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Binary Numbers

Binary Numbers. Converting Decimal to Binary Binary to Decimal. Base-Ten Place-Value System

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Binary Numbers

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  1. Binary Numbers Converting Decimal to Binary Binary to Decimal

  2. Base-Ten Place-Value System The sleek efficient number system we know today is called the base-ten number system or Hindu-Arabic system. It was first developed by the Hindus and Arabs. This used the best features from several of the systems we mentioned before. 1. A limited set of symbols (digits). This system uses only the 10 symbols:0,1,2,3,4,5,6,7,8,9. 2. Place Value. This system uses the meaning of the place values to be powers of 10. For example the number 6374 can be broken down (decomposed) as follows: The last row would be called the base-tenexpanded notation of the number 6374.

  3. Write each of the numbers below in expanded notation. • a) 82,305 = 810,000 + 21,000 + 3100 + 010 + 51 • = 8104 + 2103 + 3102 + 5100 • b) 37.924 = 310 + 71 + 9(1/10) + 2(1/100) + 4(1/1000) • = 3101 + 7100 + 910-1 + 210-2 + 410-3 • Write each of the numbers below in standard notation. • 6105 + 1102 + 4101 + 5100 = 600,000 + 100 + 40 + 5 = 600,145 • b) 7103 + 3100 + 210-2 + 810-3 = 7000 + 3 + .02 + .008 = 7003.028 Multiplying and Dividing by Powers of 10 If a number x is multiplied or divided by 10 this causes a “shift“ in the decimal point to the right (multiplication) or the left (division) since all powers of 10 are increased or decreased by 1. If x is multiplied or divided by a higher power of 10 then the decimal point is shifted by the same number of places as the power of 10.

  4. Binary Numbers Binary or Base 2 numbers are very important in today's technological world. They form the numerical representation of numbers in a computer or any digital device cell phone, ipod, DVD, etc. This is because a electronic device can best detect one of two states either electrical current is flowing or it is not. The light bulbs that are on represent the base 2 digit 1 and the ones that are off represent the base 2 digit 0. 1001012= 37 Base 2 Base 10 Dienes Blocks Light Bulbs 0002 0 1 0012 2 0102 0112 3 1002 4 1012 5 1102 6 1112 7

  5. Base Two The important details about base 2 are that the symbols that you use are 0 and 1. The place values in base 2 are (going from smallest to largest): Binary Point 2-5 () 25 (32) 24 (16) 23 (8) 22 (4) 21 (2) 20 (1) 2-1 () 2-2 () 2-3 () 2-4 () Change the base 2 number 1100112to a base 10 (decimal) number. Change the base 10 (decimal) number 47 to a base 2 (binary) number. 47  2 = 23 remainder 1 23  2 = 11 remainder 1 11  2 = 5 remainder 1 5  2 = 2 remainder 1 2  2 = 1 remainder 0 1  2 = 0 remainder 1 1100112 11 = 1 12 = 2 04 = 0 08 = 0 116 = 16 132 = 32 51 47 = 1011112

  6. Converting Fractional Parts of Numbers Find the first 5 binary digits of the fraction Convert to base 10. digits Multiplying and Dividing by Powers of 2 If a number x is multiplied or divided by 10 this causes a “shift“ in the binary point to the right (multiplication) or the left (division) since all powers of 10 are increased or decreased by 1.

  7. Repeating Base 2 binary digits Find the first 5 binary digits of the fraction Once we see that the fraction has occurred again after I multiplied by 2 this pattern will continue. This gives me the entire binary expansion of . This is what is referred to as a repeating binary number. digits Change to base 10. Move binary point 5 places Move binary point 2 places Subtract each side Solve for x

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