1. Binary Values and Number Systems
Chapter 2
Binary Values and Number Systems

2. 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

3. 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

4. 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

5. Natural Numbers How many ones are there in 642? 600 + 40 + 2 ?
Or is it ?
Or maybe… ? 4

6. 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

7. 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

8. Counting Systems in Binary/Octal/Decimal
These are not equivalent.
They are just showing the counting upwards by 1 unit

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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!

15. OCTAL SYSTEM – Base 8

16. Octal Octal is base 8 Numeric representation can be from 0-7
Then, go to the next position

17. 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

18. 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

19. Converting Decimal to Octal
What is 1988 (base 10) in base 8? 64 4
Answer is :
Backwards 80

20. Converting Decimal to Octal
Try some!
web/mathematics/real/Calculators/ BaseConv_calc_1.htm
Still a challenge? Try a calculator!

21. Check it out
You can use the calculator to convert

22. Using Calculator for Converting
Put the number in Dec first

23. Using Calculator for Converting
Click the new format, Oct

24. Apple Programmer Mode You can see the conversion for binary live
Press 10
Enter 1988
You can see the conversion for binary live

25. BINARY SYSTEM – Base 2

26. 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

27. 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

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

30. 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

31. 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

32. 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

33. 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

34. 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

35. Subtracting Binary Numbers

36. 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 = =

37. 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

38. 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

39. HEXADECIMAL SYSTEM

40. 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

41. Converting Decimal to Hexadecimal
What is 3567 (base 10) in base 16? 32 15
Backwards
D E F 21

42. Converting Binary to Hexadecimal
Mark groups of 4 (from right) =
Convert each group
A B
is AB in base 16 18

43. 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

44. CONVERTING SYSTEMS

45. 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

46. 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?