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**Binary Values and Number Systems**

Chapter 2 Binary Values and Number Systems

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**Chapter Goals Distinguish among categories of numbers**

Describe positional notation Convert numbers in other bases to base 10 Convert base-10 numbers to numbers in other bases Describe the relationship between bases 2, 8, and 16 Explain the importance to computing of bases that are powers of 2 24 6

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**Numbers Natural Numbers Negative Numbers**

Zero and any number obtained by repeatedly adding one to it. 0, 1, 2 , 3, 4 Examples: 100, 0, 45645, 32 Negative Numbers A value less than 0, with a – sign Examples: -24, -1, , -32 2

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**Numbers Integers Rational Numbers**

A natural number, a negative number, zero Examples: 249, 0, , - 32 Rational Numbers An integer or the quotient of two integers Examples: -249, -1, 0, 3/7, -2/5 3

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**Natural Numbers How many ones are there in 642? 600 + 40 + 2 ?**

Or is it ? Or maybe… ? 4

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**Natural Numbers Aha! 642 is 600 + 40 + 2 in BASE 10**

The base of a number determines the number of digits and the value of digit positions When we count by 10s we are using the Base 10 numbering system – that is the Decimal system 5

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**Different Counting Systems**

BASE – Specifies the number of digits used in the system Always begins with 0 When you add 1 more, then, you go to the next position 9 + 1 = 10 Base 2: Two digits, 0 and 1 Base 8: 8 digits; 0, 1, 2, 3, 4, 5, 6, 7 Base 10: 10 digits 0 - 9 Base 16: 16 digits 0 – 9, A-F

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**Counting Systems in Binary/Octal/Decimal**

These are not equivalent. They are just showing the counting upwards by 1 unit

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**Positional Notation 642 in base 10 positional notation is:**

Continuing with our example… 642 in base 10 positional notation is: 6 x 102 = 6 x = 600 + 4 x 101 = 4 x = 40 + 2 x 10º = 2 x = = 642 in base 10 0 is the first position The power indicates the position of the number This number is in base 10 6

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**Positional Notation dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R + d1**

R is the base of the number As a formula: dn * Rn-1 + dn-1 * Rn d2 * R + d1 n is the number of digits in the number d is the digit in the ith position in the number 101 = 1 100 = 0 642 is (63 * 102) + (42 * 101) + (21 * 100) 7

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**Positional Notation What if 642 has the base of 13? Convert backwards**

642 in base 13 is equivalent to 1068 in base 10 They represent the same number of items or units. Numbers represent the items in various ways. + 6 x 132 = 6 x = 1014 + 4 x 131 = 4 x = + 2 x 13º = 2 x = = in base 10 8 6

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**Binary Decimal is base 10 and has 10 digits: 0,1,2,3,4,5,6,7,8,9**

Binary is base 2 and has 2 digits: 0,1 For a number to exist in a given base, it can only contain the digits in that base, which range from 0 up to (but not including) the base. 9

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Bases Higher than 10 How are digits in bases higher than 10 represented? With distinct symbols for 10 and above. Base 16 has 16 digits - HEXADECIMAL: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F So you get numbers like 00FF22 Often used to represent colors in web programs Red – Green – Blue – 000, to FFFFFF 10

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**Bases What bases can these numbers be in?**

122 198 178 G1A4 Always start at 0, end at base – 1 So, 0-9 means base 10 (10-1=9) Identify the number of possible units in the first position, to give you the base number!

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OCTAL SYSTEM – Base 8

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**Octal Octal is base 8 Numeric representation can be from 0-7**

Then, go to the next position

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**Converting Octal to Decimal**

What is the decimal equivalent of the octal number 642? Remember this is Base 8! 6 x 82 = 6 x 64 = 384 + 4 x 81 = 4 x 8 = 32 + 2 x 8º = 2 x 1 = st position = 418 in base 10 11

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**Converting Decimal to Other Bases**

Algorithm for converting number in base 10 to other bases While (the quotient is not zero – so there is a remainder) Divide the decimal number by the new base Make the remainder the next digit to the left in the answer Replace the original decimal number with the quotient 19

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**Converting Decimal to Octal**

What is 1988 (base 10) in base 8? 64 4 Answer is : Backwards 80

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**Converting Decimal to Octal**

Try some! web/mathematics/real/Calculators/ BaseConv_calc_1.htm Still a challenge? Try a calculator!

