Questions and Answers for NCERT Class 10 Maths Chapter 1, Real Numbers (2023)

Questions and answers for NCERT Class 10 Maths Chapter 1, "Real Numbers":

1. What is a real number?

Answer: A real number is a number that can be represented by a point on the number line. It includes rational numbers, irrational numbers, and integers.

2. What is a rational number?

Answer: A rational number is a number that can be expressed in the form p/q, where p and q are integers and q is not equal to zero.

3. What is an irrational number?

Answer: An irrational number is a number that cannot be expressed in the form p/q, where p and q are integers and q is not equal to zero. Examples of irrational numbers include the square root of 2, pi, and e.

4. What is a terminating decimal?

Answer: A terminating decimal is a decimal that has a finite number of digits after the decimal point. For example, 0.25 is a terminating decimal.

5. What is a non-terminating decimal?

Answer: A non-terminating decimal is a decimal that has an infinite number of digits after the decimal point. For example, 0.333... is a non-terminating decimal.

6. What is a recurring decimal?

Answer: A recurring decimal is a decimal that has a repeating pattern of digits after the decimal point. For example, 0.333... is a recurring decimal, where the digit 3 repeats infinitely.

7. What is Euclid's division algorithm?

Answer: Euclid's division algorithm is a method for finding the highest common factor (HCF) of two integers. It involves repeated division and is based on the fact that the HCF of two numbers is the same as the HCF of one of the numbers and the remainder when the larger number is divided by the smaller number.

8. What is a prime number?

Answer: A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. For example, 2, 3, 5, 7, 11, and 13 are prime numbers.

9. What is a composite number?

Answer: A composite number is a positive integer greater than 1 that has more than two positive divisors. For example, 4, 6, 8, 9, and 10 are composite numbers.

10. What is the Fundamental Theorem of Arithmetic?

Answer: The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a unique product of primes, up to the order in which the primes are written. This means that every positive integer has a unique prime factorization.

11. What is a square number?

Answer: A square number is a number that is the product of an integer multiplied by itself. For example, 4, 9, 16, and 25 are square numbers.

12. What is a cube number?

Answer: A cube number is a number that is the product of an integer multiplied by itself twice. For example, 8, 27, and 64 are cube numbers.

13. What is the rationalizing factor?

Answer: The rationalizing factor is a factor that is multiplied with a given expression to eliminate any radicals in the denominator of the expression. For example, if we want to rationalize the denominator of the expression 1/√2, we can multiply both the numerator and denominator by √2 to get √2/2.

14. What is the decimal expansion of an irrational number?

Answer: The decimal expansion of an irrational number is a non-repeating, non-terminating decimal. For example, the decimal expansion of √2 is 1.41421356...

15. What is the decimal expansion of a rational number?

Answer: The decimal expansion of a rational number is either a terminating decimal or a repeating decimal. For example, the decimal expansion of 1/4 is 0.25, which is a terminating decimal, and the decimal expansion of 1/3 is 0.333..., which is a repeating decimal.

16. What is the difference between a natural number and a whole number?

Answer: A natural number is a positive integer greater than zero, while a whole number is a non-negative integer that includes zero. For example, 1, 2, 3, 4, and so on are natural numbers, while 0, 1, 2, 3, and so on are whole numbers.

17. What is the Euclidean algorithm for finding the HCF of two numbers?

Answer: The Euclidean algorithm involves dividing the larger number by the smaller number and finding the remainder. The larger number becomes the new dividend and the smaller number becomes the new divisor, and the process is repeated until the remainder is zero. The HCF is the last non-zero remainder.

18. What is the least natural number that has no common factor with any other natural number except 1?

Answer: The least natural number that has no common factor with any other natural number except 1 is 1 itself since it has no factors other than 1. This property is known as being a "unit" in the context of numbers.

19. What is the product of any two consecutive natural numbers?

Answer: The product of any two consecutive natural numbers is always one less than the square of the larger number. For example, the product of 3 and 4 is 12, which is one less than 16, the square of 4.

