Basic Mathematical Operations – Comprehensive Study Notes

Overview of the Four Basic Operations

• Mathematics is built on four fundamental operations that apply to all real (and most complex) numbers:

  • Addition — finding the sum $\left( + \right)$

  • Subtraction — finding the difference $\left( - \right)$

  • Multiplication — finding the product $\left( \times \text{ or } * \right)$

  • Division — finding the quotient $\left( \div \text{ or } / \right)$

• Equivalent fractional, decimal-fraction, and percentage representations are often encountered:

  • Fraction example: $\tfrac{9}{?}$ (illustrative)

  • Decimal example: $0.000$

  • Percentage example: $\%$

Addition

Definition

• Addition means combining two or more numbers into one.

• Denoted by the plus sign $+$.

• Commutative: $A+B = B+A$ — order is irrelevant.

• Applicable to whole, fractional, decimal, real, complex numbers, etc.

Rules for Integers

• $\text{positive} + \text{positive} \;\rightarrow\; \text{positive}$

• $\text{negative} + \text{negative} \;\rightarrow\; \text{negative}$

• Mixed signs: subtract absolute values and keep the sign of the larger absolute value.

Additive Identity & Inverse

• Identity element: $0$ ⇒ $0+A = A+0 = A$

• Additive inverse of $A$ is $-A$.
  $A + \left(-A\right) = 0$

Subtraction

Definition

• Subtraction yields the difference between two numbers; symbol $-$.

• It is the inverse of addition and can be viewed as adding a negative:
  $A - B = A + \left(-B\right)$

• Useful for answering “how many are left?” when something is taken away.

Rules for Integers

• $\text{positive} - \text{positive}$ → sign of the larger absolute value.

• $\text{negative} - \text{negative}$ → likewise, sign of larger absolute value.

• Equivalent restatement (matching transcript wording):

  • Both operands positive ⇒ answer positive.

  • Both operands negative ⇒ answer negative.

  • Different signs ⇒ subtract magnitudes, keep sign of the larger magnitude.

Multiplication

Definition

• Multiplication is repeated addition; symbols $\times, *$.

• Involves a multiplicand and a multiplier. Their result is the product.

Rules for Integers (Sign Rules)

• $\text{positive} \times \text{positive} \;\rightarrow\; \text{positive}$

• $\text{negative} \times \text{negative} \;\rightarrow\; \text{positive}$

• $\text{positive} \times \text{negative} \;\rightarrow\; \text{negative}$

Multiplicative Identity & Inverse

• Identity element: $1$ ⇒ $1\times A = A$

• Multiplicative inverse (reciprocal) of $A \; (A\neq0)$ is $\dfrac{1}{A}$.
  $A \times \dfrac{1}{A} = 1$

Division

Definition

• Division is the inverse of multiplication; symbols $\div$ or $/$.

• Dividend ÷ Divisor = Quotient.

  • If \text{Dividend} > \text{Divisor} ⇒ quotient >1.

  • If \text{Dividend} < \text{Divisor} ⇒ quotient <1.

Rules for Integers (Sign Rules)

• $\text{positive} \div \text{positive} \;\rightarrow\; \text{positive}$

• $\text{negative} \div \text{negative} \;\rightarrow\; \text{positive}$

• Different signs ⇒ negative quotient.

Order of Operations — BODMAS

• Brackets

• Orders (powers & roots)

• Division

• Multiplication

• Addition

• Subtraction

• Evaluate left-to-right within each tier. Division & multiplication share priority; addition & subtraction share priority.

Basic Arithmetic Properties

Commutative Property (Addition, Multiplication)

• $A+B = B+A$

• $A\times B = B\times A$

• Example: $1+2 = 2+1$; $1\times2 = 2\times1$

Associative Property (Addition, Multiplication)

• $A+(B+C) = (A+B)+C$

• $A\times(B\times C) = (A\times B)\times C$

• Example: $1+(2+3) = (1+2)+3$; $1\times(2\times3) = (1\times2)\times3$

Distributive Property

• $A\times(B+C) = A\times B + A\times C$

• Example: $2\times(3+4) = (2\times3)+(2\times4) \Rightarrow 2\times7 = 6+8 \Rightarrow 14 = 14$

Relationship & Conversions: Fractions, Decimals, Percentages

• A percent is a ratio “per hundred,” hence directly convertible to fractions/decimals.

• Universal principle: Any fraction ↔ decimal ↔ percent conversion is possible.

Converting a Fraction

• To decimal: divide numerator by denominator.
  Example: $\tfrac{3}{8} = 0.375$

• To percent: convert to decimal, move decimal 2 places right, append $\%$.
  $0.375 \rightarrow 37.5\%$

Converting a Decimal

• To fraction: “read” the decimal (e.g., $0.375 = \tfrac{375}{1000}$) then reduce.
  $\tfrac{375}{1000}=\tfrac{3}{8}$

• To percent: move decimal 2 places right, add $\%$.
  $0.08 \rightarrow 8\%$

Converting a Percent

• To decimal: move decimal 2 places left, drop $\%$.
  $45\% \rightarrow 0.45$

• To fraction: write over 100, reduce.
  $45\% = \tfrac{45}{100} = \tfrac{9}{20}$

Quick Sign-Rule Cheat Sheet (Integers)

Ethical / Practical / Real-World Notes

• Healthcare relevance (context of original presenter, nursing college):

  • Accurate drug dosing often requires rapid conversion between mg (fractions), mL (decimals), and concentrations (percent).

  • BODMAS ensures correct order when calculating complex dose equations.

• Numerical literacy in nursing prevents medication errors and improves patient safety.

Practice Prompts

• Simplify $\left(6 - 3\right) \times 2 + \tfrac{1}{4}$ using BODMAS.

• Find the additive inverse and multiplicative inverse of $-\tfrac{5}{7}$.

• Convert $0.0075$ to fraction and percent.

• A drug label says “(2.5\%) solution.” How many mg are in $1\,\text{mL}$? (Hint: (2.5\% = 2.5\,\text{g}/100\,\text{mL})).

Any Questions?

• Re-visit these notes before exams; they encapsulate every key point from the transcript.