Foundations of Mathematics: Some basic mathematics
Zero:
Zero is neither positive nor negative and is considered an even number.
Numbers:
- Whole numbers initiate from zero.
- Positive numbers are greater than zero, while negative numbers are less than zero.
Natural Numbers:
- Natural numbers, also known as counting or cardinal numbers, commence from 1.
Number 1:
- 1 is a natural and whole number.
- Not prime, not composite; the only positive integer neither prime nor composite.
Numbers:
- The smallest prime number is 2.
- The smallest composite number is 4.
- Integers encompass positives, negatives, and zero but exclude decimals and fractions.
Numeral and Place Value:
- A numeral is a group of digits.
- Place value indicates the local value of a digit based on its place.
- Face value denotes the actual value of a digit.
Types of Numbers:
- Natural numbers start from 1.
- Ordinal numbers signify order (first, second, third, etc.).
- Whole numbers include zero and all natural numbers.
- Integers encompass natural, positive, negative numbers but exclude decimals and fractions.
- Non-negative integers: 0, 1, 2, 3…
- Non-positive integers: 0, -1, -2, -3…
- Prime numbers have only two factors (1 and itself).
- Composite numbers have more than two factors and are not prime.
- Even numbers are divisible by 2.
- Odd numbers are not divisible by 2.
- Co-prime numbers share a highest common factor of 1.
Rational Numbers:
- Expressible as P/Q, where P and Q (Q≠0) are integers.
- Include perfect squares, terminating decimals, and repeating decimals.
Irrational Numbers:
- Real numbers that cannot be expressed as a fraction of two integers.
- Examples: √2 and π.
Real Numbers:
- A combination of rational and irrational numbers.
- A complete representation of the number system.
Angle Concepts:
- A straight line angle measures 180 degrees.
- The sum of all angles in a triangle is 180 degrees.
- A circle has 360 degrees.
- A rectangle also has 360 degrees.
- A complete angle measures 360 degrees.
Types of Angles:
- Acute: measures less than 90 degrees.
- Right: measures exactly 90 degrees.
- Obtuse: measures between 90 and 180 degrees.
- Reflex: measures between 180 and 360 degrees.
Triangles:
- The sum of all angles in a triangle is 180 degrees.
Types of Triangles (based on Side Length):
- Equilateral: all sides are equal.
- Isosceles: two sides are equal.
- Scalene: all sides and angles are unequal.
Types of Triangles (based on Angle Measurement):
- Acute: all angles measure less than 90 degrees.
- Right: one angle measures exactly 90 degrees.
- Obtuse: one angle measures greater than 90 degrees.
Angle Relationships:
- Complementary angles sum up to 90 degrees.
- Supplementary angles sum up to 180 degrees.
- Adjacent angles share a vertex and a common arm.
Right Angles and Triangles:
- A right angle measures 90 degrees.
- A right triangle contains one right angle.
Lines and Line Segments:
- A line consists of infinite points.
- A line segment is part of a line with two endpoints.
Angle Vertex:
- The vertex of an angle is the common endpoint of two rays forming the angle.
Circle Properties:
- Circumference: distance around a circle.
- Diameter: distance from one side to the other, passing through the center.
- π (pi): ratio of the circumference to diameter, approximately 3.14.
Quadrilateral:
- A polygon with four sides and four vertices.
- Can be regular or irregular.
- Sum of interior angles is 360°.
Square:
- A quadrilateral with all four sides equal, opposite sides parallel, and all angles equal to 90°.
Rectangle:
- A quadrilateral with all four sides equal, opposite sides parallel, and all angles equal to 90°.
Parallelogram:
- Opposite sides equal and parallel.
- Opposite angles equal.
- No angle measures 90 degrees.
Rhombus:
- Four equal sides.
- Opposite sides parallel.
- Opposite angles equal.
- No angle measures 90 degrees.
Trapezium:
- Only one pair of opposite sides parallel.
Kite:
- Two pairs of adjacent equal sides.
- One pair of equal angles.
Perimeter and Area:
- Perimeter is the boundary distance.
- Area is measured in square units.
Rectangle:
- Perimeter: 2(length + width).
- Area: length × width.
Square:
- Perimeter: 4 × side or sum of all sides (a + a + a + a).
- Area: side × side.
Circle:
- Circumference (Perimeter): π × diameter.
- Area: π × radius².
Calculation Example:
Given length (L) and width (W) of a rectangle are 5.3 cm each:
- Perimeter: 2(L + W) = 2(5.3 + 5.3) = 21.2 cm.
- Area: L × W = 5.3 × 5.3 = 28.09 cm².
Conversion: Fahrenheit to Celsius:
- C = (F – 32) × (5/9).
Conversion: Celsius to Fahrenheit:
- F = (C × 9/5) + 32.
Part/Whole Percentage:
- Percentage = (part/whole) × 100.
Cross Multiplying:
- A method to find missing values in proportions.
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Some basic mathematics