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Check it out You can use the calculator to convert

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**Using Calculator for Converting**

Put the number in Dec first

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**Using Calculator for Converting**

Click the new format, Oct

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**Apple Programmer Mode You can see the conversion for binary live**

Press 10 Enter 1988 You can see the conversion for binary live

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BINARY SYSTEM – Base 2

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**Binary Numbers and Computers**

Computers have storage units called binary digits or bits Low Voltage = 0 High Voltage = all bits have 0 or 1 22

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**Binary and Computers Byte 8 bits are in 1 byte**

The number of bits in a word determines the word length of the computer, but it is usually a multiple of 8. WHY? Store & process information better/faster. 32-bit machines 64-bit machines etc. 23

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Binary Values are 0 or 1 When you get both, then go to the next position

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**Converting Binary to Decimal**

What is the decimal equivalent of the binary number ? (8 digits) 1 x 27 = 1 x 128= 128 + 1 x 26 = 1 x 64 = 64 + 1 x 25 = 1 x 32 = 32 + 1 x 24 = 1 x 16 = 16 + 1 x 23 = 1 x 8 = 8 + 1 x 22 = 1 x 4 = 4 + 1 x 21 = 1 x 2 = 2 + 1 x 2º = 1 x 1 = 1 = 256 in base 10 13

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**Converting Binary to Decimal**

What is the decimal equivalent of the binary number ? (I added the 0 in the first digit as a placeholder) 0 x 27 = 0 x 64 = 0 1 x 26 = 1 x 64 = 64 + 1 x 25 = 1 x 32 = 32 + 0 x 24 = 0 x 16 = 0 + 1 x 23 = 1 x 8 = 8 + 1 x 22 = 1 x 4 = 4 + 1 x 21 = 1 x 2 = 2 + 0 x 2º = 0 x 1 = 0 = 110 in base 10 13

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Arithmetic in Binary Remember that there are only 2 digits in binary, 0 and 1 – There are ONLY 3 options 0 + 0 = 0 1 + 0 = 0 1 + 1 = 0 with a carry (there is no 2) Carry Values 14

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**Subtracting Binary Numbers**

Remember borrowing? Apply that concept here. 1 2 2 0 2 0-0=0 1-0=1 1-1=0 0-1= borrow 2 15

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**Converting Binary to Decimal**

What is the decimal equivalent of the binary number ? 1 x 27 = 1 x 128= 128 + 1 x 26 = 1 x 64 = 64 + 1 x 25 = 1 x 32 = 32 + 1 x 24 = 1 x 16 = 16 + 1 x 23 = 1 x 8 = 8 + 1 x 22 = 1 x 4 = 4 + 1 x 21 = 1 x 2 = 2 + 1 x 2º = 1 x 1 = 1 = 256 in base 10 13

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**Subtracting Binary Numbers**

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**Special Binary Numbers**

Often you will see numbers expressed in 8 digits for many computer and network applications – , The lowest is The highest would be What are the decimal equivalents? = 0 = 255 = =

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**Converting Binary to Octal**

Mark groups of 3 (from right) = Convert each group to base 9 is 253 in base 8 + 1 x 22 = 1 x 4 = 4 + 0 x 21 = 0 x 2 = 0 + 1 x 2º = 1 x 1 = 1 17

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**Converting Binary to Decimal**

What is the decimal equivalent of the binary number ? 1 x 26 = 1 x 64 = 64 + 1 x 25 = 1 x 32 = 32 + 0 x 24 = 0 x 16 = 0 + 1 x 23 = 1 x 8 = 8 + 1 x 22 = 1 x 4 = 4 + 1 x 21 = 1 x 2 = 2 + 0 x 2º = 0 x 1 = 0 = 110 in base 10 13

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HEXADECIMAL SYSTEM

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**Converting Hexadecimal to Decimal**

What is the decimal equivalent of the hexadecimal number DEF? D x 162 = 13 x 256 = 3328 + E x 161 = 14 x 16 = + F x 16º = 15 x = = in base 10 Remember, the digits in base 16 are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F A=10, B=11, C=12, D=13, E=14, F=15

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**Converting Decimal to Hexadecimal**

What is 3567 (base 10) in base 16? 32 15 Backwards D E F 21

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**Converting Binary to Hexadecimal**

Mark groups of 4 (from right) = Convert each group A B is AB in base 16 18

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**Converting Binary to Hexadecimal**

Convert to decimal first, then to hexadec 1010 = = 10 = A 1011 = = 11 = B A B is AB in base 16 18

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CONVERTING SYSTEMS

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**On the exam, you will have to manually convert between:**

Review On the exam, you will have to manually convert between: Decimal Binary Octal Hexadecimal

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**Ethical Issues – Tenth Strand**

What is Iron Man CC2013? What is a Knowledge Unit? What is Curricula 2001? Who put together the standards? Why are these standards important?

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