**(D) Converting Numbers in Base Two, Eight and Five to Base Ten and Vice Versa**

**1.**Steps to convert numbers in base 2, 8 and 5 to base 10 are as follows.

**(a)**write the number in

**expanded notation**.

**(b)**simplify the expanded notation into a

**single number**.

**Example 1:**

Convert each of the following numbers to a number in base 10.

**(a)**10101

_{2}

**(b)**1423

_{8}

**(c)**324

_{5}

*Solution:***(a)**10101

_{2}= 1 × 2

^{4}+ 0 × 2

^{3}+ 1 × 2

^{2}+ 0 × 2

^{1}+ 1 × 2

^{0}=

**21**

_{10}**1423**

(b)

(b)

_{8 }= 1 × 8

^{3}+ 4 × 8

^{2}+ 2 × 8

^{1}+ 3 × 8

^{0}=

**787**

_{10}**324**

(c)

(c)

_{5 }= 3 × 5

^{2}+ 2 × 5

^{1}+ 4 × 5

^{0}=

**89**

_{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:(a)[BIN] 10101 [=] [ DEC ]
The screen display is: [21]
Therefore 10101
_{2} = 21_{10}(b)[OCT] 1423 [=] [ DEC ]
The screen display is: [787]
Therefore 1423
_{8} = 787 _{10} |

^{ }

^{ }

^{ }**2.**Steps to convert a number in base 10 to a number in base 2, 8 and 5 are as follows.

**(a)**perform repeated division until the

**quotient**is zero.

**(b)**write the number in new base by referring to the

**remainders**from

**bottom**to the

**top**.

**Example 2:**

Convert 61

_{10 }to a number in**(a)**Base two

**(b)**base eight

**(c)**base five

*Solution:***(a)**

**(b)**

**(c)**

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:(a)[DEC] 61 [=] [ BIN ]
The screen display is: [111101
_{2}]Therefore 61
_{10} = 111101_{2}(b)[DEC] 61 [=] [ OCT ]
The screen display is: [75]
Therefore 61
_{10} = 75 _{8} |

^{ }