AP SSC Class 10 Maths Chapter 1 Real Numbers

Real numbers are a set of rational and irrational numbers. In the AP SSC Class 10 Maths Chapter 1 Real Numbers, we learn about various theorems that are used to explore the properties of rational and irrational numbers. We also, study about a type of function known as logarithms and see their applications in science and everyday life.

Theorems On Real Numbers

  • The Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorised) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.
  • Let x = \(\frac{p}{q}\) be a rational number, such that the prime factorization of q is of the form \(2^{n}5^{m}\), where n, m are non-negative integers. Then x has a decimal expansion which terminates.
  • Let x = \(\frac{p}{q}\) be a rational number, such that the prime factorization of q is not of the form \(2^{n}5^{m}\), where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).

Below, we have provided a few chapter questions with solutions,

  1. Express each number as a product of its prime factors.
    1. 140 – 140 can be expressed as a product of \(2\times 2\times 5\times 7=140\)
    2. 3825 – 3825 can be expressed as a product of \(5\times 5\times 3\times 3\times 17=3825\)
    3. 5005 – 5005 can be expressed as a product of \(5\times 7\times 11\times 13=5005\)
  1. Using the stated theorems, without actual division, state whether the following rational numbers are terminating or non-terminating decimals.
    1. \(\frac{4}{45}\)

      Solution: \(\frac{4}{45}\) can also be written as \(\frac{4}{3^{2}\times 5^{1}}\), since it is not in the form of \(2^{n}5^{m}\), it is a non-terminating decimal.

    2. \(\frac{25}{32}\)

      Solution: \(\frac{25}{32}\) can be expressed as \(\frac{25}{2^{5}\times 5^{0}}\), since it is the form of \(2^{n}5^{m}\), it is a terminating decimal.

  1. Write the following in logarithmic form.
    1. \(3^{5}=243\)

      Solution: \(3^{5}=243\) in logarithmic form is written as \(\log_{3}(243)=5\)

    2. \(10^{-3}=0.001\)

      Solution: \(10^{-3}=0.001\) in logarithmic form is written as \(\log_{10}(0.001)=-3\)

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