Binary, Hexadecimal and Octal number system - TI-Basic Developer TI-Basic Developer L₁ L₂ L₃ L₄ L₅ L₆ ≅ ℕ º L₁ L₂ L₃ L₄ L₅ L₆ ≅ ℕ º L₁ L₂ L₃ L₄ L₅ L₆ ≅ ℕ º pop out Binary, Hexadecimal and Octal number system Binary, hexadecimal, and octal refer to different number systems. The one that we typically use is called decimal. These number systems refer to the number of symbols used to represent numbers. In the decimal system, we use ten different symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. With these ten symbols, we can represent any quantity. For example, if we see a 2, then we know that there is two of something. For example, this sentence has 2 periods on the end.. When we run out of symbols, we go to the next digit placement. To represent one higher than 9, we use 10 meaning one unit of ten and zero units of one.

This may seem elementary, but it is crucial to understand our default number system if you want to understand other number systems. For example, when we consider a binary system which only uses two symbols, 0 and 1, when we run out of symbols, we need to go to the next digit placement. So, we would count in binary 0, 1, 10, 11, 100, 101, and so on. This article will discuss the binary, hexadecimal, and octal number systems in more detail and explain their uses. End How a Number System Works Number systems are used to describe the quantity of something or represent certain information. Because of this, I can say that the word "calculator" contains ten letters. Our number system, the decimal system, uses ten symbols. Therefore, decimal is said to be Base Ten. By describing systems with bases, we can gain an understanding of how that particular system works. When we count in Base Ten, we count starting with zero and going up to nine in order.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, … Once we reach the last symbol, we create a new placement in front of the first and count that up. 8, 9, 10, 11, 12, … , 19, 20, … This continues when we run out of symbols for that placement. So, after 99, we go to 100. The placement of a symbol indicates how much it is worth. Each additional placement is an additional power of 10. Consider the number of 2853. We know this number is quite large, for example, if it pertains to the number of apples in a basket. That's a lot of apples. How do we know it is large? We look at the number of digits. Each additional placement is an additional power of 10, as stated above. Consider this chart.

103 8192+3328+176+7 11703 As you can see, placements in hexadecimal are worth a whole lot more than in any of the other three number systems. Conversion It is important to know that 364 in octal is not equal to the normal 364. This is just like how a 10 in binary is certainly not 10 in decimal. 10 in binary (this will be written as 102 from now on) is equal to 2. 108 is equal to 8. How on earth do we know this? What is 20C.38F16, and how do we find out? Here is why it is important to understand how the number systems work. By using our powers of the base number, it becomes possible to turn any number to decimal and from decimal to any number. Base to Decimal So, we know that 3648 is not equal to the decimal 364. Then what is it? There is a simple method in converting from any base to the decimal base ten.

If you remember how we dissected the numbers above, we used powers, such as 24, and ended up with a number we understand. This is exactly what we do to convert from a base to decimal. We find out the true value of each digit according to their placement and add them together. In a formula, this algorithm looks like: (1) \begin{equation} V_{10} = v_pB^p+v_{p-1}B^{p-1}+...+v_1B+v_0 \end{equation} Where V10 is the decimal value, v is the digit in a placement, p is the placement from the right of the number assuming the rightmost placement is 0, and B is the starting base. Do not be daunted by the formula! We are going to go through this one step at a time. So, let us say we had the simple hexadecimal number 2B. We want to know what this number is in decimal so that we can understand it better. How do we do this? Let us use the formula above. Define every variable first. We want to find V10, so that is unknown.

The number 2B16 has two positions since it has two digits. p therefore is one less than that 1 , so p is 1. The number is in base 16, so B is 16. Finally, we want to know what v is, but there are multiple v's. You have v1 and v0. This refers to the value of the digit in the subscripted position. v1 refers to the digit in position one (the second digit from the right). So, v1 is 2. v0 is the first digit which is B. In the case of the conversion, you must convert all the letters to what they are in decimal. B is 11 in decimal, so v0 is 11.

Now, plug all this into the formula: (2) \begin{align} V_{10} = 2(16^1)+11(16^0) \\ V_{10} = 2(16)+11(1) \\ V_{10} = 32+11 \\ V_{10} = 43 \\ \end{align} Therefore, 2B16 is equal to 43. Now, let me explain how this works. Remember how digit placement affects the actual value? For example, in the decimal number 123, the "1" represents 100 which is 1*102. The "2" is 20, or 2*101. Likewise, in the number 2B16, the "2" is 2*161, and the B is 11*160. We can determine the value of numbers in this way. For the number 3648, we will make a chart that exposes the decimal value of each individual digit. Then, we can add them up so that we have the whole.

The number has three digits, so starting from the right, we have position 0, position 1, and position 2. Since this is base eight, we will use powers of 8. 82 4 Now, we add the values together to get 244. Therefore, 3648 is equal to 24410. In the same way that for 123, we say there is one group of 100, two groups of 10, and three groups of 1, for octal and the number 364, there are three groups of 64, six groups of 8, and four groups of 1. Decimal to Base Just like how we can convert from any base to decimal, it is possible to convert decimal to any base. Let us say that we want to represent the number 23610 in binary, octal, and hexadecimal. What we need to do is pretty much reverse whatever we did above. There isn't really a good formula for this, but there is an algorithm that you can follow which will help accomplish what we want.

(3) \begin{align} (1) \hspace{6pt} Let \hspace{4pt} P = \operatorname{int}(\sqrt[B]{V}) \\ (2) \hspace{6pt} Let \hspace{4pt} v = \operatorname{int}(V \div B^P) \\ (v \hspace{4pt} is \hspace{4pt} the \hspace{4pt} next \hspace{4pt} digit \hspace{4pt} to \hspace{4pt} the \hspace{4pt} right) \\ (3) \hspace{6pt} Make \hspace{4pt} V = V-vB^p \\ (4) \hspace{6pt} Repeat \hspace{4pt} steps \hspace{4pt} 1 \hspace{4pt} through \hspace{4pt} 3 \hspace{4pt} until \hspace{4pt} p=0 \\ \end{align} This algorithm may look confusing at first, but let us go through an example to see how it can be used.

