Question

Combine and simplify. $$(-1-2 i)-(i+6)$$

Short answer

-7 - 3i

Step-by-step solution

Distribute Negative Sign

Distribute the negative sign to the second complex number, changing its sign. The expression becomes \[ -1-2i-1i-6 \].

Combine Like Terms

Combine real parts and imaginary parts separately. Real parts: \( -1 - 6 = -7 \) and imaginary parts: \( -2i - i = -3i \) to get the simplified form.

Key concepts

Complex Numbers

Complex numbers are a cornerstone of algebra and provide a powerful way to describe and interpret mathematical phenomena that go beyond the scope of real numbers. A complex number consists of two parts: a real part and an imaginary part. It is typically written in the form of a + bi, where a is the real component and bi is the imaginary component with i representing the square root of -1.

In mathematics, the imaginary unit is a fundamental mathematical quantity that is used to construct the set of complex numbers. Complex numbers are essential in various fields such as engineering, physics, and computer science, particularly when dealing with oscillations, waves, and other phenomena that can be modeled using sinusoidal functions.

Combine Like Terms

In algebra, to simplify an expression, one of the key steps is to combine like terms. Like terms are terms that have identical variable parts, such as the same variable raised to the same power. When dealing with complex numbers, this concept extends to combining real parts with real parts and imaginary parts with imaginary parts.

To visualize, consider the expression described in the exercise (-1-2i)-(i+6). Here, we have the real numbers -1 and -6, and the imaginary numbers -2i and -i. Combining like terms involves adding or subtracting the real numbers together and the imaginary numbers together, which simplifies the expression to a single complex number with a real part and an imaginary part.

Imaginary Numbers

Imaginary numbers are a fascinating and integral part of complex numbers. They are defined as real numbers multiplied by the imaginary unit i, which itself is the square root of -1. Imaginary numbers aren't 'imaginary' in the sense that they don't exist; they simply exist in a different dimension of the number system.

Unlike real numbers, which can be represented on a one-dimensional number line, imaginary numbers need a two-dimensional plane to be represented, known as the complex plane. On this plane, real and imaginary parts of complex numbers are plotted on perpendicular axes. The imaginary axis represents imaginary numbers, while the real axis represents real numbers, allowing visualization of complex numbers as coordinates on a plane.