Introduction Roman Numerals Counting and Arithmetic Converting from Base 10

2. Introduction Roman Numerals Counting and Arithmetic Converting from Base 10

3. 100’s place 10’s place 1’s place  • 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

4. Roman 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

5. Roman Numerals Rules • 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

6. Roman Numerals Rules • 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

7. Write in Roman Numerals XXI • 21 • 32 • 515 • 900 • 1005 • 1954 • 3592 XXXII DXV CM MV MCMLIV MMMDXCII

8. LXIII A. 53 B. 63 C. 113 Click on the number that matches the Roman Numeral

9. OOPS! Try again!

10. You are correct! LXIII = 50+10+3 = 63 Remember: L= 50; X=10; III=3

11. DCXXIV A. 624 B. 1624 C. 5524 Click on the number that matches the Roman Numeral

12. OOPS! Try again!

13. You are correct! DCXXIV = 500 + 100 + 20 + 4 = 624

14. CCL A. 150 B. 250 C. 550 Click on the number that matches the Roman Numeral

15. OOPS! Try again!

16. You are correct! CCL = 100 + 100 + 50 = 250

17. Roman 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.

18. Adding Roman Numerals • 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.

19. Adding Roman Numerals 23 + 58

20. Adding 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

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

22. Counting 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

23. Decimal, 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

24. Decimal Binary Octal Hexadecimal 0 0 0 0 1 1 1 1 2 10 2 2 3 11 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7 8 1000 10 8 9 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F

25. Why Binary? • Early computer design used decimal • John von Neumann proposed binary data processing (1945) • Simplified computer design • Used for both instructions and data

26. Numbers: Physical Representation • Different numerals, same number of oranges • Cave dweller: IIIII • Roman: V • Arabic: 5 • Different bases, same number of oranges • 510 • 1012 • 114

27. Number 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

28. 10’s place 1’s place Positional Notation: Base 10 43 = 4 × 101 + 3 × 100 43

29. 100’s place 10’s place 1’s place Positional Notation: Base 10 527 = 5 × 102 + 2 × 101 + 7 × 100 527

30. 1’s place 64’s place 8’s place Positional Notation:Octal =40410 6248 404

31. 4,096’s place 1’s place 16’s place 256’s place Positional Notation: Hexadecimal =26,37210 6,70416 26372

32. 4,096’s place 1’s place 16’s place 256’s place Positional Notation: Hexadecimal =69310 2B516 693

33. Positional Notation: Binary 110101102 = 21410

35. Converting from Base 10 Power

36. 22 Base 10to Base 2 2210 = 101102 Power 2 6 5 4 3 1 0 Base 32 2 64 16 8 4 2 1 Binary 1 0 1 1 0 Integer 0/1 2/2 6/8 6/4 22/16 Remainder 0 0 6 6 2

37. Remainder Quotient 22 Base 10to Base 2 Base 10 22 2 ) 22 ( 0 2 ) 11 ( 1 2 ) 5 ( 1 2 ) 2 ( 0 2 ) 1 ( 1 0 Base 2 10110

38. 42 Base 10to Base 2 4210 = 1010102 Power 6 5 4 3 2 1 0 Base 32 2 64 16 8 4 2 1 Binary 1 0 1 0 1 0 Integer 0/1 2/2 10/8 2/4 10/16 42/32 Remainder 0 0 10 10 2 2

39. Remainder Quotient 42 Base 10to Base 2 Base 10 42 2 ) 42 ( 0 2 ) 21 ( 1 2 ) 10 ( 0 2 ) 5 ( 1 2 ) 2 ( 0 1 Base 2 101010

40. Addition in Binary Carry 1 0 1 0 + 1 0 1 0 ––––––––– 1 1 0 0 1 0

41. Multiplication in Binary 1 1 × 1 1 ––––– 1 1 + 1 1 0 –––––– 1 0 1 1 0

42. Remainder Quotient 126 Base 10to Base 8 Base 10 126 8 ) 126 ( 6 8 ) 15 ( 7 8 ) 1 ( 1 Base 8 176 0

43. Remainder Quotient 126 Base 10to Base 16 Base 10 126 16 ) 126 ( 14 E 8 ) 7 ( 7 0 Base 16 7E