Contents
 To convert a binary number to decimal:
Example: Convert 100110_{2} to a base 10 (or decimal) number Rule 1: Write the place values from the right hand side Rule 2: Write each digit under its place value
Rule 3: Multiply each digit by its place value (2^{5}x1) + (2^{4}x0) + (2^{3}x0) + (2^{2}x1) + (2^{1}x1) + (2^{0}x0) Rule 4: Add up the products. The answer will be the decimal number in base 10 32+0+0+4+2+0 =38_{10} Converting binary numbers to decimal numbers Fractional numbersA decimal number which has both an integral and fractional part is called a real number. The weight of the integral part of a real number increases from right to left in factors of 10 while that of the fractional part decreases from left to right factors of 10^{x} Rules are the same as above however the place values are negated on digits with decimal Example: Convert 100110.01_{2} to a base 10 (or decimal) number
Convert 100110.01_{2} to a base 10 (or decimal) number Rule 1: Write the place values from the right hand side Rule 2: Write each digit under its place value
Rule 3: Multiply each digit by its place value (2^{5}x1) + (2^{4}x0) + (2^{3}x0) + (2^{2}x1) + (2^{1}x1) + (2^{0}x0) + (2^{1}x1) + (2^{2}x1) Rule 4: Add up the products. The answer will be the decimal number in base 10 32+0+0+4+2+0+1/2+1/4 =38.75_{10} There are two methods of converting decimal numbers to binary. i.e
The decimal number is continuously divided by 2. However at each level of the division, the remainder which is either a 1 or a 0 is written to the right of the quotient. Reading of remainder digits from bottom to top makes the binary equivalent of the number. Example: Using the long division method, convert 891_{10}
into binary 891_{10 }= 1101111011_{2} The place value methodProcedure: Write down the place values in factors of 2 up to the value immediately larger or equal to the number being considered. For example; To convert 247_{10} into binary, we write down the
place values up to 2^{8} i.e. 256. Similarly, to convert 258_{10},
write down the place values up to 2^{9} i.e. 512. If the number being
considered is itself a factor of 64, 128, 256 etc. then place values should be
written up to the number itself.
To covert 891_{10}
to binary, start from the left. Subtract the place value from the number being converted. If the difference is a positive number or a 0, place a 1 in the binary digit row. If the difference is negative, place a zero.
The binary equivalent of 891_{10} is 01101111011_{2} Converting decimal numbers to binary numbers with decimal places
Convert 69.75_{10} into binary Procedure:
Convert 69.75_{10} into binary Procedure: 1. First convert the whole part 2. then the fractional part 3. combine First convert the whole part then the fractional part Since the product of the last step of the fractional part is zero or keeps repeating itself, we stop. combine 69_{10}=1000101_{2} 0.75_{10}=0.11_{2} Thus 69.75_{10}= 1000101.11_{2} For perfection, do as many questions as you can
1> Convert the following base two numbers into denary(base10) numbers a) 0101_{2} b) 1111_{2} c) 10101101110_{2} d) 10111111_{2} e) 1011001_{2} f) 111000111_{2} g) 1101_{2} h) 101_{2} i) 001_{2} j) 1001_{2} k) 1000_{2} l) 00101_{2} m) 1000000_{2}
2> Using the place value add long division methods convert each of the following base 10 numbers to their binary equivalents. a) 10_{10} b) 43_{10} c) 365_{10} d) 512_{10} e) 143_{10} f) 954_{10} 3> What are the binary equivalents of the following? a) 0.4_{10} b) 0.5_{10} c) 0.75_{10} 4> By working out the whole numbers and fractions separately, convert the following: a) 2.4_{10} b) 5.75_{10} c) 4.5_{10} d) 10.625_{10} e) 34.450_{10} f) 2.500_{10} g) 5.1625_{10} h) 7.1875_{10} i) 69.350_{10}
5> Get the denary equivalents of the following: a) 0.11_{2} b) 0.101_{2} c) 0.0001_{2} d) 11.01_{2} e) 101.100_{2} f) 100.001_{2} g) 0.10011_{2} h) 0.0010_{2} i) 0.10101_{2} j) 11.0110_{2} k) 101.11110_{2} l) 100.110_{2}

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