Number Systems – Definitionspage 214

The radix or base 밑 refers to the number b in an expression of the form bn.

The number n is called the exponent 지수and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n".

The term power 거듭제곱 strictly refers to the entire expression, but is sometimes used to refer to the exponent.

Representation of a number in a system with base (radix) N may only consist of digits that are less than N.

If(1) M = akNk + ak-1Nk-1 + ... + a1N1 + a0 with 0 ≤ ai

< N we have a representation of M in base N system and write

  M = (akak-1...a0)N If we rewrite (1) as

(2) M = a0 + N·(a1 + N·(a2 + N·...)) the algorithm for obtaining coefficients ai becomes more obvious. For example,

a0 ≡ M (mod N) and a1 ≡ (M/N) (mod N)

Place Value Decimal System1 2 3 . 4 5 6 7

Millions

Hundred Thousands

Ten ThousandsThousandsH

undredsTensO

nes

.Tenths

Hundredths

Thousandths

Ten thousandths

Hundred Thousandth

Millionths

Ten Millionths

106 105 104 103 102 101 100 10-1 10-2 10-3 10-4 10-5 10-6 10-7

100 20 3 0.4 0.05 0.006 0.0007

100 20 3104

1005

10006

100007

Decimal Numbers

Each digit in the decimal system we use everyday represents a number times a power of 10.

123.789 = 1*102 + 2*101 + 3*100 + 7*10-1 + 8*10-2 + 9*10-3

= 100 + 20 + 3 + 0.7 + 0.08 + 0.009We use the numbers 0 9 (10 numbers)

Place Value Binary 이진법 System

sixteens

eights

fours

twos

ones

.halves

fourths

eighths

sixteenths

24 23 22 21 20 2-1 2-2 2-3 2-4

Binary NumbersEach digit in the binary system represents a

number times a power of 2.1011.012

We use the numbers 0 1 (2 numbers)We place a 2 as a subscript (lower number) to

show it is a base other than base 10 or the decimal system.

1*23 + 0*22 + 1*21 + 1*20 + 0*2-1 + 1*2-2

8 + 0 + 2 + 1 + 0 + ¼ = 11.2510

1011.012 = 11.2510

Why Use Binary Numbers?For computers, binary numbers are great stuff

because: • They are simple to work with -- no big addition

tables and multiplication tables to learn, just do the same things over and over, very fast.

• They just use two values of voltage, magnetism, or other signal, which makes the hardware easier to design and more noise resistant.

Computer Basis for Binary:• Binary, base 2, is the way that computers "know"

numbers, much as we know our numbers in base 10, also called decimal notation. Computers use binary because it correlates well with electronic switching: Off = 0, and On = 1. For example, if you have two light switches in your room you can demonstrate four numbers:

Switch A Switch B Binary No. Decimal No.

Off Off 0 0

Off On 1 1

On Off 10 2

On On 11 3

Challenge questions: We can see that we can represent the numbers 0-3 (four numbers) with two switches. How many numbers do you think we can represent with three switches? What about four, or five?

Converting Decimal to Binary• Let D= the number we wish to convert from

decimal to binary • Find P, such that 2^P is the largest power of two

smaller than D. • Repeat until P<0 • If 2^P<=D then

– put 1 into column P – subtract 2^P from D

• Else – put 0 into column P

• End if • Subtract 1 from P

Example• Now that we have an algorithm, we can use it to convert numbers

from decimal to binary relatively painlessly. Let's try the number D=55.

• Our first step is to find P. We know that 2^4=16, 2^5=32, and 2^6=64. Therefore, P=5.

• 2^5<=55, so we put a 1 in the 2^5 column: 1-----. • Subtracting 55-32 leaves us with 23. Subtracting 1 from P gives us

4. • Following step 3 again, 2^4<=23, so we put a 1 in the 2^4 column:

11----. • Next, subtract 16 from 23, to get 7. Subtract 1 from P gives us 3. • 2^3>7, so we put a 0 in the 2^3 column: 110--- • Next, subtract 1 from P, which gives us 2. • 2^2<=7, so we put a 1 in the 2^2 column: 1101-- • Subtract 4 from 7 to get 3. Subtract 1 from P to get 1. • 2^1<=3, so we put a 1 in the 2^1 column: 11011- • Subtract 2 from 3 to get 1. Subtract 1 from P to get 0. • 2^0<=1, so we put a 1 in the 2^0 column: 110111 • Subtract 1 from 1 to get 0. Subtract 1 from P to get -1. • P is now less than zero, so we stop.

