Number - Scientific Notation

Scientific notation, also known as standard form, is a method used to write very large or very small numbers in a concise format. It is written as $a \times 10^n$, where the coefficient $a$ must satisfy $1 \le a < 10$ and the exponent $n$ must be an integer. Visually, the coefficient $a$ always has exactly one non-zero digit to the left of the decimal point.

When converting a large number (greater than 10) to scientific notation, the decimal point is moved to the left. Each movement or 'jump' increases the value of the positive exponent $n$. For example, to convert $5,800,000$, imagine the decimal point jumping 6 places from the end of the number to the position between $5$ and $8$, resulting in $5.8 \times 10^6$.

To convert a small number (between 0 and 1) to scientific notation, the decimal point moves to the right. Each jump to the right results in a negative exponent $n$. If you have $0.00042$, visualize the decimal shifting 4 places to the right to sit between $4$ and $2$, which gives $4.2 \times 10^{-4}$.

When multiplying numbers in scientific notation, you multiply the coefficients ($a$) and add the exponents ($n$) according to the laws of indices. If the resulting coefficient is not between 1 and 10, you must adjust it. For example, $(4 \times 10^2) \times (3 \times 10^5)$ initially gives $12 \times 10^7$, but must be corrected to $1.2 \times 10^8$ by shifting the decimal left and increasing the exponent.

When dividing numbers in scientific notation, divide the coefficients and subtract the exponent of the divisor from the exponent of the dividend. This follows the index law $\frac{x^m}{x^n} = x^{m-n}$. For instance, $(6 \times 10^8) \div (2 \times 10^3)$ results in $3 \times 10^5$.

Addition and subtraction require the exponents of the powers of 10 to be the same before the operation can be performed. Visualize this as aligning the place values. If the exponents differ, you must rewrite one of the numbers. For example, to add $2 \times 10^3$ and $3 \times 10^2$, you could change $3 \times 10^2$ to $0.3 \times 10^3$ so that you can add the coefficients: $(2 + 0.3) \times 10^3 = 2.3 \times 10^3$.

To convert from scientific notation back to ordinary (standard) decimal form, look at the exponent $n$. If $n$ is positive, move the decimal point $n$ places to the right (filling gaps with zeros). If $n$ is negative, move the decimal point $n$ places to the left, which visually makes the number smaller and closer to zero.