#### To determine

**To express:** The given number as a ratio of integers.

#### Answer

The number 0.8¯ can be expressed as 89.

#### Explanation

**Given:**

0.8¯=0.8888...

**Result used:**

The geometric series ∑n=1∞arn−1 (or) a+ar+ar2+⋯ is convergent if |r|<1 and its sum is a1−r, where *a* is the first term and *r* is the common ratio of the series.

**Calculation:**

Rewrite the number and express 0.8¯ as follows,

0.8¯=0.8888...=0.8+0.08+0.008+0.0008+⋯=810+8102+8103+8104+⋯=∑n=1∞810n

Clearly, it is geometric series with first term of the series is a=810 and common ratio is

r=110.

Since |r|<1 and the Result stated above, the geometric series ∑n=1∞810n is convergent.

Obtain the sum of the geometric series.

Since a=810 and r=110,

∑n=1∞810n=8101−110=810910=810⋅109=89

Thus, the series is convergent to the sum is ∑n=1∞810n=89.

Therefore, the number 0.8¯ can be expressed as 89.