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Binary Conversion Binary Conversion In everyday life, we normally use a numbering system that is constructed on multiples of ten. We call this numbering system the Base-10 or decimal numbering system. Base-10 numbering systems dictate that the numbering scheme begins to repeat after the tenth digit (in our case, the number 9). When we count, we usually count "0, 1, 2, 3, 4, 5 , 6, 7, 8, 9, 10, 11, 12, ‌" There's more to the numbering scheme than just counting, though. In grade school, we all were taught that each digit to the left and right of the decimal point is given a name which identifies that digit's placeholder. For right now, let's just consider digits to the left of the decimal, or positive numbers. Remember that the first digit to the left of the decimal point is called the "ones" digit. It is followed by the "tens" digit, followed by the "hundreds", followed by the "thousands", and on and on. What they probably didn't tell you in grade school is that each placeholder (ones, tens, hundreds, thousands, etc.) actually represents a multiple of ten (remember – "Base-10"?). Each placeholder can be represented by an exponent of ten. For instance, the expression 100 represents the "ones" position, the expression 101 represents the "tens" position, the expression 102 represents the "hundreds" position and so on. We can begin to see this more clearly if we break down a number into exponents of ten. Let's take a look at the following number: 7408. Starting at the decimal point, we'll work our way left. Know More About :- Identity Property of Real Numbers

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Conversion to and from other numeral systems Decimal ;- To convert from a base-10 integer numeral to its base-2 (binary) equivalent, the number is divided by two, and the remainder is the least-significant bit. The (integer) result is again divided by two, its remainder is the next least significant bit. This process repeats until the quotient becomes zero. Conversion from base-2 to base-10 proceeds by applying the preceding algorithm, so to speak, in reverse. The bits of the binary number are used one by one, starting with the most significant (leftmost) bit. Beginning with the value 0, repeatedly double the prior value and add the next bit to produce the next value. This can be organized in a multi-column table. For example to convert 100101011012 to decimal: Prior value × 2 + Next Bit Next value 0 ×2+ 1=1 1 ×2+ 0=2 2 ×2+ 0=4 4 ×2+ 1=9 9 ×2+ 0 = 18 Hexadecimal :- Binary may be converted to and from hexadecimal somewhat more easily. This is because the radix of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 24, so it takes four digits of binary to represent one digit of hexadecimal, as shown in the table to the right. To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits: 3A16 = 0011 10102 E716 = 1110 01112 To convert a binary number into its hexadecimal equivalent, divide it into groups of four bits. If the number of bits isn't a multiple of four, simply insert extra 0 bits at the left (called padding). To convert a hexadecimal number into its decimal equivalent, multiply the decimal equivalent of each hexadecimal digit by the corresponding power of 16 and add the resulting values: C0E716 = (12 × 163) + (0 × 162) + (14 × 161) + (7 × 160) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,38310 Read More About :- Whole Numbers Basics

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Binary Conversion