quit introduction roman numerals counting and arithmetic converting from base 10
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Introduction
Roman Numerals
Counting and Arithmetic
Converting from Base 10
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• We use a base 10 system with 10 digits, they are
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• This is the decimal place – value system.
437 = 4 × 102 + 3 × 101 + 7 × 100
10’s place100’s place 1’s place
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Roman NumeralsRoman Numerals
• There is no place-value
• The letters have fixed values
• They are ordered from largest to smallest
• If a letter representing a smaller value comes before a larger one it is subtracted
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• The letters should be arranged from largest to smallest.• 1510 is written MDX, largest to smallest
• Only powers of ten can be repeated. • Don’t repeat a letter more than three times in
a row.• 100 is written LL, not XXXXXXXXXX
Roman Numerals RulesRoman Numerals Rules
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• Numbers can be written using subtraction. A letter with a smaller value precedes one of the larger value. The smaller number is then subtracted from the larger number.• Only powers for ten (I, X, C, M) can be subtracted.• The smaller letter must be either the first letter or
preceded by a letter at least ten times greater than it.• CCXLIII = 100 + 100 + (50 – 10) + 1 + 1 + 1= 243
Roman Numerals RulesRoman Numerals Rules
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Write in Roman NumeralsWrite in Roman Numerals• 21• 32• 515• 900• 1005• 1954• 3592
XXIXXXIIDXVCMMVMCMLIVMMMDXCII
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A. 53B. 63
C. 113Click on the number that matches the Roman Numeral
LXIII
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OOPS! Try again!
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You are correct! LXIII = 50+10+3 = 63
L= 50; X=10; III=3
Remember:
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A. 624
B. 1624
C. 5524
DCXXIV
Click on the number that matches the Roman Numeral
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OOPS! Try again!
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You are correct! DCXXIV
= 500 + 100 + 20 + 4= 624
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A. 150B. 250
C. 550
CCL
Click on the number that matches the Roman Numeral
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OOPS! Try again!
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You are correct! CCL
= 100 + 100 + 50 = 250
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Roman vs. Indo-Arabic NumeralsRoman vs. Indo-Arabic Numerals
• Indo-Arabic Numerals are the numbers that we use today.• 1, 2, 3, 4, 5, 6, 7, 8, 9, 0
• Roman Numerals are used today, but not in everyday writing.• I, V, X, L, C, D, M
• Roman Numerals don’t have a symbol for zero.
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• Write down the two numbers you are adding right next to each other
• Rearrange the letters so they start with the largest and end with the smallest.
• Then start combining similar letters.• Check your answer by adding the Indo-Arabic
numbers.
Adding Roman NumeralsAdding Roman Numerals
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23 + 58
Adding Roman NumeralsAdding Roman Numerals
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Adding Roman NumeralsAdding Roman Numerals
Step 1. 23 + 58
Step 2. XXIII + LVIII
Step 3. XXIIILVIII
Step 4. LXXVIIIIII
Step 4. IIIIII = VI
Step 5. LXXVVI
Step 6. VV = X
Step 7. LXXXI = 81
Step 8. 23 + 58 = 81
Roman Numeral Number
I 1
V 5
X 10
L 50
C 100
D 500
M 1000
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• 1. 10 + 15• 2. 225 + 130• 3. 5 + 4• 4. 100 + 215• 5. 30 + 50• 6. 100 + 200
Adding Roman NumeralsAdding Roman Numerals
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Counting and ArithmeticCounting and Arithmetic• Decimal or base 10 number system
• Origin: counting on the fingers• “Digit” from the Latin word digitus meaning
“finger”• Base: the number of different digits including zero in
the number system• Example: Base 10 has 10 digits, 0 through 9
• Binary or base 2: 2 digits, 0 and 1 • Octal or base 8: 8 digits, 0 through 7• Hexadecimal or base 16: 16 digits, 0 – 9 and A – F
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Decimal, Binary, Octal, HexadecimalDecimal, Binary, Octal, Hexadecimal• Binary (base 2)
• The number system is used directly by computers
• Hexadecimal (base 16)
• The number system that is used by computers to communicate with programmers eg colouring of webpages
• Octal (base 8)
• The number system that is used by either human or by computers to communicate with programmers
• Decimal (base 10)
• The number system that we are using
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Decimal
14
15
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Hexadecimal
E
F
D
C
B
A
9
8
7
6
5
4
3
2
1
0
1110
1111
1101
1100
1011
1010
1001
1000
111
110
101
100
11
10
1
Binary
0
Octal
16
17
15
14
13
12
11
10
7
6
5
4
3
2
1
0
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On Off
+ –
True False
6V 0V
Yes No
1 0
North South
Why Binary?Why Binary?