We want to represent 236 in binary, octal, and hexadecimal. So, let's try getting it to binary first. The first step is to make p equal to $\operatorname{int}(\sqrt[B]{V})$. B is the base we want to convert to which is 2. The V is the number we want to convert, 236. Essentially, we are taking the square root of 236 and disregarding the decimal part. Doing this makes p become 7. Step two says to let v equal our number V divided by Bp. Bp is 27, or 128, and the integer part of 236 divided by 128 is 1. Therefore, our first digit on the left is 1. Now, we actually change V to become V minus the digit times the Bp. So, V will now be 236-128, or 108. We simply repeat the process until the p becomes a zero. When p becomes zero, we complete the steps a last time and then end. So, since V is now 108, p becomes 6.

108 divided by 26 is 1. The 1 goes to the right of the 1, so now we have 11. V becomes 44 since 108-64 is 44. How? Now you might be asking yourself how to read these numbers. Well, that’s not so difficult. First, I’ll give a general mathematical explanation which can be fit into one formula: (4) \begin{equation} V=vB^P \end{equation} In human language: the value of the cipher in the number is equal to the value of the cipher on its own multiplied by the base of the number system to the power of the position of the cipher from left to right in the number, starting at 0. Read that a few times and try to understand it. Thus, the value of a digit in binary doubles every time we move to the left. (see table below) From this follows that every hexadecimal cipher can be split up into 4 binary digits. In computer language: a nibble. Now take a look at the following table: Binary Numbers Binary Number System Binary Number System A Binary Number is made up of only 0s and 1s.

110100 Example of a Binary Number There is no 2, 3, 4, 5, 6, 7, 8 or 9 in Binary! A "bit" is a single binary digit. The number above has 6 bits. Binary numbers have many uses in mathematics and beyond. In fact the digital world uses binary digits . How do we Count using Binary? Binary But then there is no symbol for 2 ... what do we do? Well how do we count in Decimal? This is the last digit in Decimal So we start back at 0 again, but add 1 on the left The same thing is done in binary ... Now start back at 0 again, but add 1 on the left ••• When we run out of digits, we ... ... start back at 0 again, but add 1 on the left And that is what we do in binary ... Start back at 0 again, but add 1 on the left ••• start back at 0 again, and add one to the number on the left... ... but that number is already at 1 so it also goes back to 0 ... ...

and 1 is added to the next position on the left ••••• Start back at 0 again (for all 3 digits), add 1 on the left ••••••••• See how it is done in this little demonstration (press play button): Decimal vs Binary Here are some equivalent values: Decimal: Here are some larger equivalent values: Decimal: "Binary is as easy as 1, 10, 11." Now see how you can use Binary to count past 1,000 on your fingers: In the Decimal System there are Ones, Tens, Hundreds, etc In Binary there are Ones, Twos, Fours, etc, like this: This is 1×8 + 1×4 + 0×2 + 1 + 1×(1/2) + 0×(1/4) + 1×(1/8) = 13.625 in Decimal Numbers can be placed to the left or right of the point, to show values greater than one and less than one.

10.1 The number to the left of the point is a whole number (such as 10) As we move further left, every number place gets 2 times bigger. The first digit on the right means halves (1/2). As we move further right, every number place gets 2 times smaller (half as big). Example: 10.1 The "10" means 2 in decimal, The ".1" means half, You can do conversions at Binary to Decimal to Hexadecimal Converter . Words The word binary comes from "Bi-" meaning two. We see "bi-" in words such as "bicycle" (two wheels) or "binocular" (two eyes). When you say a binary number, pronounce each digit (example, the binary number "101" is spoken as "one zero one", or sometimes "one-oh-one"). This way people don't get confused with the decimal number. A single binary digit (like "0" or "1") is called a "bit". For example 11010 is five bits long.

The word bit is made up from the words "binary digit" How to Show that a Number is Binary To show that a number is a binary number, follow it with a little 2 like this: 1012 This way people won't think it is the decimal number "101" (one hundred and one). Examples Example: What is 11112 in Decimal? The "1" on the left is in the "2×2×2" position, so that means 1×2×2×2 (=8) The next "1" is in the "2×2" position, so that means 1×2×2 (=4) The next "1" is in the "2" position, so that means 1×2 (=2) The last "1" is in the ones position, so that means 1 Answer: 1111 = 8+4+2+1 = 15 in Decimal Example: What is 10012 in Decimal?

The "1" on the left is in the "2×2×2" position, so that means 1×2×2×2 (=8) The "0" is in the "2×2" position, so that means 0×2×2 (=0) The next "0" is in the "2" position, so that means 0×2 (=0) The last "1" is in the ones position, so that means 1 Answer: 1001 = 8+0+0+1 = 9 in Decimal Example: What is 1.12 in Decimal? The "1" on the left side is in the ones position, so that means 1. The 1 on the right side is in the "halves" position, so that means 1×(1/2) So, 1.1 is "1 and 1 half" = 1.5 in Decimal Example: What is 10.112 in Decimal?

The "1" is in the "2" position, so that means 1×2 (=2) The "0" is in the ones position, so that means 0 The "1" on the right of the point is in the "halves" position, so that means 1×(1/2) The last "1" on the right side is in the "quarters" position, so that means 1×(1/4) So, 10.11 is 2+0+1/2+1/4 = 2.75 in Decimal "There are 10 kinds of people in the world, those who understand binary numbers, and those who don't." Binary - learn.sparkfun.com Binary Share on Twitter Share on Facebook Pin It ABC's of 1's and 0's Prepare to have your perception of reality shattered. Everything you ever thought you knew about something so simple as numbers is going to change. All your life you thought having 100 somethings meant having this many: When, really, it only meant having this many: No, Rod Serling’s not about to cross into the frame, and you haven’t swallowed any red pills .

