quit introduction roman numerals counting and arithmetic converting from base 10

Quit

Quit

Introduction

Roman Numerals

Counting and Arithmetic

Converting from Base 10

Quit

• 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

Quit

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

Quit

• 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

Quit

• 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

Quit

Write in Roman NumeralsWrite in Roman Numerals• 21• 32• 515• 900• 1005• 1954• 3592

XXIXXXIIDXVCMMVMCMLIVMMMDXCII

Quit

A. 53B. 63

C. 113Click on the number that matches the Roman Numeral

LXIII

Quit

OOPS! Try again!

Quit

You are correct! LXIII = 50+10+3 = 63

L= 50; X=10; III=3

Remember:

Quit

A. 624

B. 1624

C. 5524

DCXXIV

Click on the number that matches the Roman Numeral

Quit

OOPS! Try again!

Quit

You are correct! DCXXIV

= 500 + 100 + 20 + 4= 624

Quit

A. 150B. 250

C. 550

CCL

Click on the number that matches the Roman Numeral

Quit

OOPS! Try again!

Quit

You are correct! CCL

= 100 + 100 + 50 = 250

Quit

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.

Quit

• 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

Quit

23 + 58

Adding Roman NumeralsAdding Roman Numerals

Quit

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

Quit

• 1. 10 + 15• 2. 225 + 130• 3. 5 + 4• 4. 100 + 215• 5. 30 + 50• 6. 100 + 200

Adding Roman NumeralsAdding Roman Numerals

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

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

Quit

22 Base 1022 Base 10 to Base 2to Base 2

0

( 112 )

10110Base 2

( 022 )

( 152 )

( 1 112 )

( 0222 )

Remainder

Quotient 22Base 10

Quit

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

Quit

42 Base 1042 Base 10 to Base 2to Base 2

1

( 022 )

101010Base 2

( 152 )

( 0102 )

( 1 212 )

( 0422 )

Remainder

Quotient 42Base 10

Quit

1

Addition in BinaryAddition in Binary

1 0 1 0

1 0 1 0+

–––––––––01 0

Carry

1 0

Quit

1

–––––

Multiplication in BinaryMultiplication in Binary

1 1

1 1

×

01+

1 1

1 1001––––––

Quit

126 Base 10126 Base 10 to Base 8to Base 8

176Base 80

( 118 )

( 7 158 )

( 61268 )

Remainder

Quotient 126Base 10

Quit

126 Base 10126 Base 10 to Base 16to Base 16

7EBase 16

0

( 7 78 )

( 12616 )

Remainder

Quotient 126Base 1014E

Yes No

Do you want to end show?