**(E) Converting from One Base to Another**

**1.**The following steps are used to convert a number from one base to another base.

**(a)**convert the number to a number in

**base 10**by using expended notation.

**(b)**use

**repeated division**to convert the number in base 10 to the respective bases.

Example 1:

Example 1:

Convert

**(a)**110101

_{2}to a number in base 5

**(b)**43

_{5}to a number in base 2

**(c)**313

_{8}to a number in base 5

**(d)**422

_{5}to a number in base 8

**(e)**100111

_{2}to a number in base 8

**(f)**157

_{8}to a number in base 2

Solution:Solution:

**(a)**110101

_{2}

= 1 × 2

^{5}+ 1 × 2^{4}+ 0 × 2^{3}+ 1 × 2^{2}+ 0 × 2^{1}+ 1 × 2^{0}=

**53**← (Convert from base 2 to base 10)_{10 }**(b)**43

_{5}

= 4 × 5

^{1}+ 3 × 5^{0}=

**23**← (Convert from base 5 to base 10)_{10}**(c)**313

_{8}

= 3 × 8

^{2}+ 1 × 8^{1}+ 3 × 8^{0}=

**203**← (Convert from base 8 to base 10)_{10}**422**

(d)

(d)

_{5 }

= 4 × 5

^{2}+ 2 × 5^{1}+ 2 × 5^{0}=

**112**← (Convert from base 5 to base 10)_{10}**100111**

(e)

(e)

_{2}

= 1 × 2

^{5}+ 0 × 2^{4}+ 0 × 2^{3}+ 1 × 2^{2}+ 1 × 2^{1}+ 1 × 2^{0}=

**39**← (Convert from base 2 to base 10)_{10 }**(f)**157

_{8 }

= 1 × 8

^{2}+ 5 × 8^{1}+ 7 × 8^{0}=

**111**← (Convert from base 8 to base 10)_{10}Calculator Computation1. Set the calculator to the ‘BASE’ mode by pressing: [MODE] [MODE] [3 (BASE)]
2. Set the calculator to the desired number system by pressing:[BIN] → for base 2
[DEC] → for base 10
[OCT] → for base 8
Key in the following [For (e) and (f) only]:(e)[ BIN ] 100111 [ = ] [ OCT ]
The screen display is: [47
_{8}]Therefore 100111
_{2} = 47_{8}(f)[ OCT ] 157 [ = ] [ BIN ]
The screen display is: [1101111]
Therefore 157
_{8} = 1101111_{2} |