You’ve entered the binary zone and have just encountered base numbering systems. Number Systems and Bases Number systems are the methods we use to represent numbers. Since grade school, we’ve all been mostly operating within the comfy confines of a base-10 number system, but there are many others. Base-2, base-8, base-16, base-20, base…you get the point. There are an infinite variety of base-number systems out there, but only a few are especially important to electrical engineering. The really popular number systems even have their own name. Base-10, for example, is commonly referred to as the decimal number system. Base-2, which we’re here to talk about today, also goes by the moniker of binary. Another popular numeral system, base-16, is called hexadecimal. The base of a number is often represented by a subscripted integer trailing a value. So in the introduction above, the first image would actually be 10010 somethings while the second image would be 1002 somethings. This is a handy way to specify a number’s base when there’s ever any possibility of ambiguity. Why Binary? Why binary you ask?

Well, why decimal? We’ve been using decimal forever and have mostly taken for granted the reason we settled on the base-10 number system for our everyday number needs. Maybe it’s because we have 10 fingers, or maybe it’s just because the Romans forced it upon their ancient subjugates. Regardless of what lead to it, tricks we’ve learned along the way have solidified base-10’s place in our heart; everyone can count by 10’s. We even round large numbers to the nearest multiple of 10. We’re obsessed with 10! Computers and electronics are rather limited in the finger-and-toe department. At the lowest level], they really only have two ways to represent the state of anything: ON or OFF, high or low, 1 or 0. And so, almost all electronics rely on a base-2 number system to store, manipulate, and math numbers. The heavy reliance electronics place on binary numbers means it’s important to know how the base-2 number system works. You’ll commonly encounter binary, or its cousins, like hexadecimal, all over computer programs. Analysis of Digital logic circuits and other very low-level electronics also requires heavy use of binary.

In this tutorial, you’ll find that anything you can do to a decimal number can also be done to a binary number. Some operations may be even easier to do on a binary number (though others can be more painful). We’ll cover all of that and more in this tutorial. Suggested Reading We’re reaching pretty deep into the concepts bin. A lot of this tutorial builds on mathematical concepts like addition, multiplication, division (including remainders), and exponents. No previous knowledge of electronics is required (except for knowing how the base-10 system works, which most people do), but we do observe how binary is used in Arduino programming, and knowing something about data types could come in handy. In addition, knowledge of digital logic will help to supplement all of this binary business. Counting and Converting The base of each number system is also called the radix. The radix of a decimal number is ten, and the radix of binary is two. The radix determines how many different symbols are required in order to flesh out a number system.

In our decimal number system we’ve got 10 numeral representations for values between nothing and ten somethings: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each of those symbols represents a very specific, standardized value. In binary we’re only allowed two symbols: 0 and 1. But using those two symbols we can create any number that a decimal system can. Counting in binary You can count in decimal endlessly, even in your sleep, but how would you count in binary? Zero and one in base-two should look pretty familiar: 0 and 1. From there things get decidedly binary. Remember that we’ve only got those two digits, so as we do in decimal, when we run out of symbols we’ve got to shift one column to the left, add a 1, and turn all of the digits to right to 0. So after 1 we get 10, then 11, then 100. Let’s start counting… Decimal 31 11111 Does that start to paint the picture? Let’s examine how we might convert from those binary numbers to decimal.

Converting binary to decimal There’s no one way to convert binary-to-decimal. We’ll outline two methods below, the more “mathy” method, and another that’s more visual . We’ll cover both, but if the first uses too much ugly terminology skip down to the second. Method 1 There’s a handy function we can use to convert any binary number to decimal: There are four important elements to that equation: an, an-1, a1, etc., are the digits of a number. These are the 0’s and 1’s you’re familiar with, but in binary they can only be 0 or 1. The position of a digit is also important to observe. The position starts at 0, on the right-most digit; this 1 or 0 is the least-significant. Every digit you move to the left increases in significance, and also increases the position by 1. The length of a binary number is given by the value of n, actually it’s n+1. For example, a binary number like 101 has a length of 3, something larger, like 10011110 has a length of 8.

Each digit is multiplied by a weight: the 2n, 2n-1, 21, etc. The right-most weight - 20 equates to 1, move one digit to the left and the weight becomes 2, then 4, 8, 16, 32, 64, 128, 256,… and on and on. Powers of two are of great importance to binary, they quickly become very familiar. Let’s get rid of those n’s and exponents, and carry out our binary positional notation equation out eight positions: Taking that further, let’s plug in some values for the digits. What if you had a binary number like: 10011011? That would mean an values of: For the sake of this tutorial, assume that the right-most value is always the least-significant. The least-significant digit in a number is the digit that has the smallest influence on a number’s ultimate value. The significance of the digits is arbitrary - part of a convention called endianness.

A binary number can be either big-endian, where the most-significant digit is the left-most, or little-endian which we’ll use in this tutorial (and you’ll usually see binary numbers written this way). Now, plug those digits into our binary to decimal equation. Because our number is little-endian the least-significant value should be multiplied by the smallest weight. 1 * 27 + 0 * 26 + 0 * 25 + 1 * 24 + 1 * 23 + 0 * 22 + 1 * 21 + 1 * 20 And we can simplify it down to find our decimal number: = 1 * 128 + 0 * 64 + 0 * 32 + 1 * 16 + 1 * 8 + 0 * 4 + 1 * 2 + 1 * 1 = 128 + 16 + 8 + 2 + 1 = 155 You’ll quickly notice that it takes many more digits to represent a number in binary than it does in decimal, but it’s all done with just two digits!

