**Real Numbers **

A real number is any number that can be found in the real world. Numbers are all around us. Natural numbers are traditionally used to count objects, rational numbers are used to represent fractions, irrational numbers are used to compute the square root of a number, and integers are used to measure temperature, etc. Together, these types of numbers form a real number collection.Real numbers have the property of being represented over a number line. Imagine a horizontal line. Imagine that it has its origin at zero. The positive points are located on the right, and the negative points are located on the left. Those points can be considered real numbers.

Among these numbers you will find a rational one like 34% and a rational one like 72.3, and you will also find an irrational one like pi. These numbers are in a line, making them easy to compare.In addition to being greater or less than another, they can also be ordered, and you can multiply, divide, and add them..

## Basic properties of Real Numbers

- Positive or negative real numbers are non-zero.
- Two non-negative real numbers summed together, or their product, is another non-negative real number, which is defined by a positive cone. This provides a linear order of the numbers on a number line.
- It is true that there are uncountably infinitely many real numbers, but there is only a countably infinite number of natural numbers; in other words, real numbers cannot be injected into natural numbers. Accordingly, more elements are in a countable set than there are real numbers.
- The real numbers are composed of a hierarchy of countably infinite subsets, such as integers, rationals, algebraics, and computables, each one being a separate subset of the next one in the hierarchy. It is uncountably infinite for all successive sets of irrational, transcendental, and non-computable real numbers in the reals.
- Measuring continuous quantities with real numbers is possible. In decimal representation, the number may have an infinite series of digits to the right of the decimal point. These are expressed like 324.823122147…, where the ellipsis (three dots) indicates that there are more to follow. The foregoing hints at the fact that only a fraction of real numbers can be precisely represented with finitely many symbols.

**Real Numbers Class 10 MCQ Online Test**

**Every Irrational Number is a Real Number**

(a) True

(b) False

### Every Real Number is an Irrational Number True or False

(a) True

(b) False

**Is -2 a Real Number**

(a) Yes

(b) No

**Which pair of complex numbers has a real number product**

A) (1 + 3i)(6i)

B) (1 + 3i)(2 – 3i)

C) (1 + 3i)(1 – 3i)

D) (1 + 3i)(3i)

**The decimal expansion of 22/7 is**

(a) Terminating

(b) Non-terminating and repeating

(c) Non-terminating and Non-repeating

(d) None of the above

**For some integer n, the odd integer is represented in the form of:**

(a) n

(b) n + 1

(c) 2n + 1

(d) 2n

**HCF of 26 and 91 is**

(a) 15

(b) 13

(c) 19

(d) 11

**Which of the following is not irrational?**

(a) (3 + √7)

(b) (3 – √7)

(c) (3 + √7) (3 – √7)

(d) 3√7

**The addition of a rational number and an irrational number is equal to:**

(a) rational number

(b) Irrational number

(c) Both

(d) None of the above

**The multiplication of two irrational numbers is:**

(a) irrational number

(b) rational number

(c) Maybe rational or irrational

(d) None

**If set A = {1, 2, 3, 4, 5,…} is given, then it represents:**

(a) Whole numbers

(b) Rational Numbers

(c) Natural numbers

(d) Complex numbers

**If p and q are integers and is represented in the form of p/q, then it is a:**

(a) Whole number

(b) Rational number

(c) Natural number

(d) Even number

The largest number that divides 70 and 125, which leaves the remainders 5 and 8, is:

(a) 65

(b) 15

(c) 13

(d) 25

**The least number that is divisible by all the numbers from 1 to 5 is:**

(a) 70

(b) 60

(c) 80

(d) 90

** The sum or difference of of two irrational numbers is always**

(a) rational

(b) irrational

(c) rational or irrational

(d) not determined

**The decimal expansion of the rational number 23/(22 . 5) will terminate after**

(a) one decimal place

(b) two decimal places

(c) three decimal places

(d) more than 3 decimal places

**Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy**

(a) 1 < r < b

(b) 0 < r ≤ b

(c) 0 ≤ r < b

(d) 0 < r < b

**For some integer m, every even integer is of the form**

(a) m

(b) m + 1

(c) 2m

(d) 2m + 1

**Using Euclid’s division algorithm, the HCF of 231 and 396 is**

(a) 32

(b) 21

(c) 13

(d) 33

Answer: (d) 33

** If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is**

(a) 4

(b) 2

(c) 1

(d) 3

**The prime factorisation of 96 is**

(a) 25 × 3

(b) 26

(c) 24 × 3

(d) 24 × 32

** n² – 1 is divisible by 8, if n is**

(a) an integer

(b) a natural number

(c) an odd integer

(d) an even integer

** For any two positive integers a and b, HCF (a, b) × LCM (a, b) =**

(a) 1

(b) (a × b)/2

(c) a/b

(d) a × b

**The values of the remainder r, when a positive integer a is divided by 3 are**

(a) 0, 1, 2

(b) Only 1

(c) Only 0 or 1

(d) 1, 2