$\color{#55F}\text{Politics is for the present,}$ $\color{#55F}\text{but an equation is for eternity.}$

• ⚛️ $\color{#56E}\text{Albert Einstein}$

🌍 Standard Notation

Derived from Arabic numerals, whose origins trace back to classical India 🐘. This system revolutionized counting and brought clarity to mathematics. Just as we learned that precise logic underpins every mathematical theory, standard notation provides the exact language needed to represent numbers and communicate quantitative ideas unambiguously.

🧮 In Standard Notation, we use base ten (decimal), which is the number system we're all most familiar with.

📕 The digits represent quantities from zero to nine.

\color{#666}
\begin{matrix}
0  & := &  \\
1 & :=  & 𓃉   \\
2 &  := &  𓃊   \\
3 &  := &  𓃋   \\
4 &  := &  𓃌   \\
5 &  := &  𓃍   \\
6 &  := &  𓃎   \\
7 &  := &  𓃏   \\
8 &  := &  𓃐  \\
9 &  := &  𓃑   \\
10 &  := &  𓃉𓃑   \\
\end{matrix}

📏 Position & Power:

📗 Each digit’s position indicates a quantity of a power of 10, and together, they create large (or small) numbers by summing those quantities:

\color{#675}
\begin{matrix}
1   & : &   Unit &       (10⁰) \\
10  & : &   Tens &       (10¹) \\
100 & : &   Hundreds &   (10²) \\
1⠀000 & :  & Thousands  & (10³) \\
⁝
\end{matrix}

📓 Grouping digits

📔 We typically group digits in sets of three using a separating spacer like , ., ,, _, or '.

• Example (English): 1,000,000
• Example (Spanish): 1.000.000 This convention makes large numbers easier to parse and work with.

📓 Decimal numbers:

We use a different separator (usually . or ,) to indicate decimals (numbers with negative powers of 10), separating whole numbers from fractional parts.

\color{#684}
1.000.000.000,50 = \text{ One billion and fifty-hundredths.}

Decimal notation allows us to represent parts of a whole with clarity and precision.

🧑‍🔬 Scientific Notation

📕 When dealing with extremely large or small numbers, scientific notation becomes a powerful tool. Instead of writing a giant string of zeros, we use powers of 10 to condense everything neatly.

📕 The number is written as

\color{#693}
a \times 10^b

\color{#693}
a· ⏨b

\color{#693}
a· EⅩ^b

• Where $a$ is a number between 1 and 10 ($1≤a<10$)
• And $b$ is the integer exponent that indicates how many times to multiply or divide by 10.

📔 $\color{#6A2}2,300,000. = 2.3 · 10^6 = 2.3 EⅩ^6 = 2.3 ⏨6$

🪐 Large Number:

📔 $\color{#6B1}7.4 \times 10^9 = 7,400,000,000$

• This notation means the number 7.4 is multiplied by 10 raised to the power of 9.

🦠 Small Number:

📔 $\color{#6C0}7.4 \times 10^{-9} = 0.0000000074$

• Here, $7.4$ is divided by $10^9$, shifting the decimal point 9 places to the left.

Scientific notation condenses lengthy numerical strings into a compact, efficient form! Ideal for scientific calculations.

🔄 Converting Between Standard and Scientific Notation

📔 From Scientific Notation to Standard:

Positive exponent

📔 Move the decimal point to the right.

• $\color{#6DF}4.7 \times 10^5 \text{→ Move the decimal 5 spaces to the right →} 470,000$.

Negative exponent

📔 Move the decimal point to the left.

• $\color{#6EE}4.7 \times 10^{-5}\text{ → Move the decimal 5 spaces to the left → } 0.000047$

📔 From Standard to Scientific Notation:

• Count how many places you have to move the decimal point so that there is only a number between 1 and 10 that remains.

• The number of places is your exponent for $10$.

📒 Example:

• $\color{#6FD}320,000 \text{ becomes }3.2 \times 10^5$.
• $\color{#7FC}0.00032 \text{ becomes } 3.2 \times 10^{-4}$.

🧠 Operations in Scientific Notation

📓 Multiplying Numbers:

📘 When multiplying two numbers in scientific notation, multiply the coefficients and add the exponents:

$\color{#70C}(2 × 10^4 ) (3 × 10^5 )$
$\color{#71B}= 2 • 10^4 • 3 • 10^5$
Keep the base 10 and add the exponents $\color{#72A}10^{(4+5)} = 10^9$
$\color{#739}= 2 × 3 × 10^9$
$\color{#748}= 6 × 10^9$

📓 Dividing Numbers:

📙 When dividing two numbers in scientific notation, divide the coefficients and subtract the exponents:

\color{#757}
\frac{8 \times 10^9}{4 \times 10^6} = \frac{8}{4} \times 10^{9-6} = 2 \times 10^3

📔 Keep the base 10 and subtract the exponents : $\color{#766}10^{9 - 6} = 10^3$ .

🔢 Why Use Scientific Notation?

📕 Scientific notation is a lifesaver when dealing with extreme values:

• Large scales (e.g., the distance between planets 🌍🪐)
• Tiny scales (e.g., the size of atoms ⚛️)

📕 It’s compact, efficient, and perfect for the digital age 💻, making calculations faster and cleaner while avoiding those long strings of zeros.

The precision of standard notation mirrors the logical rigor we explored earlier. Just as our language of mathematics builds from clear definitions and logical sentences, our numerical notation—whether standard or scientific—provides the exact tools for representing, manipulating, and communicating quantitative ideas. This consistency across all levels of mathematics enables us to build complex theories on a foundation of unambiguous clarity.

Embrace these notational tools as the next step in your journey of discovery, ensuring that every number and every calculation speaks with the precise language of mathematics.