1.0 numbering system - lecturer

7/30/2019 1.0 Numbering System - Lecturer

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NUMBERING SYSTEM


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NUMBERING SYSTEM

Decimal

Binary

Octal Changing Numbering System Process

Basic Operation


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DECIMAL

A decimal number (based on the number 10)

contains a decimal point.

To understand decimal numbers you must first

know about place value.

When we write numbers, the position (or

place) of each number is important.


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In the number 327:

o The 7 is in the Units position, meaning just 7 (or 7 1s)

o The 2 is in the Tens position meaning 2 tens (or twenty)

o The 3 is in the Hundreds position meaning 4 hundreds


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As we move left, each position is 10 times bigger!

From Units, to Tens, to Hundreds, to Thousands,

to Ten-Thousands, to Hundred-Thousands, to

Millions!

As we move right, each position is 10 timessmaller.

From Millions, to Hundred-Thousands, to Ten-

Thousands, to Thousands, to Hundreds, to Tens,to Units.


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But what if we continue past Units?

What is 10 times smaller than Units?

But we must first write a decimal point, so we know exactly

where the Units position is.

are!(Tenths)th10

1


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The decimal point is the most important part of a decimal

number. It is exactly to the right of the Units position.Without it, we would be lost and not know what each position

meant.

Now we can continue with smaller and smaller values, from

tenths, to hundredths, and so on, like in this example:


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So, our decimal system lets us write numbers

as large or as small as we want, using the

decimal point.

Numbers can be placed to the left or right of a

decimal point, to indicate values greater than

one or less than one.


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The number to the left of the decimal point isa whole number (17 for example).

As we move further left, every number place

gets 10 times bigger. The first digit on the right means tenths

(1/10).

As we move further right, every number placegets 10 times smaller (one tenth as big).


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The word decimal really means based on

10.

From Latin decima: a tenth part.


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A decimal fraction is a fraction where the

denominator (the bottom number) is a number such

as 10, 100, 1000, etc (in other words a power of ten).

10

23:thislikelookwould"2.3"So

100

1376:thislikelookwould"13.76"So


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Or, you could think of a decimal number as a

whole number plus a decimal fraction.

10

3and2:thislikelookwould"2.3"So

100

76and13:thislikelookwould"13.76"So


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BINARY

A binary number is made up of only 0s and 1s.

110100Example of a binary number

There is no 2, 3, 4, 5, 6, 7, 8 or 9 in binary!


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How do we count using binary?

Binary

1 We start 0

2 Then 1

??? But then there is no symbol for 2. What do we do?


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Well, how do we count in decimal?

0 Start at 0

Count 1, 2, 3, 4, 5, 6, 7, 8 and then

9 This is the last digit in decimal

10 So we start back at 0 again, but add 1 on the left


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The same thing is done in binary:

Binary

0 Start at 0

1 Then 1

10 Now start back at 0 again, but add 1 on the left

11 1 more

??? But now what?


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What happens in decimal?

99 When we run out of digits,

100 We start back at 0 again, but add 1 on the left


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Binary

0 Start at 0 1 Then 1

10 Start back at 0 again, but add 1 on the left

11

100 1is added to the next position on the left

101

110

111

1000 Start back at 0 again and add 1 on the left

1001 And so on!


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Decimal VS Binary


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In the decimal system there are the Units, Tens,

Hundreds, etc. In binary, there are Units, Twos, Fours, etc, like this:


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The word binarycomes form bi- meaning two. Wesee bi- in words such as bicycle (two wheels) or

binocular (two eyes).

When you say a binary number, pronounce each digit(example, the binary number 101 is spoken as one

zero one, or sometimes one-oh-one). This waypeople dont get confused with the decimal number.

A single binary digit (like 0 or 1) is called a bit. Forexample 11010 is five bits long. The word bit is madeup from the words binary digit.


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To show that a number is a binary number,follow it with a little 2 like this:

This way people wont think it is the decimal

number 101 (one hundred and one).

2101


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Value of a Number in Binary

The value of the n-th digit in a number in base two is

equal to .21n

First

Second

Third

Fourth

Fifth

Sixth

n-th

02

12

22

32

42

52

12

n


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State the value of the underlined digit in each of the

following numbers in

a)

b)

c)

21011100

4212

20010111621

4

210101016421

6


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Binary in Expanded Notation

A number in base two can be written in expanded

notation as the sum of the product of the digit and

its digit value.