Method 2 Another, more visual way to convert binary numbers to decimal is to begin by sorting each 1 and 0 into a bin. Each bin has a successive power of two weight to it, the 1, 2, 4, 8, 16,… we’re used to. Carrying it out eight places would look something like this: 128 And then add up any remaining weights to get your number! Converting from decimal to binary Just like going from binary to decimal, there’s more than one way to convert decimal to binary. The first uses division and remainders, and the second uses subtraction. Try both, or stick to one you’re comfortable with! Method 1 It isn’t quite as simple to convert a decimal number to binary. This conversion requires repeatedly dividing the decimal number by 2, until you’ve reduced it to zero. Every time you divide the remainder of the division becomes a digit in the binary number you’re creating. Don’t remember how to do remainders? If it’s been a while, remember that, since we’re dividing by two, if the dividend is even, the remainder will be 0; an odd dividend means a remainder of 1.

For example, to convert 155 to binary you’d go through this process: 155 ÷ 2 = 77 R 1 (That’s the right-most digit, 1st position) 77 ÷ 2 = 38 R 1 (2nd position) 38 ÷ 2 = 19 R 0 (3rd position) 19 ÷ 2 = 9 R 1 9 ÷ 2 = 4 R 1 4 ÷ 2 = 2 R 0 2 ÷ 2 = 1 R 0 1 ÷ 2 = 0 R 1 (8th position) The first remainder is the least-significant (right-most) digit, so read from top-to-bottom to flesh out our binary number right-to-left: 10011011. Match it up to the example above…that’s a bingo! Method 2 If dividing and finding remainders isn’t your thing, there may be an easier method for converting decimal to binary. Start by finding the largest power of two that’s still smaller than your decimal number, and subtract it from the decimal.

Then, continue to subtract by the largest possible power of two until you get to zero. Every weight-position that was subtracted, gets a binary 1 digit; digits that weren’t subtracted get a 0. Continuing with our example, 155 can be subtracted by 128, producing 27: 155 - 128 = 27 We’ve got a binary number! Conversion calculators Fortunately, there are tons of binary-to-decimal and vice-versa calculators out there. You shouldn’t often need to resort to long-hand conversion. Here! Plug some numbers into these boxes, and the other should magically change. Binary: Decimal: Make sure you only put 1’s or 0’s in the binary box, and 0-9 in the decimal one. Have fun! Bits, Nibbles, and Bytes In discussing the make of a binary number, we briefly covered the length of the number. The length of a binary number is the amount of 1’s and 0’s it has. Common binary number lengths Binary values are often grouped into a common length of 1’s and 0’s, this number of digits is called the length of a number.

Common bit-lengths of binary numbers include bits, nibbles, and bytes (hungry yet?). Each 1 or 0 in a binary number is called a bit. From there, a group of 4 bits is called a nibble, and 8-bits makes a byte. Bytes are a pretty common buzzword when working in binary. Processors are all built to work with a set length of bits, which is usually this length is a multiple of a byte: 8, 16, 32, 64, etc. To sum it up: Byte 10110101 Word is another length buzzword that gets thrown out from time-to-time. Word is much less yummy sounding and much more ambiguous. The length of a word is usually dependent on the architecture of a processor. It could be 16-bits, 32, 64, or even more. Padding with leading zeros You might see binary values represented in bytes (or more), even if making a number 8-bits-long requires adding leading zeros. Leading zeros are one or more 0’s added to the left of the most-significant 1-bit in a number.

You usually don’t see leading zeros in a decimal number: 007 doesn’t tell you any more about the value of a number 7 (it might say something else). Leading zeros aren’t required on binary values, but they do help present information about the bit-length of a number. For example, you may see the number 1 printed as 00000001, just to tell you we’re working within the realm of a byte. Both numbers represent the same value, however, the number with seven 0’s in front adds information about the bit-length of a value. Bitwise Operators There are a number of ways to manipulate binary values. Just as you can with decimal numbers, you can perform standard mathematical operations - addition, subtraction, multiplication, division - on binary values (which we’ll cover on the next page). You can also manipulate individual bits of a binary value using bitwise operators. Bitwise operators perform functions bit-by-bit on either one or two full binary numbers. They make use of boolean logic operating on a group of binary symbols. These bitwise operators are widely used throughout both electronics and programming. Complement (NOT) The complement of a binary value is like finding the exact opposite of everything about it.

The complement function looks at a number and turns every 1 into a 0 and every 0 becomes a 1. The complement operator is also called NOT. For example to find the complement of 10110101: NOT 10110101 (decimal 181) -------- = 01001010 (decimal 74) NOT is the only bitwise operator which only operates on a single binary value. OR OR takes two numbers and produces the union of them. Here’s the process to OR two binary numbers together: line up each number so the bits match up, then compare each of their bits that share a position. For each bit comparison, if either or both bits are 1, the value of the result at that bit-position is 1. If both values have a 0 at that position, the result also gets a 0 at that position.

The four possible OR combinations, and their outcome are: 0 OR 0 = 0 1 OR 0 = 1 1 OR 1 = 1 For example to find the 10011010 OR 01000110, line up each of the numbers bit-by-bit. If either or both numbers has a 1 in a column, the result value has a 1 there too: 10011010 OR 01000110 -------- = 11011110 Think of the OR operation as binary addition, without a carry-over. 0 plus 0 is 0, but 1 plus anything will be 1. AND AND takes two numbers and produces the conjunction of them. AND will only produce a 1 if both of the values it’s operating on are also 1. The process of AND'ing two binary values together is similar to that of OR. Line up each number so the bits match up, then compare each of their bits that share a position. For each bit comparison, if either or both bits are 0, the value of the result at that bit-position is 0.

