Practice converting fractions to decimals: To convert Fractions of Decimal Numbers in base ten to base two, the given fraction needs to be split into smaller fractions in such a way that when added will still give you back your original fraction, and all the numerators of the new fractions formed must be factors of the denominator of the original fraction given.

## Practice converting fractions to decimals: **Example 1:** convert ⅞ base ten to base two.

Solution:

*To convert Fractions of Decimal Numbers*

**First step** is to split the given fraction into smaller fractions.

⅞ = (4 + 2 + 1)/8

= 4/8 + 2/8 + 1/8

Reduce each fraction to its lowest terms.

= ½ + ¼ + ⅛

*Note that when you add all those fractions, you will still arrive at the original fraction given.*

Let us proceed

½ + ¼ + ⅛ = 2–¹ + 4–¹ + 8–¹

*Recall that any number raised to power minus 1 is one over that particular number, that is x–¹ = ¹/x.*

#### Second step is to make sure they have a common base

½ + ¼ + ⅛ = 2–¹ + 4–¹ + 8–¹

= 2–¹ + 2²(–¹) + 2³(–1)

*Opening of the parentheses*

= 2–¹ + 2–² + 2–³

#### Third step is to provide a multiplication partners to each and every digits of the above fractions that will keep them unchanged when they are being multiplied.

= 2–¹ + 2–² + 2–³

= 1 x 2–¹ + 1 x 2–² + 1 x 2–³

The collection of your multiplicands is the answer in base two.

⅞ base ten = 0.111 base two.

## Practice converting fractions to decimals: **Example 2:** Convert 9⅜ base ten to base two.

#### First is to convert 9 base ten to its base two. See how to convert to base two here

9 base ten = 1001 base two.

*To convert Fractions of Decimal Numbers*

⅜ = (2 + 1)/8 = 2/8 + ⅛

= ¼ + ⅛ = 4–¹ + 8–¹

**Making the base uniform**

= 2²(–¹) + 2³(–¹)

= 2–² + 2–³

= 1 x 2–² + 1 x 2–³

*Check very well you will see that something is missing which we have to provide in that number.*

In 1 x 2–² + 1 x 2–³, we have (-2) and (-3) without having (-1). So we have to fix it in the exact place it is supposed to be by multiplying it by zero.

= 0 x 2–¹ + 1 x 2–² + 1 x 2–³

= .011 base two

9⅜ ten = 9 + ⅜

= (1001 + .011) base two

= 1001.011 base two.

### QUESTIONS AND SOLUTIONS:

#### Q1. Convert (17.25) base ten to base two.

Solution:

First we convert 17 base ten to base two.

17 base ten = 10001 base two.

Now, let us convert (.25) to base two as well.

.25 base ten

= 25/100 = ¼

0.25 = ¼ = 4–¹ = 2–² = 1 x 2–²

We can see that something is missing which is (-1). We have to fix and multiply it by zero.

0.25 base ten = 0 x 2–¹ + 1 x 2–² = 0 1

Therefore 17.75 base ten = 10001.01 base two.

#### Q2. Convert (9.75) base ten to base two.

Solution:

First we convert the integral part to base two.

9 base ten = 1001 base two.

Convert the decimal part to base five as well.

0.75 base ten = 75/100

= (3/4) = (2/4) + (1/4)

= ½ + ¼

= 2–¹ + 4–¹ = 2–¹ + 2–²

= 1 x 2–¹ + 1 x 2–²

= 1 1

So, (9.75) base ten = 1001.11 base two.

#### Q3. Convert 1011.011 base two to fraction.

Solution:

First we convert the integral part to base ten.

1011 base two

= 1 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2°

= 8 + 0 + 2 + 1

= 11 base ten.

Now, convert the decimal part to base ten too.

.011 base two

= 0 x 2–¹ + 1 x 2–² + 1 x 2–³

= 0 + 2–² + 2–³

= (1/2²) + (1/2³)

= 1/4 + 1/8

= ⅜ base ten

So, 1011.011 base two

= 11⅜ base ten.

#### Q4. Convert (3⅞) base ten to bicimal.

Solution:

First we convert the integral part to base two.

3 base ten = 11 base two

Convert the ⅞ base ten to base two

⅞ = (4 + 2 + 1)/8

= 4/8 + 2/8 + 1/8

= 1/2 + 1/4 + 1/8

= 2–¹ + 2–² + 2–³

= 1 x 2–¹ + 1 x 2–² + 1 x 2–³

= .111 base two

Thus, (3⅞) base ten = 11.111 base ten.

#### Q5. Convert 57/64 base ten to bicimal.

Solution:

57/64 = (32 + 16 + 8 + 1)/64

= 32/64 + 16/64 + 8/64 + 1/64

= 1/2 + 1/4 + 1/8 + 1/64

= 2–¹ + 4–¹ + 8–¹ + 64–¹

= 2–¹ + 2–² + 2–³ + 2^–6

As you can see that (-4) and (-5) are missing, we have to provide them by multiplying them by zero.

= 1 x 2–¹ + 1 x 2–² + 1 x 2–³ + 0 x 2–⁴ + 0 x 2^–5 + 1 x 2^–6

= 0.111001 base two

#### Q6. Convert 0.123 base four to bicimal.

*Note before that decimal is to base ten as bicimal is to base two*

Solution:

First convert 0.123 base four to base ten.

0.123 base four

= 1 x 4–¹ + 2 x 4–² + 3 x 4–³

= 1 x ¼ + 2 x ¼² + 3 x ¼³

= ¼ + ⅛ + 3/64

= 27/64 base ten.

Now, convert the base ten to bicimal

27/64 base ten

= ¼ + ⅛ + 3/64

= ¼ + ⅛ + 2/64 + 1/64

= ¼ + ⅛ + 1/32 + 1/64

= 4–¹ + 8–¹ + 32–¹ + 64–¹

= 2–² + 2–³ + 2^–5 + 2^–6

Do you notice that (-1) and (-4) are missing?, We have to provide them the way we use to.

= 0 x 2–¹ + 1 x 2–² + 1 x 2–³ + 0 x 2–⁴ + 1 x 2^–5 + 1 x 2^–6

= 0.011011 base two

#### Q7. Convert 34 base five to binary.

Solution:

We have to convert to base ten first.

34 base five

= 3 x 5¹ + 4 x 5°

= 3 x 5 + 4 x 1

= 15 + 4 = 19 base ten

Now, convert to the required base.

34 base five = 10011 base two.

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