1 week 2: binary, octal and hexadecimal numbers reading: chapter 2

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Positional Notation:BaseQuick Re-cap:

•1. Conversion from other bases to base 10, where b is the base of the original number, di is the digit at the i-th position :

• 2. Conversion from base 10 to other bases

EECS 1520 -- Computer Use: Fundamentals

dn-1 * bn-1 + … + d2 * b2 + d1 * b1 + d0 * b0

Step 1: Always divide the decimal number by the new base, write down the quotient and the reminder

Step 2: Divide the quotient by the new base, write down the new quotient and the new reminder

Step 3: Repeat step 2 until the quotient is 0

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Positional Notation:Base• In general, we can convert numbers in any bases with the following method:


(number)A (number)10 (number)B

where A ≠ B ≠ 10

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Positional Notation:Adding binary numbers• Basic idea in adding digits in decimal: 1 + 2 = 3, 4 + 1 = 5….


• When you try to add two numbers whose sum is equal to or greater than the base value (i.e. 10), we reuse the same digits and rely on position.

• The rules of adding binary numbers are similar, but we run out of digits much faster (since we only have 2 digits)

0+ 1 1

1+ 0 1

0+ 0 0

1+ 1 1 0

9+ 1 1 0

1 is carried into the next position to the left

1 is called the “carry bit”

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Positional Notation:Adding binary numbers• Example 1: what is the sum of the following two binary numbers?


• We can confirm that the above sum is correct by converting each of the above binary numbers to their decimal representations

0011+ 1001 1100

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Positional Notation:Adding binary numbers


0011

1001

convert to base 10

convert to base 10

1*21 + 1*20 = 3

1100 convert to base 10

1*23 + 1*20 = 9

1*23 + 1*22 = 12

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Positional Notation:Adding binary numbers• Example 2: what is the sum of the following two binary numbers?


0111+ 1101 10100

0111

1101

convert to base 10

convert to base 10

1*22 + 1*21 + 1*20 = 4+2+1 = 7

10100 convert to base 10

1*23 + 1*22 + 1*20 = 8+4+1 = 13

1*24 + 1*22 = 16+4 = 20

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Positional Notation:Power-of-2 number systems


Decimal (base 10) Binary (base 2) Octal (base 8)

0 0000 0

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

8 1000 10

9 1001 11

10 1010 12

99

Positional Notation:Power-of-2 number systems• An unique relationship between binary and octal numbers, as 8 is a power of

2 (i.e. 8 = 23)

• Given an octal number, you can convert it directly to binary by replacing each digit in the octal number with the binary representation of that digit (note: the largest digit can only be “7”)


Decimal Binary Octal

0 000 0

1 001 1

2 010 2

3 011 3

4 100 4

5 101 5

6 110 6

7 111 7

1010



• Example: what is the binary representation of the octal “154” ?

1 5 4

100101001binary

octal

1111

Positional Notation:Power-of-2 number systems• Now, if we convert the binary number to base 10:


• We got 108, same as the number we got when we converted the octal number “154” to its base 10 representation:

dn-1 * bn-1 + … + d2 * b2 + d1 * b1 + d0 * b0

1 * 82 + 5 * 81 + 4 * 80

= 64 + 40 + 4 = 108

001 101 100 1 * 26 + 1 * 25 + 1* 23 + 1 * 22

= 64 + 32 + 8 + 4 = 64 + 44 = 108

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Positional Notation:Power-of-2 number systems• Convert from binary number to octal number:


Step 1: start at the rightmost binary digit and mark the digits in groups of threesStep 2: convert each group of three into its octal value

Note: Add 0s to the front if the leftmost group does not have 3 binary digits

• Example: what is the octal representation of the binary number: 1101111110 ?

• Note: any digits > 7 do not exist in octal, if you get an octal value > 7, something is wrong.

001 101 111 110

1 5 7 6

binary

octal

1313


0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

• Recall that, in base-16, we have:


• The previous “grouping” method can be applied to convert a binary number to its hexadecimal number (since 16 = 24)

0000, ……………………………………….., 1111 binary

hexadecimal

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Positional Notation:Power-of-2 number systems• Convert from binary number to hexadecimal number:


Step 1: start at the rightmost binary digit and mark the digits in groups of foursStep 2: convert each group of four into its hexadecimal value

Note: Add 0s to the front if the leftmost group does not have 4 binary digits

• Example: what is the octal representation of the binary number: 1101111110 ?

0011 0111 1110

3 7 E

binary

hexadecimal

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Positional Notation:Power-of-2 number systems• Conversion between octal and hexadecimal numbers are much easier now

as we can utilize the intermediate binary digits “grouping” method


• Example: what is the hexadecimal representation of the octal number: 4567 ?

100 101 110 111

4 5 6 7 octal

binary

1001 0111 0111

hexadecimal 9 7 7

1616

Positional Notation:Adding octal numbers


• When you try to add two numbers whose sum is equal to or greater than the base value (i.e. 8), we reuse the same digits and rely on position.

(7)8

+ (1)8

(1 0)8


• Example 1: what is the sum of the following two octal numbers?

(27)8

+ (35)8

(64)8

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Positional Notation:Adding octal numbers


(27)8

(35)8

convert to base 10

convert to base 10

2*81 + 7*80 = 16+7 = 23

(64)8 convert to base 10

3*81 + 5*80 = 24+5 = 29

6*81 + 4*80 = 48+4 = 52

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Positional Notation:Adding hexadecimal numbers


• Same concept is applied to adding two hexadecimal numbers.

• When you try to add two hex numbers whose sum is equal to or greater than the base value (i.e. 16), we reuse the same digits and rely on position.

(F)16

+ (1)16

(1 0)16


(F)16

+ (2)16

(1 1)16

(F)16

+ (3)16

(1 2)16

(F)16

+ (4)16

(1 3)16

and so on…..

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Positional Notation:Adding hexadecimal numbers


• Example: (2F)16

+ (73)16

(A2)16

(2F)16

(73)16

convert to base 10

convert to base 10

2*161 + 15*160 = 32+15 = 47

(A2)16 convert to base 10

7*161 + 3*160 = 112+3 = 115

10*161 + 2*160 = 160+2 = 162

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Positional Notation:Subtracting two base 10 numbers


• When you try to subtract a larger digit from a smaller one, we have to “borrow one” from the next left digit of the number from which you are subtracting. More precisely, we borrow one power of the base, such as in base 10, we borrow 10.

52- 38 1 4 Borrow “10” from 5, so

we have 10- 8 + 2 = 4

• Example:

2121

Positional Notation:Subtracting two binary numbers


• In base 2, when borrow one power of the base to do subtraction, we borrow 2. In fact, we only need to borrow a 2 when we subtract 0 from 1.

1101- 0110

• Example:

0111

(1101)2

(0110)2

convert to base 10

convert to base 10

13


6

7

2222

Positional Notation:Subtracting two octal numbers


• In base 8, when borrow one power of the base to do subtraction, we borrow 8.

(153)8

- (67)8

• Example:

(64)8

(153)8

(67)8

convert to base 10

convert to base 10

107


55

52

2323

Positional Notation:Subtracting two hex numbers


• In base 16, when borrow one power of the base to do subtraction, we borrow 16.

(153)16

- (67)16

• Example:

(EC)16

(153)16

(67)16

convert to base 10

convert to base 10

339

(EC)16 convert to base 10

103

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1 week 2: binary, octal and hexadecimal numbers reading: chapter 2

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