20. What is the relationship between the HCF and LCM of two numbers?

Answer: The product of the HCF and LCM of two numbers is equal to the product of the two numbers themselves. For example, if the two numbers are 12 and 18, their HCF is 6 and their LCM is 36, and 6 x 36 = 12 x 18 = 216.

21. What is prime factorization?

Answer: A prime factorization is a representation of a number as the product of prime numbers. For example, the prime factorization of 60 is 2 x 2 x 3 x 5.

22. What is the Fundamental Theorem of Arithmetic?

Answer: The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a unique product of prime numbers, up to the order in which the primes are written.

23. What is an irrational number?

Answer: An irrational number is a number that cannot be expressed as a ratio of two integers. Irrational numbers have non-repeating, non-terminating decimal expansions, and examples include √2 and π.

24. What is a rational number?

Answer: A rational number is a number that can be expressed as a ratio of two integers. Rational numbers have either terminating or repeating decimal expansions, and examples include 1/2, -3/4, and 0.8.

25. What is the square root of a number?

Answer: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 x 4 = 16.

26. What is the cube root of a number?

Answer: The cube root of a number is a value that, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because 3 x 3 x 3 = 27.

27. What is an even number?

Answer: An even number is a number that is divisible by 2. Examples of even numbers include 2, 4, 6, 8, and so on.

28. What is an odd number?

Answer: An odd number is a number that is not divisible by 2. Examples of odd numbers include 1, 3, 5, 7, and so on.

29. What is a composite number?

Answer: A composite number is a number that has more than two factors. Examples of composite numbers include 4, 6, 8, 9, and so on.

30. What is a prime number?

Answer: A prime number is a number that has exactly two factors: 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, and so on.

31. What is a perfect square?

Answer: A perfect square is a number that can be expressed as the square of an integer. Examples of perfect squares include 1, 4, 9, 16, 25, and so on.

32. What is a perfect cube?

Answer: A perfect cube is a number that can be expressed as the cube of an integer. Examples of perfect cubes include 1, 8, 27, 64, and so on.

33. What is a multiple of a number?

Answer: A multiple of a number is a number that can be obtained by multiplying that number by any integer. For example, 12 is a multiple of 3 because 12 = 3 x 4.

34. What is a factor of a number?

Answer: A factor of a number is a number that divides the original number exactly, without leaving any remainder. For example, 3 and 5 are factors of 15, because 15 ÷ 3 = 5 and 15 ÷ 5 = 3.

35. What is a common factor?

Answer: A common factor is a factor that two or more numbers have in common. For example, 2 and 3 are common factors of 6 and 12.

36. What is the greatest common factor?

Answer: The greatest common factor (GCF) of two or more numbers is the largest factor that they have in common. For example, the GCF of 12 and 18 is 6, because 6 is the largest factor that both numbers have in common.

37. What is the least common multiple?

Answer: The least common multiple (LCM) of two or more numbers is the smallest multiple that they have in common. For example, the LCM of 4 and 6 is 12, because 12 is the smallest multiple that both numbers have in common.

38. What is a common multiple?

Answer: A common multiple is a multiple that two or more numbers have in common. For example, 12 and 24 are common multiples of 4 and 6.

39. What is a fraction?

Answer: A fraction is a number that represents a part of a whole. It consists of a numerator (top number) and a denominator (bottom number) and is written in the form of a/b, where a is the numerator and b is the denominator.

40. What is a proper fraction?

Answer: A proper fraction is a fraction where the numerator is smaller than the denominator. For example, 1/2 and 3/4 are proper fractions.

41. What is an improper fraction?

Answer: An improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, 5/4 and 7/3 are improper fractions.

42. What is a mixed number?

Answer: A mixed number is a combination of a whole number and a proper fraction. For example, 3 1/2 is a mixed number, where 3 is the whole number and 1/2 is the proper fraction.

43. What is a decimal?

Answer: A decimal is a number that is written using a decimal point to separate the whole number from the fractional part. For example, 3.5 and 0.25 are decimals.

44. What is a terminating decimal?

Answer: A terminating decimal is a decimal that has a finite number of digits after the decimal point. For example, 0.75 and 0.6 are terminating decimals.

