© 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