There are 2 methods: (A) Reverse of Binary-To-Digital Method4510 = 32 + 0 + 8 + 4 +0 + 1  = 25+0+23+22+0+20= 1 0 1 1 0 12 (B) Repeat Division

This method uses repeated division by 2. Eg. convert 2510 to binary

25 / 2 = 12+ remainder of 1 1 (Least Important Digit)

12 / 2 = 6 + remainder of 0 0

6 / 2 = 3 + remainder of 0 0

3 / 2 = 1 + remainder of 1 1

1 / 2 = 0 + remainder of 1 1(Most Important Digit)

Result 2510 = 1 1 0 0 12

Practice Converting Decimal to Binarypage 219

Convert the decimal number to binary:1. 22. 113. 384. 1305. 256

101011

100110

10000010

100000000

Adding Binary Numbers

To add binary numbers just add the columns. If a sum equals 1 or 0 write it, it if equals 2, write a 0 and carry a 1.

1 0 1 1+ 1 0 0 11 0 1 0 0

Practice Adding Binary Numberspage 220

1. 1 0 0 1 0 1 + 1 1 0 0 1 1

2. 1 0 0 1 1 1 + 1 0 1 0 1 0

3. 1 0 0 0 0 0 1 + 1 1 0 1 1 0

1 0 1 1 0 0 0

1 0 1 0 0 0 1

1 1 1 0 1 1 1

Subtracting Binary Numbers

To subtract binary numbers just subtract the columns. To borrow, subtract one from a column and add 2 to the column to the right.

1 1 1 0- 1 0 1

0 2

1 0 0 1

Practice Subtracting Binary Numberspage 221

1. 1 0 0 1 0 1 - 1 0 0 1 1

2. 1 0 0 1 1 1 - 1 1 0 1 0

3. 1 0 0 0 0 0 1 - 1 1 0 1 1 0

1 0 0 1 0

1 1 0 1

1 0 1 1

Multiplication is Simple

1 1 1 0 14 ● 1 0 1 ● 5 1 1 1 0 7010 = 10001102

0 0 0 0 0 1 1 1 0 0 01 0 0 0 1 1 0

Practice page 222

Other Number Systems

Binary-To-Octal / Octal-To-Binary ConversionEach Octal digit is represented by three bits

of binary digit.Example100 111 0102 = (100) (111) (010)2 = 4728 The same methods as Decimal to Binary work –

the two different processes

Binary-To-Hexadecimal /Hexadecimal-To-Binary Conversion

Hexadecimal Digit 0 1 2 3 4 5 6 7

Binary Equivalent 0000 0001 0010 0011 0100 0101 0110 0111

Hexadecimal Digit 8 9 A B C D E F

Binary Equivalent 1000 1001 1010 1011 1100 1101 1110 1111

Decimal (10) Hexadecimal(16) Octal (8) Binary (2)

0 0 0 01 1 1 12 2 2 103 3 3 114 4 4 1005 5 5 1016 6 6 1107 7 7 1118 8 10 10009 9 11 100110 A 12 101011 B 13 101112 C 14 110013 D 15 110114 E 16 111015 F 17 1111

Decimal Hexadecimal Octal Binary

16 10 20 10000

17 11 21 10001

18 12 22 10010

19 13 23 10011

20 14 24 10100

21 15 25 10101

22 16 26 10110

23 17 27 10111

24 18 30 11000

25 19 31 11001

26 1A 32 11010

27 1B 33 11011

28 1C 34 11100

29 1D 35 11101

31 1F 37 11111

Octal-To-Hexadecimal /Hexadecimal-To-Octal Conversion

1) Convert Octal (Hexadecimal) to Binary first.2a) To Octal - Regroup the binary number in 3 bits a group starts from the LSD .

2b) Convert to Hexadecimal - Regroup the binary number in 4 bits a group from the LSD .

(LSD = Least Significant Digit)

Practice

• page 228

page 228

1017313368

Pages 223-2251. 11101 27+2=29 =111012

2. 1100 7 + 5=12 =11002

3. 1100000 81+5=86 = 11000002

4. 10110110 175+7=182 =101101102

5. 11001 51 – 2 =49 =110012

6. 10010 23-5=18 =100102

7. 1000010 81-15=66 =10000102

8. 1101000 175-71=104 =11010002

9. 11000 12 x 2 =24 =110002

10. 1011111 19x5=95 =10111112

11. 100111011 45x7=315 =1001110112

12. 101100101 21x17=357 =1011001012

13. 101014. 1000115. 1111116. 10000117. 1000100