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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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17.591
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:
notation?expandedin1001isWhat 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:
notation?expandedin10.11isWhat 2
aswrittenbecan10.112
21
01
2
112
112021
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Example 5:
notation?expandedin110.01isWhat 2
aswrittenbecan110.012
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:
notation?expandedin261isWhat 8
aswrittenbecan2618
012 818682
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Example 2:
notation?expandedin4271isWhat 8
aswrittenbecan42718
0123 21878284
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Example 3:
notation?expandedin5.2isWhat 8
aswrittenbecan5.28
1
0
81285
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Example 4:
notation?expandedin3.21isWhat 8
aswrittenbecan3.218
21
0
811
81223
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Example 5:
notation?expandedin45.17isWhat 8
aswrittenbecan45.178
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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decimalto1111Convert 2
10
0123
2
15
1248
212121211111
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Convert Octal into Decimal
decimalto235Convert 8
10
012
8
157
524128
858382235
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decimalto55Convert 8
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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Convert Decimal into Binary
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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Convert Decimal into Binary
binaryinto9Convert 10
1R
0
12
0R
1
22
0R
2
42
1R
4
92
210 10019
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Convert Decimal into Octal
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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Convert Decimal into Octal
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