If both values have a 1 at that position, the result also gets a 1 at that position. The four possible AND combinations, and their outcome are: 0 AND 0 = 0 1 AND 0 = 0 1 AND 1 = 1 For example, to find the value of 10011010 AND 01000110, start by lining up each value. The result of each bit-position will only be 1 if both bits in that column are also 1. 10011010 AND 01000110 -------- = 00000010 Think of AND as kind of like multiplication. Whenever you multiply by 0 the result will also be 0. XOR XOR is the exclusive OR. XOR behaves like regular OR, except it’ll only produce a 1 if either one or the other numbers has a 1 in that bit-position.

The four possible XOR combinations, and their outcome are: 0 XOR 0 = 0 For example, to find the result of 10011010 XOR 01000110: 10011010 XOR 01000110 -------- = 11011100 Notice the 2nd bit, a 0 resulting from two 1’s XOR’ed together. Bit shifts Bit shifts aren’t necessarily a bitwise operator like those listed above, but they are a handy tool in manipulating a single binary value. There are two components to a bit shift - the direction and the amount of bits to shift. You can shift a number either to the left or right, and you can shift by one bit or many bits. When shifting to the right, one or more of the least-significant bits (on the right-side of the number) just get cut off, shifted into the infinite nothing. Leading zeros can be added to keep the bit-length the same.

For example, shifting 10011010 to the right two bits: RIGHT-SHIFT-2 10011010 (decimal 154) -------- = 00100110 (decimal 38) Shifting to the left adds pushes all of the bits toward the most-significant side (the left-side) of the number. For each shift, a zero is added in the least-significant-bit position. For example, shifting 10011010 to the left one bit: LEFT-SHIFT-1 10011010 (decimal 154) -------- = 100110100 (decimal 308) That simple bit shift actually performs a relatively complicated mathematical function. Shifts to the left n bits multiplies a number by 2n (see how the last example multiplied the input by two?), while a shift n bits to the right will do an integer divide by 2n.

Shifting to the right to divide can get weird - any fractions produced by the shift division will be chopped off, which is why 154 shifted right twice equals 38 instead of 154/4=38.5. Bit shifts can be a powerfully fast way to divide or multiply by 2, 4, 8, etc. These bitwise operators provide us most of the tools necessary to do standard mathematical operations on binary numbers. Binary in Programming At their very lowest-level, binary is what drives all electronics. As such, encountering binary in computer programming is inevitable. Representing binary values in a program In Arduino, and most other programming languages, a binary number can be represented by a 0b preceding the binary number. Without that 0b the number will just be a decimal number.

For example, these two numbers in code would produce two very different values: a = 0b01101010; // Decimal 106 c = 01101010; // Decimal 1,101,010 - no 0b prefix means decimal Bitwise operators in programming Each of the bitwise operators discussed a few pages ago can be performed in a programming language. AND bitwise operator To AND two different binary values, use the ampersand, &, operator. For example: x = 0b10011010 & 0b01000110; // x would equal 0b00000010 AND’ing a binary value is useful if you need to apply a bit-mask to a value, or check if a specific bit in a binary number is 1. The AND bitwise operator shouldn’t be confused with the AND conditional operation , which uses the double-ampersand (&&) and produces a true or false based on the input of multiple logic statements. OR bitwise operator The OR bitwise operator is the pipe | (shift+\, the key below backspace).

For example: y = 0b10011010 | 0b01000110; // y would equal 0b11011110 OR’ing a binary value is useful if you want to set one or more bits in a number to be 1. As with AND, make sure you don’t switch up the OR bitwise operator with the OR conditional operator - the double-pipe (||). NOT bitwise operator The bitwise NOT operator is the tilde ~ (shift+`, the key above tab). As an example: z = ~(0b10100110); // z would equal 0b01011001 XOR bitwise operator To XOR two values use the caret (^) between them: r = 0b10011010 ^ 0b01000110; // r would equal 0b11011100 XOR is useful for checking if bits are different, because it’ll only result in a 1 if it operates on both a 0 or 1.

Shifting left and right To shift a binary number left or right n bits, use the <<n or >>n operators. A couple examples: i = 0b10100101 << 4; // Shift i left 4 bits // i would equal 0b101001010000 j = 0b10010010 >> 2; // Shift j right 2 bits // j would equal 0b00100100 Shift’s are an especially efficient way to multiply or divide by powers of two. In the example above, shifting four units to the left multiplies that value by 24 (16). The second example, shifting two bits to the right, would divide that number by 22 (4). Resources and Going Further Binary is the building block of all computations, calculations, and operations in electronics. So there are many places to go from here. Are you interested in learning more about binary, and other important numeral systems? Binary Blaster Hookup Guide – If you’re looking to practice your binary conversions, the Binary Blaster Kit can be a great help. This is a soldering kit.

Once you’ve put it together, use it to test your knowledge of how binary, hex, and decimal are related. Hexadecimal - Learn about this base-16 numeral system and how it relates to binary and decimal. Now that you can convert between decimal and binary, you can apply that knowledge to understanding how characters are encoded universally: Triadic Numbers Triadic Numbers Introduction Triadic numbers have digits with values -1, 0, and +1. The centrality of zero is possible with (and only with) any odd-numbered base. In bases 5, 7, 9, etc. the sides "weigh more" than the center. In base 5, for instance, there are two negative values, two positive values, and zero by itself at the center. With base 3, the "mass" of the center is equal to the "mass" of its left and right (- and +) sides. Each of the three parts (-, 0, and +) holds an equal fraction of the whole. Bases other than ten We are accustomed to using and thinking with base ten, though other numeric bases are part of our every-day world as well.

We have a dozen and a gross (base twelve), minutes and seconds (base sixty, whether for time or angular measure). The computer's binary (base two) numbers and the derivative octal and hexadecimal have made the concept familiar to others. In any base the number of distinct digits is equal to the base. Base two uses zero and one, base ten uses zero through nine. In numbers with two or more digits, the digits to the left of the initial digit indicate multiples of the base to some power. In base two, two is expressed as 10-one times two (to the first power) plus zero times one. Four is 100 (two to the second power), eight is 1000 (two to the third power). In any base x, the value x is expressed as 10-one times x plus zero times one. The number 100 represents the base to the second power, 1000 is the base to the third power. Digits to the right of the units digit (separated by a "base point") represent fractions. Base Two: Binary (base 2) numbers have only two digits, 0 and 1.

