# Where does binary arithmetic/manipulation enter the mathematics/engineering curriculum?

Binary arithmetic is both an educational basis for elementary logic and an pervasive tool for practical mechanics in managing computer systems (at a very particular level). That is the state of affairs. And the history of it is well known: Leibniz introduced it (in 1703), Claude Shannon introduced it as the mathematics behind circuit design in the 1937 (yes, neither invented the concepts, but instead introduced or popularized the notions).

Except I don't know how individuals are presented the material, if at all, in the modern school and university curricula.

In my (very possibly dated) experience, often in late elementary school (in the US system, 4th or 5th grade), a section is taught on alternate number systems (Mayan, Babylonian, binary, maybe even Roman). Basically all that is presented are some different ways of writing digits, and you learn how to write a number and that's about as far as it goes, no binary addition.

It is fairly common practice now to have computer classes in high school, but only to the extent of teaching the simplest of programming.

In the university setting, it is assumed as a matter of course that binary, and even hexadecimal, manipulation is understood, with no mention of the basics. The students deal with it with no problem at all (as far as I can tell).

My questions are:

• Does the above description match others (in detail, or in the general observation that binary is never explicitly taught)? That is, in the US secondary/tertiary system, is binary arithmetic taught at a particular stage? If not, is it considered so elementary?
• Is it taught in the school system in other countries, and if so at what stage, and with what extent?
• Additional question: Is it taught at all in university classes? (in my experience not in Discrete Math or in elementary computer engineering)
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There is no "US system" of education the way there is e.g. a Japanese system or a Korean system. The US federal government has no control over course curricula at any level. This is left to the states, who often set only guidelines (the details are left up to individual school boards). So it is difficult, if not misleading, to say what the "US system" does or did. But in the 1960s-80s at least it was quite common for grade school children to learn a bit of "arbitrary base" arithmetic as they learned base 10. This is less common now but many high school CS classes do cover binary. – leslie townes May 9 '12 at 1:46

## 2 Answers

As a teacher and tutor who's often introduced binary to students with good results, I found this question intriguing, so I did a little research on the US (I can't speak for other countries).

The closest thing to a nationwide US mathematics K-12 curriculum is the common core state standard for mathematics (also see the Wikipedia article for background)

Guess what -- neither "binary" nor "base two" appears in the standard at any grade level!

So I looked at the ACM Model Curriculum for K-12 Computer Science (The ACM is the major professional and academic society for Computer Science, roughly equivalent to the AMA for Mathematics). In that curriculum, binary is introduced in Level II, which is grades 9-10.

Despite all this, it does appear that binary is taught in many state and local mathematics school curricula. As I said, I have found that students pick up the concepts of binary nicely even at early grade levels, and it helps them gain perspective on a number of conceptual issues in school mathematics. I also use it as an entry point to simple digital logic (gates, adders, etc) and I find many students absolutely love that material, especially brighter ones with a bent toward math -- makes the abstract stuff come alive.

That said, there is a ton of crucial topics that is typically covered in school mathematics curricula, and I can understand why binary is farther down the list. It is too bad, though.

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About "other countries": This is about the situation in Rheinland-Pfalz, Germany (other states may differ). It's from memory and about ~15 years out of date, but I don't think much has changed now.

I was taught binary in connection with different number systems in, I think, 6th grade. We may have been taught binary addition as well. Other number systems like octal were mentioned, but not elaborated upon.

On the other hand, starting from 11th grade, there are elective computer science classes that cover more than just the basics of programming. Part of the curriculum included a somewhat more thorough treatment of binary arithmetic - addition was definitely covered, and two's complement was mentioned, in connection with a short course on Boolean logic and digital circuits.

The computer science classes in university did assume some familarity with the binary system, but binary arithmetic was taught from the ground up.

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