Real‐line‐decimals‐significant‐figures - itnett/FTD02H-N GitHub Wiki

Her er en gjennomgang av alle bilder og vedlegget med optimal bruk av GitHub Wiki-markdown, LaTeX (ved bruk av $-symbolene for matematiske uttrykk), og emoji. Jeg vil inkludere relevante forklaringer og strukturer markdownen på en måte som gjør det enkelt å forstå og følge.

Markdown-output:

📘 Real Numbers, Decimals, and Significant Figures

  1. The Real Line and Number Systems 🧮

We visualize the real numbers ($\mathbb{R}$) as a continuous number line, called the real line, with zero in the middle, negative numbers to the left, and positive numbers to the right.

Numbers can be:

Natural Numbers: $\mathbb{N} = {0, 1, 2, 3, \dots}$

Integers: $\mathbb{Z} = {0, \pm1, \pm2, \pm3, \dots}$

Rationals: $\mathbb{Q} = {\frac{a}{b} \mid a, b \in \mathbb{Z}, b \neq 0}$

Irrationals: Real numbers that cannot be written as a fraction, such as $\sqrt{2}$, $\pi$.

Reals: $\mathbb{R}$, a system that includes all rational and irrational numbers.

  1. Decimal Expansions and Recurring Decimals 🔢

Decimal Expansion of a Real Number:

A non-negative real number $\alpha$ has a decimal expansion:

\alpha = b_n b_{n-1} \dots b_2 b_1 \cdot a_1 a_2 a_3 \dots

Recurring Decimal Expansions:

A real number is rational if and only if it has a recurring decimal expansion. For example:

\frac{6}{7} = 0.\overline{857142}

  1. Significant Figures and Scientific Notation 🚀

Significant Figures:

When quoting a number with limited precision, significant figures are used:

The number of digits counted from the leftmost positive digit is the number of significant figures.

Examples:

$26.103$ has 5 significant figures.

$0.00304$ has 3 significant figures.

$0.003040$ has 4 significant figures.

Scientific Notation:

A positive real number $\alpha$ is expressed in scientific notation as:

\alpha = b \cdot a_1 a_2 \dots a_m \times 10^k

Examples:

$193.034$ becomes $1.93034 \times 10^2$.

$0.003040$ becomes $3.040 \times 10^{-3}$.

  1. Accuracy Rules 📏

Addition and Subtraction:

When adding or subtracting, the answer should be rounded to the least number of decimal places in the given data.

Example:

9.4 + 2.13 = 11.53 \approx 11.5

Multiplication and Division:

When multiplying or dividing, the answer should be rounded to the least number of significant figures in the given data.

Example:

9.4 \times 2.13 = 20.022 \approx 20 $$ (rounded to 2 significant figures)

5. Radius of the Earth 🌍

  • The radius of Earth is approximately: $$ 6,370,000 \text{ m} = 6.37 \times 10^6 \text{ m}

70,000,000 \text{ m} = 7.0 \times 10^7 \text{ m}

  1. Detailed Examples 📊

Example 1: Adding Measurements with Different Decimal Places

Given:

$\alpha = 9.4$ (1 decimal place)

$\beta = 2.13$ (2 decimal places)

The sum is:

\alpha + \beta = 9.4 + 2.13 = 11.53 \approx 11.5 $$ (rounded to 1 decimal place)

The difference is: $$ \alpha - \beta = 9.4 - 2.13 = 7.27 \approx 7.3 $$ (rounded to 1 decimal place)

Example 2: Multiplying Measurements with Different Significant Figures

Given:

  • $\alpha = 9.4$ (2 significant figures)
  • $\beta = 2.13$ (3 significant figures)

The product is: $$ \alpha \times \beta = 9.4 \times 2.13 = 20.022 \approx 20 $$ (rounded to 2 significant figures)

The division is: $$ \frac{\alpha}{\beta} = \frac{9.4}{2.13} = 4.41314554 \approx 4.4 $$ (rounded to 2 significant figures)

7. Visual Representation and Additional Rules ✍️

Rule 1: When adding or subtracting, quote the result to the least number of decimal places.

Rule 2: When multiplying or dividing, quote the result to the least number of significant figures.

Dette markdown-formatet med GitHub Wiki vil gjøre det enkelt å legge til bilder, bruke matematisk formatering via LaTeX, og samtidig inkludere relevante emojier for å gjøre materialet mer visuelt tiltalende.