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: 6 thousands3 hundreds7 tens4 ones 6000+ 300+ 70+ 4 6  1000+ 3  100+ 7  10 + 4 6  10 3 + 3  10 2 + 7  10 1 + 4 The last row would be called the base-ten expanded 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  10 4 + 2  10 3 + 3  10 2 + 5  10 0 b) 37.924 = 3  10 + 7  1 + 9  (1/10) + 2  (1/100) + 4  (1/1000) = 3  10 1 + 7  10 0 + 9  10 -1 + 2  10 -2 + 4  10 -3 Write each of the numbers below in standard notation. a)6  10 5 + 1  10 2 + 4  10 1 + 5  10 0 = 600,000 + 100 + 40 + 5 = 600,145 b) 7  10 3 + 3  10 0 + 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. 100101 2 = 37 Base 2Base 10Dienes BlocksLight Bulbs 000 2 001 2 010 2 011 2 100 2 101 2 110 2 111 2 0 1 2 3 4 5 6 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): 2 0 (1) 2 1 (2)2 (4) 2 3 (8) 2 4 (16) 2 5 (32) Change the base 2 number 110011 2 to a base 10 (decimal) number. 110011 2 1  1 = 1 1  2 = 2 0  4 = 0 0  8 = 0 1  16 = 16 1  32 = 32 51 Change the base 10 (decimal) number 47 to a base 2 (binary) number. 47  2 = 23 remainder 1 23  2 = 11 remainder1 11  2 = 5 remainder1 5  2 = 2 remainder1 2  2 = 1 remainder0 1  2 = 0 remainder1 47 = 101111 2 Binary Point

6. Converting Fractional Parts of Numbers 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. digits Repeating Base 2 binary digits Subtract each side Solve for x Move binary point 5 places Move binary point 2 places