Numerical Operations

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infoWhy this? Standard form provides an efficient and unambiguous way to represent, compare, and calculate with extremely large or small numbers. It strengthens place-value and index reasoning and is particularly useful in scientific and technological contexts.

scheduleWhy now? This unit builds on powers of ten, negative indices, decimal operations, and significant figures from earlier number work. These foundations now allow us to perform all four operations in standard form accurately and interpret the scale of results.

Standard form, also known as scientific notation, is a way of writing very large or very small numbers using powers of ten.
Numbers in standard form are written in the format `a \times 10^n`, where `1 \leq a < 10` and `n` is an integer.
To convert a large number to standard form, move the decimal point left until only one non-zero digit remains before the decimal point; the number of places moved becomes the positive power of ten.
To convert a small number (less than 1) to standard form, move the decimal point right until only one non-zero digit remains before the decimal point; the number of places moved becomes the negative power of ten.
When ordering numbers in standard form, compare the powers of ten first; if the powers are equal, compare the decimal parts.
When adding or subtracting numbers in standard form, the powers of ten must be the same before combining the decimal parts.
When multiplying numbers in standard form, multiply the decimal parts and add the powers of ten.
When dividing numbers in standard form, divide the decimal parts and subtract the powers of ten.

rocket_launchYou must be able to

Convert large numbers (e.g., 4,500,000) to and from standard form (e.g., `4.5 \times 10^6`).
Convert small numbers (e.g., 0.00032) to and from standard form (e.g., `3.2 \times 10^{-4}`).
Order a list of numbers given in standard form, from smallest to largest or vice versa.
Add and subtract numbers in standard form, ensuring the powers of ten are equal before combining the decimal parts.
Multiply and divide numbers in standard form, correctly applying the rules for multiplying or dividing the decimal parts and adding or subtracting the powers of ten.

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