Solving multi-choice questions generally acts as a good practise test for any student. It is pertinent to mention that students can easily score full marks in these objective type Maths questions based on Number System with a little dedication, hard work and practice.
MCQ Questions for Class 9 Maths Number System provided in this website will act as a helping hand to students to revise the fundamental concepts quickly and secure excellent marks in their upcoming CBSE Class 9th Maths examination. But let us first understand the fundamental concepts, as per the syllabus.
- What is a number?
Ans: Any combination of digits makes a Number. The size of a Number depends on the count of digits that are used for its creation.
- What is a number system?
Ans: Also called the numeration system, it is a method of representing numbers on the Number Line with the help of a set of Symbols and rules. These symbols range between 0-9 and are called digits.
The numbers are represented uniquely and allow mathematical computations ranging from great scientific calculations to simple calculations like counting the number of students in a classroom. Number Systems can be of multiple types depending on the base value for their digits.
Numbers can be:
Real Number:
Any number represented on a number line or by an infinite decimal expansion is a real number. A real number is any rational or irrational number, including positive and negative whole numbers, decimals, integers, fractions, pi (π), Euler’s number (e), etc. We normally use and apply for these numbers while dealing with real-world applications.
Real Numbers can be:
- Natural numbers.
- Whole numbers.
- Integers.
- Fractions.
- Rational numbers.
- Irrational numbers, etc.
Natural Numbers: They are the most basic type of Numbers that range from 1 to infinity. These numbers are also called Positive or Counting Numbers. They are represented by the symbol N.
Eg: 1, 2, 3, 4, 5, 6, 7, etc.
Whole Numbers: They are the Natural Numbers, plus ‘zero’. Whole numbers are represented by the symbol W.
Eg: 0, 1, 2, 3, 4, etc.
Positive Numbers: Numbers represented on the right side of the zero (on a number line) are called Positive Numbers. The value of these numbers rises on moving towards the right from left. Example: 1, 2, 3, 4, …
Negative Numbers: Numbers represented on the left side of the zero (on a number line) are called Negative Numbers. The value of these numbers comes down while moving towards the left. Example: -1, -2, -3, -4, …
Integers: Integers include whole numbers and negative whole numbers. Integers can be positive, negative, or zero.
Integers don’t include fractional numbers, or they can’t be written in the form p/q, where p and q are two natural numbers. The range of Integers is from negative infinity to positive infinity, including zero. The symbol Z represents integers.
Eg: 1, -1, 0, 101 and -101, etc.
The sign of the integers
It is negative (-) for all numbers to the left of the origin (0) and positive (-) for all numbers to the right (+) of 0.
Positive integers are those with a (+) sign, and Negative ones are those with a (-) sign. Positive integers are generally written without their signs.
Absolute Value of an Integer
It is defined as the distance of that number from the origin (0). Since both 4 and -4 are 4 units (steps) away from the origin, the absolute value of both numbers is 4. The absolute value of -4 is 4, because it is three steps away from zero on the number line.
Fractions: Fractions are the numbers represented in the form of p/q, where ‘p’ belongs to the set of Whole numbers, while ‘b’ belongs to the set of Natural Numbers, i.e., b can’t be 0 (Zero).
The upper part of the fraction is called Numerator, whereas the lower part is called the Denominator. Hence, “p” and “q” are the numerator and the denominator, respectively.
E.g., 1/3, 4/7, 8/3, 3/17, etc.
Rational Numbers: Rational numbers are the numbers that can be written in the fractional form, such as “p/q”. Here, p and q are both integers and p≠0.
Example: -2/7, 0.74, 1/35, 13/8, etc.
Irrational Numbers: Irrational numbers are the numbers that can’t be represented in the form of fractions or as the quotient of two integers,i.e. they can not be written as p/q, where ‘p’ and ‘q’ are two Integers.
E.g. There is no number among integers and fractions equal to the square root of 2,3,5,7, etc.
An irrational number can be expressed as an infinite decimal expansion with no regularly repeating digit or group of digits.
Example: √2, √3, π, etc.
In other words, it can be a decimal number that does not terminate and does not repeat. E.g. 0.102003000400005…
Important tips to determine a Rational number:
It is already known that normal rational numbers can be written in the p/q form. But, some rational numbers, when written in decimal form, don’t stop a few places after the decimal.
E.g. ⅓ or 0.333……, 3/13 or 0.230769230769…, etc.
Thus, when a rational number is written in decimal form, we obtain either a decimal that terminates after a finite number of terms or a pattern that repeats.
