computer scientists. Explore binary representation, and learn to represent and manipulate positive, negative, and fractional numbers stored in computers.

Get introduced to the number systems that are essential for computer scientists, and take a deep dive into binary representation. Learn how to represent and manipulate, positive, negative, and fractional numbers in binary, and learn how numbers are stored and represented in computers.

The only background this course requires is grade-school arithmetic. No programming knowledge is needed.

Number Systems For Computer Scientists

So far in this course we have looked at the decimal, binary, octal, and hexadecimal systems. We have seen how the properties of range and place values work for the base-10, base-2, base-8, and base-16 number systems. This might make us think that we could pick an arbitrary base and make our own number system. We wouldn't be wrong $-$ the aforementioned patterns can most certainly extend to *any* $n$-base number system! 

Let's see how, and generalize what we know about number systems to the rest of this lesson.

## Symbols

The decimal system has ten base symbols, the binary system has two, and the hexadecimal system has sixteen. We can recognize a pattern here.

>For an $n$-base number system, we would need $n$ base symbols. 

Each base symbol would need to inherently represent some value. In the case where we are borrowing base symbols from the decimal system, this value is apparent. 
> For example, the binary system uses the symbols $0$ and $1$, which have the values zero and one respectively. 

In other cases, we have to explicitly define the values of the symbols. 

>For example, the last six symbols of the hexadecimal system are the letters *A*, *B*, *C*, *D*, *E*, and *F*. These inherently don't have any value in the decimal system, so we have to explicitly define what they represent.

It is important to note that when we do end up borrowing symbols from the decimal system, we don't use the same pronunciation conventions we follow in the decimal system. For example, for the octal number $148$ we would not call it "one hundred and forty-eight." Instead we would say "one four eight." Similarly, $101$ in binary would be "one zero one" and not "one hundred and one."



Learn about the generalized properties of any n-base number system.

Introduction

Essential Number Systems

Some Binary Operations

Signed Binary Numbers

Representing Numbers With Fractional Parts

Representation in a Computer

Conclusion

N-Base Number System

An arbitrary number system