Application of bases

2. Application of Bases
o Decimal Numbers
o Base 10 systems
oexpresses a number as a string of digits in which each
digit’s position indicates the power of 10 by which it is
multiplied.
5,049 = 5· (1,000) + 0· (100) + 4· (10) + 9· (1).
Using exponential notation, this equation can be
rewritten as
5,049 = 5·103 + 0·102 + 4·101 + 9·100.
a sum of products of the form
d ·10n

3. Bases
o Binary Numbers
o base 2 notation, or binary notation,
osignals used in modern electronics are always in one of
only two states.
od ·2n,
owhere each n is an integer and each d is one of the
binary digits (or bits) 0 or 1.
27 = 16 + 8 + 2 + 1
= 1·24 + 1·23 + 0·22 + 1·21 + 1·20.
=110112

4. Conversion
o Converting a Binary to a Decimal Number
– Represent 1101012 in decimal notation.
1101012 = 1·25 + 1·24 + 0·23 + 1·22 + 0·21 + 1·20
= 32 + 16 + 4 + 1
= 5310
o Converting a Decimal to a Binary Number
o Represent 209 in binary notation.
20910 = 128 + a smaller number.
20910 = 128 + 64 + a smaller number.
=128 + 64 + 16 + 1
= 1·27 + 1·26 + 0·25 + 1·24 + 0·23 + 0·22 + 0·21 + 1·20.

5. Binary Addition
o Add 11012 and 1112 using binary notation.
12
+ 12
102
1 1 0 12
+ 1 1 12
12
+ 12
+ 12
112

6. Subtraction in Binary Notation
o Add 11012 and 1112 using binary notation.
102
- 12
12
1 1 0 0 02
- 1 01 12
1 1 0 1

7. Two’s Complements and the Computer Representation
of Negative Integers
o Given a positive integer a, the two’s
complement of a relative to a fixed bit length
n is the n-bit binary representation of 2n − a.
o (28 − 27)10 = (256 − 27)10 = 22910 = (128 + 64 + 32 +
4 + 1)10 = 111001012
1. 28 − a = [(28 − 1) − a] + 1.
2. The binary representation of 28 − 1 is 111111112.
3. Subtracting an 8-bit binary number a from 111111112
just switches all the 0’s in a to 1’s and all the 1’s to 0’s.
(The resulting number is called the one’s complement
of the given number.)

8. Two’s Complements and the Computer Representation
of Negative Integers
o To find the 8-bit two’s complement of a
positive integer a that is at most 255:
1. Write the 8-bit binary representation for a.
2. Flip the bits (that is, switch all the 1’s to 0’s and
all the 0’s to 1’s).
3. Add 1 in binary notation.

9. Two’s Complements and the Computer Representation
of Negative Integers
o Find the 8-bit two’s complement of 19.
1. Write the 8-bit binary representation for 19,
2. Switch all the 0’s to 1’s and all the 1’s to 0’s, and add
1.
3. 1910 = (16 + 2 + 1)10 = 000100112
4. Flip the bits→11101100
5. add→1 11101101
o To check this result, note that
111011012= (128 + 64 + 32 + 8 + 4 + 1)10 = 23710 = (256 −
19)10
= (28 − 19)10, which is the two’s complement of 19.

10. Two’s Complements and the Computer Representation
of Negative Integers
o To add two integers in the range −128 through 127
whose sum is also in the range −128 through 127:
o Convert both integers to their 8-bit representations
(representing negative integers by using the two’s
complements of their absolute values).
o Add the resulting integers using ordinary binary
addition.
o Truncate any leading 1 (overflow) that occurs in the
28th position.
o Convert the result back to decimal form (interpreting 8-
bit integers with leading 0’s as nonnegative and 8-bit
integers with leading 1’s as negative).

11. Two’s Complements and the Computer Representation
of Negative Integers
o compute 72 − 54.
o =72+(-54)

12. Hexadecimal Notation
o Hexadecimal notation is also called base 16
notation.
o Any integer can be uniquely expressed as a sum of
numbers of the form
o d ·16n,
owhere each n is a nonnegative integer and each d is
one of the integers from 0 to 15.
oThe integers 10 through 15 are represented by the
symbols A, B, C, D, E, and F.

13. Hexadecimal Notation

14. Converting from Hexadecimal to
Decimal Notation
o Convert 3CF16 to decimal notation.

15. Converting from Hexadecimal to Binary
Notation
o To convert an integer from hexadecimal to
binary notation:
1. Write each hexadecimal digit of the integer in 4-
bit binary notation.
2. Juxtapose the results.

16. Converting from Hexadecimal to Binary
Notation
o Convert B09F16 to binary notation.
B16 = 1110 = 10112, 016 = 010 = 00002, 916 = 910 =
10012, and F16 = 1510 =
11112. Consequently,
B 0 9 F
1011 0000 1001 1111
and the answer is 10110000100111112.

17. Converting from Binary to Hexadecimal
Notation
o To convert an integer from binary to
hexadecimal notation:
1. Group the digits of the binary number into sets
of four, starting from the right and adding
leading zeros as needed.
2. Convert the binary numbers in each set of four
into hexadecimal digits. Juxtapose those
hexadecimal digits

18. Converting from Binary to Hexadecimal
Notation
o Convert 1001101101010012 to hexadecimal
notation.
0100 1101 1010 1001.
4 D A 9
Then juxtapose the hexadecimal digits.
4DA916