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A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. It is significant because prime numbers are the building blocks of all natural numbers, as every composite number can be expressed as a product of prime numbers.
Two numbers are coprime if their highest common factor (HCF) is 1. This means they do not share any prime factors other than 1.
The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of prime numbers, except for the order of the factors.
Irrational numbers cannot be expressed as a ratio of two integers and have non-terminating, non-repeating decimal expansions. In contrast, rational numbers can be expressed as a fraction of two integers and may have terminating or repeating decimal expansions.
The relationship between HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of two numbers a and b is given by the equation: HCF(a, b) × LCM(a, b) = a × b.
For n^a to end with the digit 0, the prime factorization of 'a' must include both 2 and 5, as 10 is the product of these two primes.
A factor tree is a visual representation of the prime factorization of a number, showing how it can be expressed as a product of prime numbers. It is used to simplify calculations involving HCF and LCM.
Terminating rational numbers are those that can be expressed as a fraction where the denominator is a power of 10, resulting in a decimal that ends after a finite number of digits, such as 0.5 or 0.71.
To prove that √3 is irrational, assume it can be expressed as a fraction p/q in simplest form. Squaring both sides leads to a contradiction, showing that both p and q must be divisible by 3, which contradicts the assumption that p/q is in simplest form.
Non-terminating repeating decimals indicate that a rational number can be expressed as a fraction, where the decimal part repeats indefinitely, such as 1/3 = 0.333... This shows that rational numbers can have complex decimal representations.
If the HCF of two numbers is 1, it implies that the numbers are coprime, meaning they do not share any common factors other than 1, which can simplify calculations involving these numbers.
Natural numbers are the set of positive integers starting from 1 (1, 2, 3, ...), while whole numbers include all natural numbers plus zero (0, 1, 2, 3, ...).
A non-repeating decimal expansion is identified by its inability to form a repeating pattern in its digits. Examples include numbers like π or √2, which continue infinitely without repeating.
Prime factorization allows for the identification of common and unique prime factors, which can be used to calculate the HCF by taking the product of common prime factors with the lowest exponents, and the LCM by taking the product of all prime factors with the highest exponents.
Real numbers include all rational and irrational numbers, encompassing integers, whole numbers, and natural numbers. They can be represented on the number line and include both terminating and non-terminating decimals.
The number 2 is significant as it is the only even prime number. All other even numbers can be divided by 2, making them composite.
A non-terminating decimal can often be expressed as a fraction by identifying the repeating part and using algebraic methods to derive the fraction form, such as setting the decimal equal to a variable and manipulating the equation.
Understanding coprime numbers is important for simplifying fractions, calculating HCF and LCM, and solving problems involving divisibility and number theory.
Examples of irrational numbers include π (approximately 3.14159...) and √2 (approximately 1.41421...). Their decimal expansions are non-terminating and non-repeating.
The LCM of two coprime numbers is calculated by simply multiplying the two numbers together, as they do not share any common factors.
The decimal expansion is significant because rational numbers can have either terminating or repeating decimal expansions, while irrational numbers have non-terminating and non-repeating decimal expansions.