Real Numbers and Their Properties from "summary" of NCERT Class 10 Mathematics Solutions by JagranJosh
Real numbers include all rational numbers and irrational numbers. Rational numbers are numbers that can be expressed as a ratio of two integers, whereas irrational numbers cannot be expressed as a ratio of two integers. The set of real numbers is denoted by the symbol R. Real numbers can be represented on the number line. The distance between two points on the number line is called the absolute value. The absolute value of a real number is always greater than or equal to zero. Real numbers have certain properties that help us perform operations on them. The properties of real numbers include closure property, commutative property, associative property, distributive property, additive identity property, multiplicative identity property, additive inverse property, multiplicative inverse property, and the property of 0. These properties help us simplify expressions and solve equations involving real numbers. The closure property states that the sum or product of two real numbers is also a real number. The commutative property states that the order of real numbers can be changed without affecting the result of addition or multiplication. The associative property states that the grouping of real numbers can be changed without affecting the result of addition or multiplication. The distributive property states that the product of a real number with the sum of two real numbers is equal to the sum of the products of the real number with each of the two real numbers. The additive identity property states that the sum of any real number and zero is the real number itself. The multiplicative identity property states that the product of any real number and one is the real number itself. The additive inverse property states that the sum of a real number and its additive inverse is zero. The multiplicative inverse property states that the product of a real number and its multiplicative inverse is one. The property of zero states that the product of any real number and zero is zero. Understanding the properties of real numbers is important in mathematics as they help us solve equations, simplify expressions, and perform operations on real numbers efficiently. By applying these properties, we can manipulate real numbers in a systematic way and arrive at accurate results. Real numbers and their properties form the foundation of algebra and other branches of mathematics. They are fundamental concepts that are used extensively in various mathematical applications.
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