• Early computer design used decimal• John von Neumann proposed binary
data processing (1945)• Simplified computer design• Used for both instructions and data
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Numbers: Physical RepresentationNumbers: Physical Representation
• Different numerals, same number of oranges• Cave dweller: IIIII• Roman: V• Arabic: 5
• Different bases, same number of oranges • 510
• 1012
• 114
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Number SystemNumber System• Roman: position independent• Modern: based on positional notation (place value)
• Decimal system: system of positional notation based on powers of 10.
• Binary system: system of positional notation based on powers of 2
• Octal system: system of positional notation based on powers of 8
• Hexadecimal system: system of positional notation based on powers of 16
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Positional Notation: Base 10Positional Notation: Base 10
Place 101 100
Value 10 1
Evaluate 4 × 10 3 × 1
Sum 40 3
1’s place10’s place
43 = 4 × 101 + 3 × 100
43
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Positional Notation: Base 10Positional Notation: Base 10
Place 102 101 100
Value 100 10 1
Evaluate 5 × 100 2 × 10 7 × 1
Sum 500 20 7
1’s place10’s place
527 = 5 × 102 + 2 × 101 + 7 × 100
100’s place
527
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Positional Notation:OctalPositional Notation:Octal6248
Place 82 81 80
Value 64 8 1
Evaluate 6 × 64 2 × 8 4 × 1
Sum 384 16 4
64’s place 8’s place 1’s place
= 40410
404
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Positional Notation: HexadecimalPositional Notation: Hexadecimal
6,70416
Place 163 162 161 160
Value 4,096 256 16 1
Evaluate 6 × 4,096 7 × 256 0 × 16 4 × 1
Sum 24,576 1,792 0 4
4,096’s place 256’s place 1’s place16’s place
= 26,37210
26372
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Positional Notation: HexadecimalPositional Notation: Hexadecimal
2B516
Place 163 162 161 160
Value 256 16 1
Evaluate 2 × 256 11 × 16 5 × 1
Sum 512 176 5
4,096’s place 256’s place 1’s place16’s place
= 69310
693
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Positional Notation: BinaryPositional Notation: Binary
Place 27 26 25 24 23 22 21 20
Value 128 64 32 16 8 4 2 1
Evaluate 1 × 128 1 × 64 0 × 32 1 × 16 0 × 8 1 × 4 1 × 2 0 × 1
Sum 128 64 0 16 0 4 2 0
110101102 = 21410
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Binary
Number
Equivalent Decimal
Number8’s (23) 4’s (22) 2’s (21) 1’s (20)
0 0 × 20 0
1 1 × 20 1
10 1 × 21 0 × 20 2
11 1 × 21 1 × 20 3
100 1 × 22 4
101 1 × 22 1 × 20 5
110 1 × 22 1 × 21 6
111 1 × 22 1 × 21 1 × 20 7
1000 1 × 23 8
1001 1 × 23 1 × 20 9
1010 1 × 23 1 × 21 10
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Converting from Base 10Converting from Base 10
Base8 7 6 5 4 3 2 1 0
2256 128 64 32 16 8 4 2 1
832,768 4,096 512 64 8 1
1665,536 4,096 256 16 1
Power
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0
1
0
64
6
Integer
Remainder
1101Binary
24816322
12345Base
2210
22 Base 1022 Base 10 to Base 2to Base 2
Power
22/16
6
6/8
6
6/4
2 0
2/2 0/1
0
= 101102
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22 Base 1022 Base 10 to Base 2to Base 2
0
( 112 )
10110Base 2
( 022 )
( 152 )
( 1 112 )
( 0222 )
Remainder
Quotient 22Base 10
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06 12345
0
164
42/32Integer
Remainder
10101Binary
24816322
Base
4210
42 Base 1042 Base 10 to Base 2to Base 2
10
Power
10/16
10
10/8
2
2/4
2 0
2/2 0/1
0
= 1010102
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42 Base 1042 Base 10 to Base 2to Base 2
1
( 022 )
101010Base 2
( 152 )
( 0102 )
( 1 212 )
( 0422 )
Remainder
Quotient 42Base 10
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1
Addition in BinaryAddition in Binary
1 0 1 0
1 0 1 0+
–––––––––01 0
Carry
1 0
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1
–––––
Multiplication in BinaryMultiplication in Binary
1 1
1 1
×
01+
1 1
1 1001––––––
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126 Base 10126 Base 10 to Base 8to Base 8
176Base 80
( 118 )
( 7 158 )
( 61268 )
Remainder
Quotient 126Base 10
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126 Base 10126 Base 10 to Base 16to Base 16
7EBase 16
0
( 7 78 )
( 12616 )
Remainder
Quotient 126Base 1014E
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