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


Imagine you are a physicist, and you need to do some calculations. You want to multiply the speed of light, which is $300000000$, with the tiny number $0.0000001$. You can see that both of them take up a lot of space even in the decimal notation, and writing them out and multiplying them would be difficult. To tackle this problem in the decimal system, we use *scientific notation*.


To represent a decimal number in scientific notation, we set the decimal point to the right of the first non-zero digit in the number and set the power of $10$ accordingly.

> In scientific notation, $300000000$ is $3 \times 10^8$, and $0.0000001$ is $1 \times 10^{-7}$. To multiply them, we simply multiply the base numbers, i.e., $1 \times 3$, and then add the powers of $10$. Thus, our answer is $30$.

The method above using scientific notation makes the computation incredibly easy, and also provides a way to represent large numbers using fewer digits and less space, because we only need to store the **exponent** and the number before the multiplication sign (called the **mantissa**).

Space and speed of computation are both incredibly important in computers too; we generally have a limited amount of memory, and we want to get results much faster from computers than we can compute ourselves.

# What about binary numbers?

It definitely makes sense that we would want a system similar to scientific notation for binary numbers too. This is exactly what the **floating point** notation is. We will study it in detail in the rest of this chapter.

Learn why floating point representation is essential.

Introduction

Essential Number Systems

Some Binary Operations

Signed Binary Numbers

Representing Numbers With Fractional Parts

Representation in a Computer

Conclusion

Why Do We Need Floating Point Numbers?

Scientific notation