Left from the "binary point" are the one's, two's, four's, eight's, etc. positions. Going to the right of the "binary point" are the half's, fourth's, eighth's, etc. The binary number 1101.101 represents 8 + 4 + 0 + 1 + 1/2 + 0/4 + 1/8, or 13.625 (1*10 + 3 + 6/10 + 2/100 + 5/1000) in decimal form. A triadic base, centered on zero A variation on the usual way of expressing numbers works only in odd number bases. That is to have digits with negative value, as well as digits with positive value. In base three, the digits would have values one, zero, and minus one. For lack of a better notation, I'll use the symbols [ + 0 - ], respectively. Using these digits, there is no need for a minus sign; its purpose is implicit in one of the digits.

Numbers in this version of base three have the same one's, three's, and nine's places (three to the power zero, one, and two) as with the "usual" base three. Thirteen is expressed +++ (nine plus three plus one). Negative thirteen is expressed ---. Seven is expressed +-+ (nine minus three plus one). One property of this form of number (with the range of digit values centered on zero) that differs from conventional number systems is that truncating a number is identical to rounding it. If we lop off the fractional part of a decimal number, getting, for instance, 12, the unknown digits might represent a number closer to 13 (as with 12.6 or 12.95) or to 12 (as with 12.1 or 12.495). We have no way of knowing. The same sort of uncertainty occurs at any level of precision. That doesn't happen with the new scheme. The number +.---...

(1 - 1/3 - 1/9 - 1/27 - ...) when extended to any finite number of places will never get as small as 1/2, though one can get as close to 1/2 as one wants by using enough digits. Likewise, the number +.+++... (1 + 1/3 + 1/9 + 1/27 + ...) never gets as large as 1 1/2, though, again, one can get arbitrarily close with enough digits. A zero-centered, auto-rounding base 3 number system In the May, 1993 issue of Dramatic University, Anthony Blake made brief mention of a triadic computer in his talk (p. 36, first paragraph). I proposed to Tony that a number system such as what I have just discussed might be right for a triadic computer. If a complex calculation were carried out using this number system, each component calculation could be carried out concurrently, feeding into the final result. Components having less effect on the overall result could be calculated to less precision than the more important components. Perhaps nature works this way.

A puzzle solved using a base 3 number system I applied the above form of base 3 to a math puzzle discussed by Ken Pledge in the May, 1993 issue of Dramatic University (what he refers to as a 'reversed notation' on pages 23-25). The puzzle gives you a balance scale and some coins, from which you are to distinguish which coin is counterfeit and whether it is heavy or light. And, you're to use as few weighings with the scale as possible. If the scale balances, you know all the coins on the scale are of the proper weight. If it does not, then either a coin on the heavy side is too heavy, or a coin on the light side is too light. Either could cause the imbalance. As I re-phrased the problem: If you have 13 coins, of which no more than one is counterfeit (too heavy or too light), and a 14th coin, known to be authentic, how many weighings will it take to determine if a coin is bad, and if so which, and whether it is heavy or light.

There are 27 possible outcomes (all coins OK, coin one to thirteen light, coin one to thirteen heavy). Three uses of the scale produces 3*3, or 27, results - making it theoretically possible to solve the above problem in three weighings. A hint to solving the puzzle: arrange the digits expressing one to thirteen in three columns. Each column (9's, 3's, 1's) determines what gets weighed at one time. Each row determines whether a given coin will be weighed, and if so, on which side of the scale. To see the rest of the solution, go to the answer page . Octal and Hexadecimal Numeration : Numeration Systems - Electronics Textbook Octal and Hexadecimal Numeration Octal and Hexadecimal Numeration Chapter 1 - Numeration Systems Because binary numeration requires so many bits to represent relatively small numbers compared to the economy of the decimal system, analyzing the numerical states inside of digital electronic circuitry can be a tedious task.

Computer programmers who design sequences of number codes instructing a computer what to do would have a very difficult task if they were forced to work with nothing but long strings of 1’s and 0’s, the “native language” of any digital circuit. To make it easier for human engineers, technicians, and programmers to “speak” this language of the digital world, other systems of place-weighted numeration have been made which are very easy to convert to and from binary. One of those numeration systems is called octal, because it is a place-weighted system with a base of eight. Valid ciphers include the symbols 0, 1, 2, 3, 4, 5, 6, and 7. Each place weight differs from the one next to it by a factor of eight. Another system is called hexadecimal, because it is a place-weighted system with a base of sixteen. Valid ciphers include the normal decimal symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, plus six alphabetical characters A, B, C, D, E, and F, to make a total of sixteen.

As you might have guessed already, each place weight differs from the one before it by a factor of sixteen.

Let’s count again from zero to twenty using decimal, binary, octal, and hexadecimal to contrast these systems of numeration: Number Decimal Binary Octal Hexadecimal ------ ------- ------- ----- ----------- Zero 0 0 0 0 One 1 1 1 1 Two 2 10 2 2 Three 3 11 3 3 Four 4 100 4 4 Five 5 101 5 5 Six 6 110 6 6 Seven 7 111 7 7 Eight 8 1000 10 8 Nine 9 1001 11 9 Ten 10 1010 12 A Eleven 11 1011 13 B Twelve 12 1100 14 C Thirteen 13 1101 15 D Fourteen 14 1110 16 E Fifteen 15 1111 17 F Sixteen 16 10000 20 10 Seventeen 17 10001 

21 11 Eighteen 18 10010 22 12 Nineteen 19 10011 23 13 Twenty 20 10100 24 14 Octal and hexadecimal numeration systems would be pointless if not for their ability to be easily converted to and from binary notation.