Imaginary Numbers: An imaginary number is a number that, when squared, will result in a negative number. Essentially, an imaginary number, representing the square root of a negative number, doesn’t have a tangible value.
While it is not a real number, i.e. it can’t be quantified on the number line, imaginary numbers are “real” because they exist and are used in math. These numbers are also called complex numbers and are generally represented by “i” (√-1) except in electronics where they are represented by “j”, since “i” already denotes current.
E.g.: √-2, √-5,…
Imaginary numbers are used in cases where what is being measured relies on a sine or cosine wave.
Another thing: Explanation of dark matter and dark energy phenomena in astrophysics has become possible only after proving the principle of the physical reality of imaginary numbers discovered in mathematics 500 years ago.
As a corollary to this, it follows that we live in Multiverse rather than in Monoverse. All parallel universes of the Multiverse are mutually invisible.
Prime Numbers and Composite Numbers:
The numbers which don’t have any factor other than 1 (one) and the number itself are called Prime Numbers.
All the numbers other than Prime Numbers, except Zero, are Composite Numbers. Zero is neither a prime nor a composite number.
E.g., 2, 3, 5, 7, 11, 13, 17.. are prime numbers while 4, 6, 8, 9, 12,… are composite numbers.
- What are the main types of Number Systems?
Ans: There are various types of number systems, the most commonly used ones based on base and the number of digits allowed are:
- Decimal number system (of base 10),
- Binary number system (of base 2),
- Octal number system (of base 8) and
- Hexadecimal number system (of base 16).
Decimal Number System:
The Number system with a base value of 10 is known as the Decimal number system. It is also called the Hindu-Arabic number system or Arabic number system in mathematics. It uses the ten digits: 0,1,2,3,4,5,6,7,8 and 9, with the base number as 10, to create numbers.
Here, the place value is counted from right to left — the first place value is called units, second to the left as Tens, 3rd as Hundreds, 4th as Thousands, etc.
In a base-10 system or decimal system, the number 634.71 represents the sum (6 × 102) + (3 × 101) + (4 × 100) + (7 × 10−1) + (1 × 10−2).
In real life, we use the decimal number system to represent numbers. Interestingly, if any number is represented without a base, it indicates that its base is 10. For example, 72450, 3710, 822720 are some examples of numbers in the decimal number system.
Binary Number System
A method of representing numbers with 2 as their base and using only the two digits 0 and 1 for the creation of numbers is called the Binary Number System. Each successive digit represents a power of 2.
The binary number system is very useful when dealing with electronic devices, information theory and computer systems mainly due to the compact and reliable manner in which 0s and 1s can be represented in electromechanical devices with two states — viz. “ON-OFF”, “OPEN – CLOSED,” or “GO–no-GO.”
Decimal Numbers from 0 to 9 are represented in binary as: 0, 1, 10, 11, 100, 101, 110, 111, 1000, and 1001.
The binary number system is commonly used in computer languages like Java, C++, etc.
E.g. 2 in decimal notation is represented as (2)10 while the binary notation for 2 is represented as (10)2.
Octal Number System
The Octal Number System or Oct in short form is the base-8 number system and uses the digits 0 to 7, i.e. here, 10 represents 8 in the decimal system, and 100 represents 64 in the decimal system.
In this system, the position of every digit has a weight, which is a power of 8. Or each position in the Octal system is 8 times more significant than the previous state.
Thus, the numeric value of an octal number is determined by multiplying each digit by the value of the position in which the digit appears and then adding the products.
Therefore, it is also a positional (or weighted) number system. Let us dig a little further:
In the decimal system, each place is a power of ten.
Eg: 8510 = 8 x 101 + 5 x 100
In the octal system, each place is a power of eight.
Eg. 345 8 = 3 x 82 + 4 x 81 + 5 x 80
By performing the calculation above in the decimal system, we see that 345 in octal equals 192 + 32+ 5 = 229 in decimal.
Hexadecimal Number System
The Number System with a base value of 16 is known as Hexadecimal Number System. Here, the digits from 0-9 are taken, similar to those in the decimal number system, but the digits from 10-15 are represented as A-F.
This means 10 is represented as A, 11 as B, 12 as C, 13 as D, 14 as E, and 15 as F. Hexadecimal Numbers are useful for handling memory address locations.
Conclusion
Therefore, we hope you have a clear idea regarding the number system. Thoroughly practice to ace your exams and questions regarding this topic.
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