Math Unit 1

Overview: This unit focuses on the fundamental properties of real numbers and the mathematical principles required for advanced calculations. It specifically addresses irrational numbers, standardized scientific notation, and the laws governing exponents.

Definition: An irrational number is a real number that cannot be written as a simple fraction $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .
Characteristics of Irrational Numbers:
- Non-Terminating Decimals: The decimal representation of an irrational number continues infinitely without ever ending.
- Non-Recurring Decimals: The digits in the decimal expansion do not settle into a repeating pattern or cycle.
Examples of Irrational Numbers:
- Square Roots of Non-Square Numbers: Many square roots are irrational. For instance, $\sqrt{2}$ cannot be expressed as a ratio of integers; its decimal expansion starts as $1.41421356...$ .
- Pi ( $\pi$ ): The ratio of a circle's circumference to its diameter is a famous irrational constant, approximately equal to $3.14159265...$ .
- Euler's Number ( $e$ ): The base of the natural logarithm, which is approximately $2.71828...$ .
Distinction from Rational Numbers:
- Rational numbers can always be written as $\frac{p}{q}$ . Their decimals either terminate (e.g., $0.25$ ) or repeat periodically (e.g., $0.333...$ ).

Definition: Standard form, also known as scientific notation, is a method of writing numbers that are either very large or very small in a compact and standardized way.
General Structure: A number in standard form is written as $A \times 10^n$ .
Mathematical Constraints:
- The value of the coefficient $A$ (the mantissa) must satisfy the condition: 1 \le |A| < 10. This means there is only one non-zero digit before the decimal point.
- The exponent $n$ must be an integer (a whole positive or negative number).
Direction of the Decimal Shift:
- Large Numbers (n > 0): For values greater than or equal to $10$ , the decimal point is moved to the left, and $n$ is positive.
- Small Numbers (n < 0): For values between $0$ and $1$ , the decimal point is moved to the right, and $n$ is negative.
Examples:
- The speed of light is approximately $300,000,000\,m/s$ , which is written in standard form as $3 \times 10^8\,m/s$ .
- The diameter of a human cell might be $0.00001\,m$ , written as $1 \times 10^{-5}\,m$ .

Definition: An index (plural: indices), also referred to as an exponent or power, indicates how many times a base number is multiplied by itself.
Key Notation: In the expression $x^n$ , $x$ is the base and $n$ is the index.
Fundamental Laws of Indices:
- Multiplication Law: When multiplying terms with the same base, add their indices.
- Formula: $a^m \times a^n = a^{m+n}$ .
- Division Law: When dividing terms with the same base, subtract the index of the divisor from the index of the dividend.
- Formula: $\frac{a^m}{a^n} = a^{m-n}$ .
- Power of a Power Law: When a term raised to a power is itself raised to another power, multiply the indices.
- Formula: $(a^m)^n = a^{m \times n}$ .
Special Index Laws:
- Zero Index: Any non-zero base raised to the power of zero is always equal to $1$ .
- Formula: $a^0 = 1$ , where $a \neq 0$ .
- Negative Indices: A negative index indicates the reciprocal of the base raised to the positive version of that power.
- Formula: $a^{-n} = \frac{1}{a^n}$ .
- Fractional Indices: A fractional index represents the root of a number. Specifically, the denominator of the fraction indicates the order of the root.
- Formula: $a^{\frac{1}{n}} = \sqrt[n]{a}$ .
- General Fractional Index Formula: $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$ or $\sqrt[n]{a^m}$ .