What Is The Product Of $(2-3i)$ And $(1+5i)$ In Standard Form? Use The Keypad To Enter Your Answer In The Box. $\square$

Introduction

Complex numbers are mathematical expressions that consist of a real part and an imaginary part. They are used to represent points in a two-dimensional plane, known as the complex plane. In this plane, the real part of a complex number is represented on the x-axis, and the imaginary part is represented on the y-axis. Complex numbers have numerous applications in mathematics, physics, and engineering, including the representation of electrical circuits, signal processing, and quantum mechanics.

What are Complex Numbers?

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. The real part of a complex number is denoted by $a$, and the imaginary part is denoted by $b$. For example, the complex number $3 + 4i$ has a real part of $3$ and an imaginary part of $4$.

Multiplication of Complex Numbers

To multiply two complex numbers, we use the distributive property of multiplication over addition. Let's consider the product of two complex numbers, $(2-3i)$ and $(1+5i)$. To find the product, we multiply each term of the first complex number by each term of the second complex number.

Step 1: Multiply the Real Parts

The real part of the first complex number is $2$, and the real part of the second complex number is $1$. We multiply these two numbers to get the real part of the product.

Step 2: Multiply the Imaginary Parts

The imaginary part of the first complex number is $-3i$, and the imaginary part of the second complex number is $5i$. We multiply these two numbers to get the imaginary part of the product.

Step 3: Multiply the Real and Imaginary Parts

We also need to multiply the real part of the first complex number by the imaginary part of the second complex number, and the imaginary part of the first complex number by the real part of the second complex number.

Step 4: Simplify the Expression

After multiplying the terms, we simplify the expression to get the product of the two complex numbers in standard form.

Calculating the Product

Let's calculate the product of $(2-3i)$ and $(1+5i)$.

Step 1: Multiply the Real Parts

The real part of the product is $2 \cdot 1 = 2$.

Step 2: Multiply the Imaginary Parts

The imaginary part of the product is $-3i \cdot 5i = -15i^2 = 15$.

Step 3: Multiply the Real and Imaginary Parts

We also need to multiply the real part of the first complex number by the imaginary part of the second complex number, and the imaginary part of the first complex number by the real part of the second complex number.

The real part of the first complex number multiplied by the imaginary part of the second complex number is $2 \cdot 5i = 10i$.

The imaginary part of the first complex number multiplied by the real part of the second complex number is $-3i \cdot 1 = -3i$.

Step 4: Simplify the Expression

After multiplying the terms, we simplify the expression to get the product of the two complex numbers in standard form.

The product of $(2-3i)$ and $(1+5i)$ is $(2 + 10i - 3i + 15i^2)$. Since $i^2 = -1$, we can substitute this value into the expression to get $(2 + 10i - 3i - 15)$.

Simplifying the expression further, we get $(2 - 15 + 10i - 3i) = (-13 + 7i)$.

Conclusion

In this article, we have calculated the product of two complex numbers, $(2-3i)$ and $(1+5i)$, in standard form. We used the distributive property of multiplication over addition to multiply the terms, and then simplified the expression to get the product in standard form. The product of the two complex numbers is $(-13 + 7i)$.

Final Answer

The final answer is $\boxed{-13 + 7i}$.

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and computer science. However, complex numbers can be challenging to understand, especially for beginners. In this article, we will answer some frequently asked questions about complex numbers to help you better understand this concept.

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$.

Q: What is the imaginary unit $i$?

A: The imaginary unit $i$ is a mathematical concept that is used to extend the real number system to the complex number system. It is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, their sum is $(a + c) + (b + d)i$.

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you simply subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, their difference is $(a - c) + (b - d)i$.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you use the distributive property of multiplication over addition. For example, if you have two complex numbers $a + bi$ and $c + di$, their product is $(ac - bd) + (ad + bc)i$.

Q: How do I divide complex numbers?

A: To divide complex numbers, you multiply the numerator and the denominator by the conjugate of the denominator. For example, if you have two complex numbers $a + bi$ and $c + di$, their quotient is $\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a + bi$ is $a - bi$. The conjugate of a complex number is used to simplify expressions and to eliminate the imaginary part.

Q: What is the modulus of a complex number?

A: The modulus of a complex number $a + bi$ is $\sqrt{a^2 + b^2}$. The modulus of a complex number is a measure of its distance from the origin in the complex plane.

Q: What is the argument of a complex number?

A: The argument of a complex number $a + bi$ is the angle between the positive real axis and the line segment joining the origin to the point $(a, b)$ in the complex plane.

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you use the following formulas: $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.

Q: How do I convert a complex number from polar form to rectangular form?

A: To convert a complex number from polar form to rectangular form, you use the following formulas: $a = r\cos\theta$ and $b = r\sin\theta$.

Conclusion

In this article, we have answered some frequently asked questions about complex numbers to help you better understand this concept. Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. We hope that this article has been helpful in clarifying any doubts you may have had about complex numbers.

Final Answer

The final answer is $\boxed{0}$, but the real answer is that complex numbers are a fascinating and powerful tool in mathematics, and we hope that this article has inspired you to learn more about them.