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Example 1:

notation?expandedin1111isWhat 2

aswrittenbecan11112

0123 21212121


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Example 2:


aswrittenbecan10012

0123 21202021


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Example 3:

notation?expandedin1.1isWhat 2

aswrittenbecan1.12

1

0

2

1121


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Example 4:


aswrittenbecan10.112

21

01

2

112

112021


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Example 5:



21

012

2

112

10202121


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OCTAL

The octal numeral system, or oct for short, is

the base-8 number system, and uses the digits

0 to 7.

So, what if we had eight fingers, or for some

other reason, we decided to start over every

eighth number instead of every tenth?


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In this system, there are eight symbols to workwith:

0 1 2 3 4 5 6 7

We dont need an 8 or a 9 at all: out of justthose eight symbols above, we are going to

represent every possible number!


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So, we start by listing all the symbols after the

zero.1

2

34

5

67


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When we get to that point, were out of symbols. So

what do we do? We go all the way down to zero, and add a one to our

left: we write one-zero (10).

It means the number that comes after seven, or what

we normally call eight. When we write one-oh (10) in base eight, we dont

mean ten, we mean the number eight.

The numbers in base eight look just like our normal

numbers (except that they never use the symbols 8 or9), but they dont mean the same things.


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Now we start counting on the right again: one-

one, one-two, one-three, and so on (11, 12,

13, ).

Soon we hit one-seven (17) and we run out of

digits again, so we have to increment on the

left: two-zero, or 20. Every eighth number, we

start over again.


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This system works great until we get to 77,

and then we cant increment the left-handed

digit any more. So we move to the left again

and write one-zero-one (100).

Its important to remember again that this

doesnt mean the same thing we normally call

one hundred so its best not to call it that:

call it one-zero-one.


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Value of a Number in Octal

The value of the n-th digit in a number in base eight

is equal to .81n

First

Second

Third

Fourth

Fifth

Sixth

n-th

08

18

28

38

48

58

18

n


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State the value of the underlined digit in each of the

following numbers in

a)

b)

c)

83214576481

2

875321221024512282

3

8576332122884096383

4


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Octal in Expanded Notation

A number in base eight can be written in expanded

notation as the sum of the product of the digit and

its digit value.


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Example 1:


aswrittenbecan2618

012 818682


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Example 2:


aswrittenbecan42718

0123 21878284


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Example 3:


aswrittenbecan5.28

1

0

81285


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Example 4:


aswrittenbecan3.218

21

0

811

81223


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Example 5:



21

01

817

8118584


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What does 1235 base eight really mean?

So we can say 1235 base eight equals 669

base ten.

669

51216428315

81828385

1235

3210

8


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CHANGING NUMBERING

SYSTEM PROCESS

Converting to Decimal (base 10)

Convert from Decimal (base 10)


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Converting to Decimal

Convert Binary into Decimal

Convert Octal into Decimal


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Convert Binary into Decimal

decimalto1011Convert 2

10

0123

2

11

1208

212120211011


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10

0123

2

15

1248

212121211111


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Convert Octal into Decimal


10

012

8

157

524128

858382235


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10

01

8

45

540

858555


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Convert from Decimal

Convert Decimal into Binary

Convert Decimal into Octal


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binaryinto5Convert 10

1R

0

12

0R

1

22

1R

2

52

1) Divide 2 into the number you are trying

to convert.

2) Write the quotient (the answer) with a

remainder.3) Repeat this division process using the

whole number from the previous

quotient.

4) Continue repeating this division until the

number in front of the remainder is only

zero.5) The answer is the remainders read from

the bottom up.

210 1015


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binaryinto9Convert 10

1R

0

12

0R

1

22

0R

2

42

1R

4

92

210 10019


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octalinto140Convert 10

2R

0

28

1R

2

178

4R

17

1408

1) Divide 8 into the number you are trying

to convert.

2) Write the quotient (the answer) with a

remainder.3) Repeat this division process using the

whole number from the previous

quotient.

4) Continue repeating this division until the

number in front of the remainder is only

zero.5) The answer is the remainders read from

the bottom up.

810 214140


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octalinto202Convert 10

3R

0

38

1R

3

258

2R

25

2028

810 312202


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BASIC OPERATION

Addition of two numbers in binary

Subtraction of two numbers in binary


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Addition of Two Number in Binary

222 1011 i.

ii.

iii.

222222 11110111

222222 10010101111


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22 11011110010

Find the sum of

i.

ii.

21001001

22 10111100111 2111110


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Subtraction of Two Number in Binary

222 1110 i.

ii.

iii.

222 10111

222 111100


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22 1011110110

Calculate

i.

ii.

2101011

22 1110110001 2100011

1.0 numbering system - lecturer

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