1. Binary, Hexadecimal, Decimal, and How They Relate.

2. Introduction When people think of binary, they often think of just a jumble of zeros and ones. But it’s actually a lot more complicated than that. And as for hexadecimal, a lot of people don’t even know what that is. And most people haven’t heard the term “decimal” since elementary school. But lucky for you, I’m here to explain what all that is.

3. Decimal and Number Systems Before I give you the really complicated stuff, I have to teach you the basics. The BASIC basics. Decimals aren’t just the digits to the right of the decimal point. They’re actually just any numbers in the base ten number system. Number systems are different systems used to interpret numbers different ways. The number system that most people use and that everybody learns in primary (elementary) school is called base ten, because there are ten different symbols that each digit can be (1, 2, 3, 4, 5, 6, 7, 8, 9, and 0). With any number system, the amount of symbols is specified in the name. In base two, there are two symbols that each digit could be. In base sixteen, there are sixteen different symbols that each digit could be. Some number systems have other names that are used more often. In this way, base ten is called decimal, base two is called binary, and base sixteen is called hexadecimal (or just “hex”). In this presentation, I will use text for decimal numbers, (one, two, three, four); numbers for binary, (1, 2, 3, 4); and I will have all the hexadecimal numbers in italics, (1, 2, 3, 4).

4. Binary Binary is the little-known language used by computers. It is made up of ones and zeroes, meaning on and off, respectively. As in decimal, zero is 0 and one is 1, but instead of two being 2, two is 10, three is 11, and so on. Every time all the digits are one, all the digits are returned to zero and a one is added on the end, so four is 100, five is 101, six is 110, seven is 111, eight is 1000, and so on. However, in more advanced binary, for all the decimal numbers up to two hundred fifty-five, there are eight digits in each number, while all the digits to the left of the left-most one are zeroes. For numbers above two hundred fifty-five, there are simply nine, ten eleven, or twelve digits, or however many you need. Therefore, two hundred fifty-six is 100000000 (nine digits), five hundred twelve is 1000000000 (ten digits), and one thousand twenty is 10000000000 (eleven digits).

5. The REAL Truth (Part One) Okay, I’ll admit it. That’s not ACTUALLY how it works. What ACTUALLY happens is that the digit just to the left of the decimal point represents that digit multiplied by 2 0 (1), the next digit to the left represents that digit multiplied by 2 1 (2), the digit after that represents that digit multiplied by 2 2 (4), and so on. Each time you go to the next digit to the left, the exponent increases by one.

6. The REAL Truth (Part Two) We can use this same principle to understand the decimal (base ten) number system. The digit just to the left of the decimal point represents that digit multiplied by 10 0 (1), the next digit to the left represents that digit multiplied by 10 1 (10), the next digit to the left represents that digit multiplied by 10 2 (100), and so on. Each time you go one digit to the left, the exponent increases by one, and every time you go one digit to the right, the exponent decreases by one. The digit directly to the right of the decimal point represents that digit multiplied by 10 -1 (0.1). Following the rule that when you go one digit to the right, the exponent decreases by one, the next digit to the right represents that digit multiplied by 10 -2 (0.01), and so on. Once you have figured out what each digit represents, you must add up all of the translated digits.

7. Visual Aid for the Previous Slide

8. Hexadecimal The hexadecimal (base sixteen) number system uses a principle very similar to the one we used previously to understand binary and decimal. The only difference is that “A” represents ten, “B” represents eleven, “C” represents twelve, “D” represents thirteen, “E” represents fourteen, and “F” represents fifteen. The digit just to the left of the decimal point represents that digit multiplied by 16 0 (1), the next digit to the left represents that digit multiplied by 16 1 (16), the digit after that represents that digit multiplied by 16 2 (256), and so on.

9. Hexadecimal Look at the picture. The digit directly to the left of the decimal point is a zero. Zero multiplied by 16 0 equals zero. The digit directly to the left of that is a “C”. “C” is equivalent to twelve. Twelve multiplied by 16 1 equals two hundred forty. The digit directly to the left of that is a one. One multiplied by 16 2 equals two hundred fifty-six. Zero plus two hundred forty plus two hundred fifty-six equals four hundred ninety-six. Therefore, “1C0” is equivalent to four hundred ninety-six.