45. What is a repeating decimal?

Answer: A repeating decimal is a decimal that has a pattern of digits that repeats indefinitely after the decimal point. For example, 0.333... and 0.25 are repeating decimals.

46. What is a percent?

Answer: A percent is a number that is expressed as a fraction of 100. For example, 50% is the same as 0.50 or 1/2.

47. What is a ratio?

Answer: A ratio is a comparison between two quantities, expressed in the form of a fraction. For example, the ratio of boys to girls in a class could be 2/3, meaning that there are two boys for every three girls.

48. What is a proportion?

Answer: A proportion is a statement that two ratios are equal. For example, the statement "2/3 = 4/6" is a proportion, because the two ratios are equivalent.

49. What is a cross-product?

Answer: A cross product is a result of multiplying the numerator of one ratio by the denominator of the other, and vice versa. For example, the cross product of the ratios 2/3 and 4/5 would be 2 x 5 = 10 and 3 x 4 = 12, so the cross product would be 10/12, which can be simplified to 5/6.

50. What is a scale factor?

Answer: A scale factor is a ratio that compares the sizes of two similar figures. For example, if a small triangle is similar to a large triangle, the scale factor would be the ratio of their corresponding side lengths.

51. What is a variable?

Answer: A variable is a symbol or letter that represents a quantity that can change in value. For example, in the equation x + 2 = 5, x is the variable.

52. What is an equation?

Answer: An equation is a statement that shows that two expressions are equal. For example, 3x + 4 = 10 is an equation.

53. What is inequality?

Answer: An inequality is a statement that shows that two expressions are not necessarily equal, but one is greater than, less than, or equal to the other. For example, 3x + 4 < 10 is an inequality.

54. What is a function?

Answer: A function is a relationship between two variables, where the value of one variable depends on the value of the other. For example, y = 2x is a function, where the value of y depends on the value of x.

55. What is a domain?

Answer: The domain of a function is the set of all possible input values (usually x-values) that can be used in the function.

56. What is a range?

Answer: The range of a function is the set of all possible output values (usually y-values) that can be produced by the function.

57. What is a slope?

Answer: The slope of a line is the measure of how steeply the line rises or falls. It is calculated by dividing the change in the y-coordinates by the change in the x-coordinates.

58. What is an intercept?

Answer: An intercept is a point where a line intersects either the x-axis or the y-axis.

59. What is a coordinate plane?

Answer: A coordinate plane is a two-dimensional graph that is used to plot points and graph functions. It consists of two perpendicular number lines, the x-axis, and the y-axis.

60. What is an ordered pair?

Answer: An ordered pair is a pair of numbers that is used to represent a point on a coordinate plane. The first number represents the x-coordinate and the second number represents the y-coordinate.

61. What is a line of best fit?

Answer: A line of best fit is a straight line that is drawn through a set of data points on a scatter plot to approximate the trend of the data.

62. What is a scatter plot?

Answer: A scatter plot is a graph that is used to display the relationship between two variables. It consists of a series of data points that are plotted on a coordinate plane.

63. What is a frequency distribution?

Answer: A frequency distribution is a table that displays the number of times each value or range of values appears in a data set.

64. What is a histogram?

Answer: A histogram is a graph that displays the frequency distribution of a set of data. It consists of a series of adjacent rectangles, where the height of each rectangle represents the frequency of the corresponding data range.

65. What is a mode?

Answer: The mode is the value that appears most frequently in a data set.

66. What is a median?

Answer: The median is the middle value in a data set when the values are arranged in order.

67. What is a mean?

Answer: The mean is the sum of all the values in a data set, divided by the number of values in the data set.

68. What is a range?

Answer: The range is the difference between the highest and lowest values in a data set.

69. What is a standard deviation?

Answer: The standard deviation is a measure of the amount of variation or dispersion in a data set. It is calculated by finding the square root of the variance.

70. What is probability?

Answer: Probability is the measure of the likelihood or chance of an event occurring, expressed as a number between 0 and 1. An event with a probability of 0 is impossible, while an event with a probability of 1 is certain.