Question

A grain of sand has a mass approximately of $0.00000003$ grams. How would this number be expressed in scientific notation?

Answer 1

The mass of a grain of sand is given in the form of a decimal number whose non-zero digits reside at the end of the number. Its unit is grams.To express this given number in scientific notation, the last non-zero digit is separated from the second last digit (it can be zero or non-zero) by a decimal point. In other words, the decimal point is shifted backwards to the non-zero digits. The zero digits are then indicated in powers of 10.

Complete answer:
Scientific notation is sometimes referred to as the standard index form. The general representation of scientific notation is: $a{\text{ }} \times {\text{ }}{10^{b\;\;}}\;$ where $1{\text{ }} \leqslant {\text{ }}a{\text{ }} < {\text{ }}10$ and $b$ can be any integer. The number $b$ is known as the order of magnitude while the number $a$ is referred to as the mantissa or significant. The number $a$ is the coefficient of the scientific notation and is normally greater than or equal to 1 and less than 10.
The given number is $.00000003$ grams. The non-zero digit is 3. Then according to the condition of significant numbers, this digit can be separated from the zero digit by a decimal point. Thus we can write it as 0.3 or $a = 0.3$and the remaining zeros can be written as power of 10. Thus, $0.00000003 = \dfrac{{0.3}}{{10000000}} = 0.3 \times {10^{ - 7}}$. The power factor appears to be ${10^{ - 7}}$. Therefore, $b = - 7$. This is a negative exponent.
Therefore, $.00000003$ would be written in scientific notation as $0.3 \times {10^{ - 7}}$ grams. The unit grams can be converted into micrograms by writing $0.00000003{\text{ grams}} = 0.03 \times {10^{ - 6}}{\text{grams}} = 0.03{\text{ micro - grams}}$. (Since ${10^{ - 6}}$ denotes micro)

Note: Note that scientific notation is preferred for very large or very small numbers.Since $0.00000003$ is a very small number, therefore the preferred scientific notation for it is $0.3 \times {10^{ - 7}}$. Note that $0.00000003{\text{ grams}}$ is written in terms of smaller units as $0.03{\text{ micro - grams}}$. The prefix unit micro denotes ${10^{ - 6}}$. The prefix micro is denoted by the symbol $\mu$. Therefore $0.03{\text{ micro - grams}}$ can also be written as $0.03{\text{ }}\mu {\text{ - grams}}$.