Their primary purpose in being is to serve as a “shorthand” method of denoting a number represented electronically in binary form. Because the bases of octal (eight) and hexadecimal (sixteen) are even multiples of binary’s base (two), binary bits can be grouped together and directly converted to or from their respective octal or hexadecimal digits. With octal, the binary bits are grouped in three’s (because 23 = 8), and with hexadecimal, the binary bits are grouped in four’s (because 24 = 16): BINARY TO OCTAL CONVERSION Convert 10110111.12 to octal: . . implied zero implied zeros . | || . 010 110 111 100 Convert each group of bits ### ### ### . ### to its octal equivalent: 2 6 7 4 . Answer: 10110111.12 = 267.48 We had to group the bits in three’s, from the binary point left, and from the binary point right, adding (implied) zeros as necessary to make complete 3-bit groups.

Each octal digit was translated from the 3-bit binary groups. Binary-to-Hexadecimal conversion is much the same: BINARY TO HEXADECIMAL CONVERSION Convert 10110111.12 to hexadecimal: . . implied zeros . ||| . 1011 0111 1000 Convert each group of bits ---- ---- . ---- to its hexadecimal equivalent: B 7 8 . Answer: 10110111.12 = B7.816 Here we had to group the bits in four’s, from the binary point left, and from the binary point right, adding (implied) zeros as necessary to make complete 4-bit groups: Likewise, the conversion from either octal or hexadecimal to binary is done by taking each octal or hexadecimal digit and converting it to its equivalent binary (3 or 4 bit) group, then putting all the binary bit groups together. Incidentally, hexadecimal notation is more popular, because binary bit groupings in digital equipment are commonly multiples of eight (8, 16, 32, 64, and 128 bit), which are also multiples of 4.

Octal, being based on binary bit groups of 3, doesn’t work out evenly with those common bit group sizings. Base-n Concepts Decimal Notation (1.1) The decimal number system (called arabic numerals) is a positional number system. It only has symbols for the first ten values, including a symbol for zero. The only symbols in decimal are: 0 1 2 3 4 5 6 7 8 and 9. Contrast this with Roman Numerals where 1 has the symbol I, but 10 has the symbol X, and 100 has the symbol C, and 1000 has the symbol M. The reason that only ten symbols are needed (no matter how huge the number may be) is that the value contributed by a symbol depends on its position. The symbol is multiplied by its positional value. Thus in the decimal number, 3257, the 3 is multiplied by a thousand, the 2 is multiplied by a hundred, the 5 is multiplied by ten, and the 7 is multiplied by 1 (namely not changed at all).

Compare the decimal number 1111 with the Roman numeral MCXI and note that the decimal number only uses one symbol (1) whereas the Roman numeral requires four symbols(M,C,X,I). The positional values are powers of 10: one = 10 to zero-th power ten = 10 to first power hundred = 10 squared ten-thousand = 10 to fourth power etc IEC prefixes and symbols for binary multiples IEC prefixes and symbols for binary multiples At the dawn of computing and electronics, computer professionals have noticed that 210 is almost equal to 1000, and started loosely using SI prefix kilo to mean 1024. As time passed, and as computers were introduced into mainstream life, and as the amounts of data people were dealing with grew dramatically, a great deal of confusion arose around this. Prefixes kilo and mega became to mean different things in different contexts. For example when describing computer memory, manufacturers usually use “megabyte” to mean 220 = 1 048 576 bytes, while manufacturers of storage devices usually use the term to mean 106 = 1 000 000 bytes.

These two different meanings of the term can be found in many different areas of computing, and as if that is not enough confusion, a third “megabyte” of 1 024 000 bytes is the term used to format 90 mm (3½ inch), “1.44 MB” diskettes. As the numbers increase, so does the gap between the actual base‐two number and its SI prefix approximation: Differences between SI prefix approximations and actual base‐two numbers SI Prefix 1208925819614629174706176 21% Faced with this reality, in January 1997 the IEEE Standards Board has decided that IEEE standards will use the conventional, internationally adopted, definitions of the SI prefixes . Mega will mean 1 000 000, except that the base‐two definition may be used (if such usage is explicitly pointed out on a case‐by‐case basis) until such time that prefixes for binary multiples are adopted by an appropriate standards body.

In 1998 the 7th edition of the SI Brochure was published, which included a marginal note , explicitly stating that SI prefixes should not be used to indicate powers of 2. In December 1998 the International Electrotechnical Commission ( IEC ), the international organization for worldwide standardization in electrotechnology, approved as an IEC International Standard the names and symbols for prefixes for binary multiples for use in the fields of data processing and data transmission. The prefixes shall be used to indicate multiplication by 210n, where n = 1,2,3,4,5, or 6. The prefixes are as follows: IEC prefixes and symbols for binary multiples Number yottabinary (210)8 yotta: (103)8 Thus, a 90 mm (3½ inch) floppy disk should be described as having a capacity of 1.41 MiB (equivalent to 1.47 MB), not “1.44 MB” as labelled.

It is suggested that in English, the first syllable of the name of the binary‐multiple prefix should be pronounced in the same way as the first syllable of the name of the corresponding SI prefix , and that the second syllable should be pronounced as “bee”. As can be seen from the above table, the name of each new prefix is derived from the name of the corresponding SI prefix by retaining the first two letters of the name of the SI prefix and adding the letters “bi”, which recalls the word “binary”. Similarly, the symbol of each new prefix is derived from the symbol of the corresponding SI prefix by adding the letter ‘i’, which again recalls the word “binary”. (For consistency with the other prefixes for binary multiples, the symbol Ki is used for 210 rather than ki.) These prefixes for binary multiples, which were developed by the IEC Technical Committee (TC) 25, Quantities and Units, and Their Letter Symbols , with the strong support of CIPM , IEEE and ISO , were first adopted by the IEC as Amendment 2 to IEC International Standard IEC 60027‐2: Letter symbols to be used in electrical technology — Part 2: Telecommunications and electronics .

