Number - Scientific Notation

Scientific notation is a method of writing very large or very small numbers using powers of 10 in the format $a \times 10^{n}$. Visually, this simplifies long strings of zeros into a compact form comprising a coefficient and a power of ten.

The coefficient, represented by $a$, must be a number such that $1 \le a < 10$. Visually, this means there should be exactly one non-zero digit to the left of the decimal point (e.g., $3.5 \times 10^{2}$ is correct, but $35 \times 10^{1}$ is not).

The exponent $n$ represents the number of places the decimal point has moved and must be an integer. Visually, a positive exponent indicates a large number (greater than $10$), while a negative exponent indicates a small decimal (between $0$ and $1$).

To convert a large number like $50,000$ to scientific notation, imagine the decimal at the end and move it to the left until only one non-zero digit remains on the left. Visually, the number of 'jumps' the decimal makes determines the positive exponent. For $50,000$, we jump $4$ places to get $5 \times 10^{4}$.

To convert a small decimal like $0.0006$ to scientific notation, move the decimal to the right until it is behind the first non-zero digit. Visually, the number of rightward jumps determines the negative exponent. For $0.0006$, moving $4$ places right results in $6 \times 10^{-4}$.

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents using the index law $10^{m} \times 10^{n} = 10^{m+n}$. Visually, you are treating the decimal parts and the power parts as separate groups before combining them.

When dividing numbers in scientific notation, divide the coefficients and subtract the exponents using the index law $\frac{10^{m}}{10^{n}} = 10^{m-n}$. If the resulting coefficient is not between $1$ and $10$, you must shift the decimal and adjust the exponent to maintain standard form.