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 Maths 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 = [latex]\frac{p}{q}[/latex] be a rational number, such that the prime factorisation of q is of the form [latex]2^{n}5^{m}[/latex], where n, m are non-negative integers. Then x has a decimal expansion which terminates.
• Let x = [latex]\frac{p}{q}[/latex] be a rational number, such that the prime factorization of q is not of the form [latex]2^{n}5^{m}[/latex], 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 [latex]2\times 2\times 5\times 7=140[/latex]
   2. 3825 – 3825 can be expressed as a product of [latex]5\times 5\times 3\times 3\times 17=3825[/latex]
   3. 5005 – 5005 can be expressed as a product of [latex]5\times 7\times 11\times 13=5005[/latex]

2. Using the stated theorems, without actual division, state whether the following rational numbers are terminating or non-terminating decimals.
   1. [latex]\frac{4}{45}[/latex]

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

2. 1. [latex]\frac{25}{32}[/latex]

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

3. Write the following in logarithmic form.
   1. [latex]3^{5}=243[/latex]

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

3. 1. [latex]10^{-3}=0.001[/latex]

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

4. Find the HCF and LCM of 12 and 18 by the prime factorization method.

Solution: Given that 12 = 2 x 2 x 3 = 2² x 3¹ and  18 = 2 x 3 x 3 = 2¹ x 3²
HCF (12, 18) = 2¹ x 3¹ = 6 = Product of the smallest power of each common prime factors in the numbers.
LCM (12, 18) = 2² x 3² = 36 = Product of the greatest power of each prime factors, in the numbers.
So, here HCF (12, 18) x LCM (12, 18) = 12 x 18.

Thus, we can verify that for any two positive integers a and b, HCF (a,b) x LCM (a, b) = a x b. We can use this result to find the LCM of two positive integers, if we have already found the HCF of the two positive integers

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