The full content of Amendment 2, which has a publication date of 1999‐01, is reflected in the table above and the suggestion regarding pronunciation. Subsequently the contents of this Amendment were incorporated in the second edition of IEC 60027‐2 , which has a publication date of 2000‐11 (the first edition was published in 1972). The complete citation for this revised standard is IEC 60027‐2, Second edition, 2000‐11, Letter symbols to be used in electrical technology — Part 2: Telecommunications and electronics . In 2001 the American NIST has published Special Publication 330, 2001 Edition (US version of the 7th edition of the SI Brochure ), in which editor included an extra marginal note with a description, stating that IEC standard prefixes should be used in the field of information technology to avoid the incorrect usage of the SI prefixes. After a much heated discussion in December 2001 on linux‐kernel mailing list , the binary prefixes have been accepted by the key Linux developers, and are now extensively gaining ground across Open Source UNIX applications.

On December 11 2002 the IEEE‐SA has approved proposal P1541 as a Trial‐Use Standard for the duration of 2 years. IEEE Standard No.: 1541‐2002, Name: IEEE Standard for Prefixes for Binary Multiples , ISBN: 0‐73813385‐X. It can be purchased at IEEE Online Store . The standard was prepared by Standards Coordinating Committee (SCC) 14, Quantities, Units, and Letter Symbols. BSR at ANSI has also accepted BSR/IEEE 1541‐200x, Trial‐Use Standard for Prefixes for Binary Multiples (trial use standard) for a trial period: 1 December, 2002 through 30 June, 2004. In April 2003 CIPM has held a CCU meeting at BIPM , at which it has been agreed to also include a marginal note discussing the binary multiples in the upcoming 8th edition of the SI Brochure .

On September 18 2003 the Reuters has reported that Apple , Dell , Gateway , Hewlett‐Packard , IBM , Sharp , Sony and Toshiba have been sued in a class‐action suit in Los Angeles Superior Court for “deceiving” the true capacity of their hard drives. This of course was due to ambiguity of “GB” when used by software and hardware vendors. This precedent might prompt Apple to adapt binary prefixes in its Mac OS , as well as other companies to put pressure on Microsoft to adapt them in its Windows operating systems. On November 10 2003 the W3C has published a Working Group Note Units in MathML in which it showed how IEC prefixes should be incorporated into mathematical markup. On Feburary 20 2004 another lawsuit emerged, this time against flash drive manufacturers On March 19 2005 the IEEE‐SA Standards Board Standards Review Committee has elevated status of the 1541‐2002 standard from trial‐use to full‐use and scheduled the standard for a maintenance action in 2007.

The ANSI has followed suit on April 29 2005 accepting BSR/IEEE 1541‐200x, Prefixes for Binary Multiples as a new standard. On August 11 2005 the IEC has published the third edition of IEC 60027‐2 and made an announcement about inclusion of prefixes zebi and yobi in the standard. In May 2006 the BIPM has published the 8th edition of SI brochure with an extended marginal note that describes IEC 60027‐2 standard prefixes and advocates their use in the field of information technology to avoid incorrect usage of SI prefixes. In June 2006 Western Digital Corporation has settled the class action lawsuit entitled Orin Safier v. Western Digital Corporation et al., Case No. 05-03353 BZ Binary Calculator Home / Math Calculators / Binary Calculator Binary Calculator Use the following calculators to perform the addition, subtraction, multiplication, or division of two binary values, convert from binary value to decimal value or vice versa. Please note that due to the limitation of computer precision.

This calculator can only take up to 32 bit of binary value or up to 10 digit decimal values. Binary Calculation—Add, Subtract, Multiply, or Divide Convert Binary Value to Decimal Value Binary Value: Convert Decimal Value to Binary Value Decimal Value: Hex Calculator | IP Subnet Calculator References The binary system is a numerical system that uses only two symbols, 0 and 1. Due to its ease of implementation in digital electronic circuitry using logic gates, all modern computers use the binary system internally. The following are some typical conversions between binary values and decimal values.

decimal 0 = 0 in binary decimal 1 = 1 in binary decimal 2 = 10 in binary decimal 3 = 2 + 1 = 11 in binary decimal 4 = 22 = 100 in binary decimal 7 = 22 + 2 + 1 = 111 in binary decimal 8 = 23 = 1000 in binary decimal 10 = 23 + 2 = 1010 in binary decimal 16 = 24 = 10000 in binary decimal 20 = 24 + 22 = 10100 in binary Binary Addition The addition of binaries is similar to the decimal system. The only different is to carry over when the result is 2. 0 + 0 = 0 Binary to Decimal Converter Binary to Decimal Converter In order to use this new binary to decimal converter tool, type any binary value like 1010 into the left field below, and then hit the Convert button. You can see the result in the right field below. It is possible to convert up to 63 binary characters to decimal.

Binary to decimal conversion result in base numbers Binary System Binary is the simplest kind of number system that uses only two digits of 0 and 1. By using these digits computational problems can be solved by machines because in digital electronics a transistor is used in two states. Those two states can be represented by 0 and 1. That is why this number system is the most preferred in modern computer engineer, networking and communication specialists, and other professionals. Decimal System Decimal number system is the most commonly used and the most familiar one to the general public. It is also known as Base 10 numbering system since it is based on 10 following symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. In decimal system, every digit has its own position as well as the decimal point. I.e. the number 356.74 has 4 in the Hundredths position, 7 in the Tenths position, 6 in the Units position, 5 in the Tens position, and 3 in the Hundreds position. Decimal number system is also one of the oldest known numeral system, which is historically related to Hindu-Arabic numeral system.

Binary to decimal conversion examples (10011)2 = (19)10

Question: The binary number symbols 100 equate to what conventional number?
Answer: Four