Introduction to Real Numbers
Real numbers are the set of all rational and irrational numbers, forming the foundation of mathematics. They include integers, fractions, decimals, and square roots of non-perfect squares.
Definition of Real Numbers
Real numbers can be defined as the combination of both rational and irrational numbers. They can be integers (both positive or negative) including zero. These numbers are denoted by the symbol “R”. All the natural numbers, decimals, and fractions come under Real Numbers. See the figure, given below, which shows the classification of real numerals.
Classification of Real Numbers
- Natural Numbers:These are all counting Numbers like 1, 2,3,… to infinity
- Whole Numbers: These are numbers starting from zero(0) and have no end. Examples are 0,1,2,3,….
Integers: These are whole numbers, including positive numbers, negative numbers, and zero. Examples are -3, 0, 5, and 102.
Fractions: Fractions represent parts of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). Examples include 1/2, -3/4, and 5/7.
Decimals: Decimals are another way to represent fractions and are numbers that have a decimal point. Examples include 0.25, -3.75, and 6.123.
Irrational numbers: These are numbers that cannot be expressed as a fraction of two integers. They have non-repeating, non-terminating decimal expansions. Famous examples include π (pi) and √2 (the square root of 2).
Rational numbers: These are numbers that can be expressed as the quotient or fraction p/q of two integers, where q is not zero. All integers are rational numbers since they can be expressed as a fraction with a denominator of 1.
Real Numbers on a number line
Real numbers are represented on a horizontal line, in which zero(0) is taken in the middle of the line and is known as origin. In the right of “0” we take positive numbers whereas in the left of “0” we take negative. Keeping in we view the reference we locate all desired numbers. It shold be noted that we can locate all real numbers in their exact position except irrational numbers which can be represented at their approximate position . It is because that irrational numbers can’t be put in p/q form as shown in the figure.
Properties of Real Numbers
Real numbers have some cool properties that help us work with them easily:
Closure Property: When you add or multiply two real numbers together, you always get another real number. For example, 2 + 3 = 5, and 2 × 3 = 6, both of which are real numbers.
Commutative Property: It doesn’t matter what order you add or multiply such numbers in. For example, 2 + 3 is the same as 3 + 2, and 2 × 3 is the same as 3 × 2.
Associative Property: When you add or multiply three or more such numbers together, it doesn’t matter how you group them. For example, (2 + 3) + 4 is the same as 2 + (3 + 4), and (2 × 3) × 4 is the same as 2 × (3 × 4).
Applications of Real Numbers
Real numbers aren’t just for doing math problems in school. They’re super useful in the real world too!
Finance: Ever heard of interest rates on a loan or calculating how much money you’ll have in your savings account? Yup, that’s real numbers at work!
Science: Scientists use these numbers for measurements and calculations in fields like physics, chemistry, and biology. Whether it’s measuring the distance between planets or calculating the speed of a chemical reaction, such numbers are there.
Engineering: Engineers use such numbers to design buildings, bridges, cars, and even software. From calculating the load-bearing capacity of a structure to designing algorithms for computer programs, these numbers are essential.
Conclusion
Real numbers may seem like just a bunch of digits and decimals, but they’re so much more than that. They’re the building blocks of math and the foundation of our understanding of the world around us. So the next time you see a number, whether it’s on a calculator, a measuring tape, or a price tag, remember that it’s a real number, and it’s there to help us make sense of the world.
FAQ
- Yes, 0 is a real number. It is an integer and falls within the set of real numbers, which includes all rational and irrational numbers, along with zero.
- Real numbers are rational and irrational numbers, including integers, fractions, and decimals. They are represented by points on the number line.
- Five examples of real numbers: 3, -5/2 , π , 9.2 , 0 , √2
Yes, -√2 is a real number because it comes under irrational category
