Number - Scientific notation (Standard Form)

Definition of Standard Form: Standard form (or scientific notation) is a way of writing very large or very small numbers in the format $A \times 10^k$. The coefficient $A$ must be a number such that $1 \le A < 10$, meaning there is exactly one non-zero digit before the decimal point. The exponent $k$ must be an integer ($k \in \mathbb{Z}$).

Large Numbers and Positive Exponents: For numbers greater than 10, the exponent $k$ is positive. Visually, this represents how many places the decimal point has moved to the left to create a number between 1 and 10. For example, in $500$, the decimal moves 2 places left to become $5.0$, so it is written as $5 \times 10^2$.

Small Numbers and Negative Exponents: For numbers between 0 and 1, the exponent $k$ is negative. Visually, the decimal point moves to the right until it is positioned just after the first non-zero digit. For $0.004$, the decimal moves 3 places to the right to become $4.0$, resulting in $4 \times 10^{-3}$.

Converting to Ordinary Form: To convert from standard form back to an ordinary number, look at the exponent. If the exponent is positive, move the decimal point to the right, adding zeros as placeholders. If the exponent is negative, move the decimal point to the left. Imagine the decimal point 'jumping' over digits like a frog on a number line.

Operations - Multiplication and Division: To multiply numbers in standard form, multiply the coefficients and add the exponents using the law $10^m \times 10^n = 10^{m+n}$. To divide, divide the coefficients and subtract the exponents using the law $\frac{10^m}{10^n} = 10^{m-n}$. Always check that the final coefficient is still between 1 and 10; if not, adjust the exponent.

Significant Figures in Standard Form: Standard form is often used to clearly show the number of significant figures. In the expression $3.40 \times 10^5$, all digits in the coefficient $3.40$ are significant (3 sig figs). The power of 10 simply determines the magnitude (the scale of the number) without changing the precision.

Calculator Notation: On many scientific or graphic display calculators (GDC), scientific notation is shown using an 'E' or 'EE' symbol. For example, $6.2 \times 10^4$ might appear on the screen as $6.2E4$. It is important to translate this back to formal mathematical notation $A \times 10^k$ when writing your final answer.