More on Real Numbers

       In this section we continue to develop your computational skills on real numbers with emphasis on rational numbers. Use this section on rational numbers so that you will develope a solid foundation when you study algebbraic concepts in subsequent chapters.

Properties of Rational Numbers

Insert a Fraction between Two Given Fractions

       If a/b and c/d are any two fractions, then the fraction a+c/b+d lies between the given fractions a/b and c/d.

Alternative Method

       A rational number between 2/3 and 3/5 is 1/2(2/3 + 3/5)=19/30. That is,
2/3 > 19/30 > 3/5.

Note:    By using different methods, we may obtain different fractions as our              answer.

Operations on Rational Numbers

For any two rational numbers a/b and c/d, we define:

  1. addition: a/b + c/d= (ad + bc) ÷ bd

  2. subtraction: a/d - c/d= ()ad - bc) ÷ bd

  3. multiplication: a/b × c/d=ac/bd

  4. division: a/b ÷ c/d=a/b × d/c=ad/bc where b is not equal to 0, c is not equal to 0, and d is not equal to 0.

 

Properties of Operation on Rational Numbers

AdditionMultiplication
  1. Closure:

    a + b is unique rational number.

  2. Commutative:

    a + b=b + a

  3. Associative:

    (a + b) + c=a + (b + c)

  4. Identity:

    The additive identity of a rational number is zero (0), that is;

    a + 0=0 + a=a.

  5. Additive Inverse:

    For every positive rational number a, there is a negative rational number -a such that a + (-a)=0.

  6. Distributive:

    a (b + c)=(a × b) + (a × c)
  1. Closure:

    a × b is a unique rational number.

  2. Commutative:

    a × b=b × b

  3. Associative:

    (a × b) × (b × c)

  4. Identity:

    The multiplicative identity of a rational number is 1, that is, a × 1=1 × a=a.

  5. Multiplicative Inverse:

    For every rational number a except 0, there is a rational number which is the reciprocal of the number such that a × 1/a=1. a /b has the reciprocal 1/a=1 × b/a=b/a.

An Important Property of Rational Numbers

       There are differences between integers and rational numbers. For example, between two integers, there may be no other integer or definite number of rational numbers. For example, let us take 0 and 1 (no integers in between).

Irrational Numbers

       A number √a (square root of a) is called an irrational number if a is positive and a is not the square of rational number.

Real Numbers

       The set of Real numbers is the collection of rational numbers and irratioinal numbers.

Additional properties of real numbers

  1. The sum, difference, and product of two real numbers are real numbers.
  2. The division of a real number by a nonzero real number gives a real number.
  3. Evvery real number has a corresponding negative real number. The numbe zero is neither positive or negative. Zero is its own opposite, so +0=-0,, that is, zero steps to the right is the same sa zero steps to the left.
  4. The sum, difference, and product, and quotient of a rational nnumber and an irrational number are irrational.
  5. The sum, difference, product, and quotient of two irrational numbers need not be irrational.
  6. Given two real numbers a and b, √a × √b=√ab, √a/√b=√a/b, and (√a)2=√a × √a=√a × a=a.

Note: a cannot be negative

Rationalization

       In order to transform an irrational number into rational, we multiply the number by its rationalizing factor. This process is called rationalization. For example, 3 + √5 is an irrational number. If we multiply it by another irrational number 3 - √5.

(3 + √5)(3 - √5)=32 - (√5)2=9 - 5=4, which is a rational number.

We say that 3 - √5 is the rationalizing factor of 3 + √5. Similarly, 3 + √5 is the rationalizing factor of 3 - √5.

Note: If a + √b is a irrational number then a - √b is the rationalizing factor of a + √b. Similarly, a + √b is the rationalizing factor of a - √b. a + √b and a -√b are said to be conjugates to each other. There are innumerable irrational numbers between two irrational numbers.

Real numbers



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