Final answer:
The bar notation in 7.39 indicates that the digit 9 repeats indefinitely. To simplify the bar notation, we can write it as a fraction. The simplified form of 7.39 bar notation is 7 - 3.49/10.
Step-by-step explanation:
In this case, the number with the bar notation is 7.39. The bar notation indicates that the digit 9 repeats indefinitely. To simplify the bar notation, we can write it as a fraction. Let's call the repeating digit x. The number with the bar notation can be expressed as a fraction: 7.39 = 7 + x/10 + x/100 + x/1000 + ...
To find the value of x, we can multiply the equation above by 10: 10 * (7.39) = 10 * 7 + x/1 + x/10 + x/100 + ...
By subtracting the original equation from the new equation, we can eliminate the repeating digits: 10 * (7.39) - 7.39 = 10 * (7) + x/1 + x/10 + x/100 + ... - (7 + x/10 + x/100 + x/1000 + ...)
Simplifying the equation, we get: 73.9 - 7.39 = 70 + x/1 + 9x/10 + 9x/100 + 9x/1000 + ...
Combining like terms, we have: 66.51 = 70 + x * (1/1 + 9/10 + 9/100 + 9/1000 + ...)
Now we can solve for x by converting the infinite geometric series into a finite series. The sum of the infinite geometric series with a common ratio less than 1 is given by the formula: S = a / (1 - r), where a is the first term and r is the common ratio.
In our case, the first term a is x/1 and the common ratio r is 9/10. Plugging these values into the formula, we get: 66.51 = 70 + x / (1 - 9/10)
Simplifying the equation, we find: 66.51 = 70 + x / (1/10)
Multiplying both sides of the equation by 10, we get: 665.1 = 700 + 10x
Subtracting 700 from both sides of the equation, we have: 665.1 - 700 = 10x
Simplifying, we find: -34.9 = 10x
Dividing both sides of the equation by 10, we get: x = -3.49
Therefore, the simplified form of 7.39 bar notation is 7.39 = 7 - 3.49/10.