The age of Earth is about $4.6 \times 10^9$ years old. What is the number in decimal notation?
The given number is in scientific notation: \(4.6 \times 10^9\). Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. The standard form is \(a \times 10^n\), where \(1 \le |a| < 10\) and \(n\) is an integer. In this specific case, the coefficient \(a\) is \(4.6\) and the exponent \(n\) is \(9\).
To convert a number from scientific notation \(a \times 10^n\) to decimal notation when \(n\) is a positive integer, we shift the decimal point in the coefficient \(a\) to the right by \(n\) places. For \(4.6 \times 10^9\), this means moving the decimal point in \(4.6\) to the right by \(9\) positions. Adding zeros as needed to fill the empty spots ensures the magnitude of the number is maintained correctly.
Starting with \(4.6\):
Thus, \(4.6 \times 10^9 = 4,600,000,000\).
Comparing the calculated value \(4,600,000,000\) with the given options, we identify that the first option is the correct one. The age of Earth, given as \(4.6 \times 10^9\) years, is equivalent to 4.6 billion years or \(\boxed{4,600,000,000}\) years.
Scientific notation is essential for scientists to communicate massive scales, like the age of the Earth or the distance to stars. For instance, if an astronomer calculates the distance to a star as \(5.2 \times 10^{13}\) km, they are easily able to compare it to the Earth's radius, which is approximately \(6.37 \times 10^3\) km, without getting lost in strings of zeros. This shorthand avoids common errors in writing large datasets and makes scientific communication more accurate and efficient.