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Computer science deals with the basic structure of a computer and is more theoretical. As a result, it is more flexible in terms of specialization, with an emphasis on math and science. When, computers had just been introduced, computer science degrees were conducted within the math department. Since computers have become an essential part of society, the branch degree has since expanded, but the focus remains unchanged. 

Some of the most common programs offered in computer science include programming, numerical analysis, algorithms and data structures, computer systems principles, the software process, and theory of computers.  
Computer engineering deals with computer systems and understanding the most practical approach to computer development and use. Although, computer engineers focus on the software, a computer engineer is also required to be familiar with the hardware.

 Additionaly, Computer engineering combines electrical engineering and science, with a focus on software-hardware interactions. Some of the most common programs offered in computer engineering include processor interfacing, digital logical design, thermodynamics, power management, solid state physics, and magnetic fields. Degrees in computer engineering are highly customizable, as programs often cover a wide range of interests. Ultimately you are in control of your own degree as you focus on your interests, this customizable degree will propel your profession into your desired career direction.