Operations on Real Numbers

Table of Contents

What is a Real Number?

A real number is a number that can be represented in decimal notation, and that is not a complex number. Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

What are Mathematical Operations?

Mathematical operations are the basic arithmetic and algebraic manipulations that are used to solve mathematical problems. The most common operations are addition, subtraction, multiplication, and division, but there are many others, including exponents, roots, and logarithms. These operations can be performed on numbers, variables, and other mathematical expressions.

Addition of Two Rational Numbers

The sum of two rational numbers is another rational number. To find the sum of two rational numbers, add the two numbers together and keep the denominator the same.

Subtraction of Two Rational Numbers

To subtract two rational numbers, subtract the numerators and subtract the denominators.

Multiplication of Two Rational Numbers

To multiply two rational numbers, multiply the numerators and multiply the denominators.

For example, to multiply 3 and 5, multiply 3×5=15 and 5×3=15. The product is 15/15=1.

Division of Two Rational Numbers

The division of two rational numbers can be done by using the division algorithm.

The division algorithm states that to divide two rational numbers, divide the numerator by the denominator.

For example, to divide 5 by 2, divide 5 by 2 to get 2.5.

Addition of Two Irrational Numbers

Let’s add two irrational numbers together. We’ll use the irrational number π and the irrational number 2.

We’ll start by writing out the two irrational numbers.

π
2

Next, we’ll add the two numbers together.

π + 2

We get the answer 3.14159.

Subtraction of Two Irrational Numbers

To subtract two irrational numbers, subtract the numerators and subtract the denominators.

\frac{1}{2} – \frac{1}{3} = \frac{1}{6}

-\frac{\sqrt{3}}{2} = -\frac{1}{2}

Multiplication of Two Irrational Numbers

To multiply two irrational numbers, one must use the following steps:

1. Convert the irrational numbers to decimals.

2. Multiply the decimals.

3. Convert the answer back to irrational numbers.

For example, to multiply √2 and π, the steps would be:

1. Convert √2 to 1.41421356

2. Multiply 1.41421356 by 3.14159265

3. Convert the answer back to √2.

The answer is 4.7123889

Another example, to multiply √3 and π/2, the steps would be:

1. Convert √3 to 1.73205081

2. Multiply 1.73205081 by 1.5

3. Convert the answer back to √3.

The answer is 2.618033988

Division of Two Irrational Numbers

The division of two irrational numbers is not a rational operation.

Addition of an Irrational and a Rational Number

The sum of an irrational and a rational number is always irrational.

Subtraction of an Irrational and a Rational Number

To subtract an irrational number from a rational number, simply subtract the numerators and subtract the denominators.

Multiplication of an Irrational and a Rational Number

When multiplying two irrational numbers, the result will be irrational. For example, when multiplying \(\sqrt{2}\) and \(\sqrt{3}\), the result will be irrational. When multiplying a rational number and an irrational number, the result will be irrational. For example, when multiplying 3 and \(\sqrt{2}\), the result will be irrational.

Division of an Irrational Number with a Rational Number

To divide an irrational number with a rational number, divide the numerator and denominator of the rational number by the root of the irrational number.