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© 2003 Dr. Kevin ChouinardEdited by Jean Pacelli 2004 1
Section 4.3
Converting Between Number Bases
2
Place Values for the Decimal System
Also Called Base Ten__ __ __ __ __ __
thousands hundreds10º
tens ones10¹10²10³104
tenthousands
105
hundredthousands
Digits for Base Ten0, 1, 2, 3, 4, 5, 6, 7, 8, 9
3
Value of a multi-digit number in Base Ten
The base ten number 406,391 is 6 digits long, and is written in standard form.
It is read four hundred and six thousand, three hundred ninety one.
In expanded form, 406,391 =(4x105) + (0x104) + (6x103) + (3x102) + (9x101) + (1x100) =4x100,000 + 0x10,000 + 6x1,000 + 3x100 + 9x10 + 1x1
This number system is multiplicative, and positional. The digit is in a specific place value, and is the multiplier for that place value.
The 4 in the number 4000 has a different value from the 4 in the number 400.
4
Using Other Bases
When we work in a base other than base 10, we must calculate the place values for that base.
The place values will be powers of the base, starting with the 0th power, 1st power, 2nd power, etc… working from right to left.
5
Place Values for Base Five
Let’s try counting in base five.First, we need the place values.
_ _ _ _ _ _
twenty-fives
fives ones5¹5² 5º
6
Counting in Base Five _ _ _ _ _ _
twenty-fives
fives ones5¹5² 5º
242322212014131211104321Base 5
1413121110987654321Base 10
10210110044434241403433323130Base 5
27262524232221201918171615Base 10
7
Conversion from base 5 to base 10
1432five = _____ten
Answer:1 5³4 5²3 5¹
+ 2 5º
1 125 4 253 5
+2 1
125100
15+ 2242te
n
8
Conversion from base 10 to base 5
698ten = _____five
Answer:
Start by finding the powers of five which are 698
5º = 15¹ = 55² = 255³ = 125
How many 625’s are there in 698?698 625 = 1 (R 73)
How many 125’s are there in 73?73 125 = 0 (R 73)
How many 25’s are there in 73?73 25 = 2 (R 23)
How many 5’s are there in 23?23 5 = 4 (R 3)
How many 1’s are there in 3?3 1 = 3 (R 0)
698ten = 10243five62554
9
Place Values for Base Eight
Let’s examine the first four place values for base eight
_ _ _ __
five-hundred-twelves
sixty-fours
8ºeights ones
8¹8²8³
10
Conversion from base 8 to base 10
270eight = _____ten
Answer:2 8²7 8¹
+ 0 8º
2 64 7 8
+0 1
12856
+ 0184te
n
11
Conversion from base 10 to base 8
497ten = _____eight
Answer:
Start by finding the powers of eight which are 497
8º = 18¹ = 88² = 648³ = 512
How many 64’s are there in 497?497 64 = 7 (R 49)
How many 8’s are there in 49?49 8 = 6 (R 1)
How many 1’s are there in 1?1 1 = 1 (R 0)
497ten = 761eight
12
Place Values for Base Two
Let’s examine the first four place values for base two
_ _ _ __
eights fours2º
twos ones2¹2²2³
13
Conversion from base 2 to base 10
1011two = _____ten
Answer:1 2³0 2²1 2¹
+ 1 2º
1 8 0 41 2
+1 1
802
+ 111ten
14
Octal (base eight)01234567
Binary (base two)000001010011100101110111
Binary to Octal Conversion
15
Examples
101110001two = 561eight
11011two = 011011two
= 33eight
16
Place Values for Base SixteenLet’s examine the first four place values for base
sixteen
_ _ _ _
4096 25616º
sixteen ones16¹16²16³
17
Digits in various basesBase 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Base 8 0, 1, 2, 3, 4, 5, 6, 7
Base 5 0, 1, 2, 3, 4
Base 2 0, 1
Base 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
18
Digits in various bases (cont.)
Note that each base has digits from 0 up to a number one less than the baseAlso, note that in base 16,
A = 10B = 11C = 12D = 13E = 14F = 15
19
Binary to Hexadecimal Conversion Hexadecimal (Base 16)
0123456789ABCDEF
Binary (Base 2)0000000100100011010001010110011110001001101010111100110111101111
20
Convert to base 10
423five
317eight
1010111tw
o
= 113ten
= 207ten
= 87ten
21
Convert to the indicated base
21ten to base two
396ten to base eight
392ten to base five
= 10101two
= 614eight
= 3032five
22
Cereal Box Magic Trick135791113151719212325272931
2367
101114151819222326273031
4567
121314152021222328293031
89
1011121314152425262728293031
16171819202122232425262728293031
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