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Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

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Page 1: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Numbering Systems

Ours is not to reason why

Ours is to

Invert and Multiply…

Page 2: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Numbering Systems

Since early times mankind has used many different types of symbols to represent numbers

In North America we use the Base 10 system.

There are several different types of number systems. Each is used for different purposes and each is different, yet similar.

Binary (used in computers) Base 2 Hexadecimal(used in computers) Base 16 Decimal (used in North America) Base 10

Page 3: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Decimal (Base 10)It consists of 10 digits (hence the name decimal). The digits, from smallest to largest are:0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Now, any number can be broken down into columns. From right to left, the first column is the 1's column, then the 10's column, then the 100's, then the 1000's, etc..

0 X 1000 = 0 (0 thousands)1 X 100 = 100 (1 hundreds)3 X 10 = 30 (3 tens)7 X 1 = 7 (7 ones) 0+100 + 30 + 7 = 137

Thousands Hundreds Tens Ones

0 1 3 7

Page 4: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

An Examplelets try the number 77:

0 X 1000 = 0 (0 thousands)0 X 100 = 0 (0 hundreds)7 X 10 = 70 (7 tens)7 X 1 = 7 (7 ones)

0+0+70 + 7 = 77

Thousands Hundreds Tens Ones

0 0 7 7

Page 5: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Binary (Base 2)Your computer works using the binary numbering system. The binary numbering system is ideal for representing these two states because it consists of only two digits. Once again, any number can be broken down into columns. Using the binary numbering system, from right to left, the first column is the 1's column,the 2's column, the 4's, the 8's, the 16's column, the 32's column, etc..

Page 6: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Let's look at the number 137 again

1 X 128 = 128 (1 one hundred twenty-eights)

0 X 64 = 0 (0 sixty-fours)

0 X 32 = 0 (0 thirty-twos)

0 X 16 =0 (0 sixteen's)

1 X 8 = 8 (1 eights)

0 X 4 = 0 (0 fours)

0 X 2 = 0 (0 twos)

1 X 1 = 1 (1 ones)

128 64 32 16 8 4 2 1

1 0 0 0 1 0 0 1

Page 7: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Converting Decimal to Binary

There are a number of ways to convert between decimal and binary. Lets start with converting the decimal value 254 to binary. Method 1: Use the binary calculator. What we have been doing before. 20 21 22 23 24 25

Method 2: Divide the number by 2. Then divide what's left by 2, and so on until there is nothing left (0). Write down the remainder (which is either 0 or 1) at each division stage. Once there are no more divisions, list the remainder values in reverse order. This is the binary equivalent.

Page 8: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

254 / 2 giving 127 with a remainder of 0127 / 2 giving 63 with a remainder of 1 63 / 2 giving 31 with a remainder of 131 / 2 giving 15 with a remainder of 1 15 / 2 giving 7 with a remainder of 1 7 / 2 giving 3 with a remainder of 1 3 / 2 giving 1 with a remainder of 1 1 / 2 giving 0 with a remainder of 1 Reading in reverse order(Bottom to top)1111110 Tada.. Not too shabby!

Page 9: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Another example, 132 decimal132 / 2 giving 66 with a remainder of 0 66 / 2 giving 33 with a remainder of 033 / 2 giving 16 with a remainder of 1 16 / 2 giving 8 with a remainder of 0 8 / 2 giving 4 with a remainder of 0 4 / 2 giving 2 with a remainder of 0 2 / 2 giving 1 with a remainder of 0 1 / 2 giving 0 with a remainder of 1Thus the binary equivalent is 10000100

Page 10: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

128 + 0 + 0 + 0 + 8 + 0 + 0 + 1 = 137

Thus, the binary number 10001001 is equal to 137 decimal.

A single digit (0 or 1) is called a 'bit' (binary digit).

The table above contains 8 bits. Each column can contain either a 1 or a 0 ( 'cause there is only 2 digits in the binary numbering system).

So, as you can see, it takes 8 bits to represent the

decimal number 137.

lets try the number 77:

128 64 32 16 8 4 2 1

Page 11: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

0 X 128 = 0 (0 one hundred twenty-eights)

1 X 64 = 64 (1 sixty-fours)

0 X 32 = 0 (0 thirty-twos)

0 X 16 =0 (0 sixteen's)

1 X 8 = 8 (1 eights)

1 X 4 = 4 (1 fours)

0 X 2 = 0 (0 twos)

1 X 1 = 1 (1 ones)

0 + 64 + 0 + 0 + 8 + 4 + 0 + 1 = 77

Thus, the binary number 01001101 is equal to 77 decimal

Page 12: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Solve the first two rows using a binary calculator Then solve the last 2 rows using

division.

254 8 127 13

255 1020 397 9999

20 5 16 99

178 33 207 3578

Page 13: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Hexadecimal Number System [Base-16]

The hexadecimal number system uses SIXTEEN values to represent numbers.

The values are 0 1 2 3 4 5 6 7 8 9 A B C D E FWith 0 having the least value and F having the greatest value.

Hexadecimal is often used to represent values [numbers and memory addresses] in computer systems.

Page 14: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Converting hexadecimal to decimal Convert 176 in hexadecimal to decimal

Each column represents a power of 16,  176 = ---- 6 * 160 = 6 ----- 7 * 161 = 112 ------ 1 * 162 = 256

------ = 374

Convert 11 in hexadecimal to base 10

11 = 1 * 160

= 1 * 161 = 16 + 1 = 17

Page 15: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Practice Hex to Decimal

3DA D9E F3A1 09

6612 55 44A9 645

3D 14DE F309 B8A4

Page 16: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Converting binary to hexadecimal Convert 10110 to hexadecimal.

Each hexadecimal digit represents 4 binary bits. Split the binary number into groups of 4 bits, starting from the right.

1 0110= 1 = 6= 16 in hexadecimal.

Try 1111110 base 2 to base 16

What did you get ??

Page 17: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Practice Bin to Hex

1010101 1000100 0010010

0101101 11110111 101010101

111111101110001 11011100

1011100 1000 1110

Page 18: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Converting decimal to hexadecimal Convert 232 decimal to hexadecimal. Use the same method used earlier to divide decimal to binary, but divide by 16 this time.

232 / 16 = 14 with a remainder of 8 8 / 16 = you can’t have since 8 is smaller than 16. So the 8 becomes the last digit.So you get 14 and a 8. Remember…(14 decimal = E)  = E816

A=10, B=11, C=12, D=13, E=14, F=15

Page 19: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Practice Decimal to Hex

15 15831 902 55

6612 5589 13 91145

3 14 309 84

96 11 1449 64000

Page 20: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

NotationTo avoid confusion, we often add a suffix to indicate the number of the base.

162h h means hexadecimal

16216 16 means base 16 

162d d means decimal

16210 10 means base 10

162o o means octal

1628 8 means base 8 

101b b means binary

1012 2 means base 2

Page 21: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

More Converting

For Review, Work on the handout called “Digital 1”

Specific problems or questions come and ask me.

Copy and complete the following charts in your books.

Page 22: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Binary Hex Decimal101010102

001100112

111111112

101110102

111100102

001000112

000100012

100000002

100101012

111111112

Page 23: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Hex Binary Decimal

345

AF56

78C1

B0D5

7156

9

185

FFFF

FFF

FF

Page 24: Numbering Systems Ours is not to reason why Ours is to Invert and Multiply…

Decimal Hex Binary

345

32768

255

1289

15

1000

1024

999

1678

2000