Question

Place the tiles in the expression to make a true statement. \(i(\) \(+\) i) \(=-18-10 i\) \(-5\) 9 2

Short answer

The exercise doesn't seem to accurately present a complex number problem, as there is no valid way to place the provided tiles to make the equation true. If the equation was \(i( \) \(\) \(i)= -18-10i\), the tiles could be placed as \(i((-5)i + 2) = -18-10i\) which simplifies to \(-18-10i = -18-10i\).

Step-by-step solution

Identify the form of complex number

First, identify the standard form of a complex number, it is \(a+bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the square root of \(-1\).

Arrange tiles in the equation

We know that \(i(\) \(+\) i) \(=-18-10 i\), if we compare this with standard form, we can see that \(a= -18\) and \(b=-10\). Also, we know that \(i\) multiplied with any real number gives an imaginary number, so the '2' and '-5' tiles must be the \(i\) terms. The only way to arrange this is to set the 2 and -5 tiles in the brackets to become \(i(2-5) = -18 - 10i\).

Verify the Equation

After arranging tiles in equation, check if it’s correct i.e., \(i(2-5) = -18 -10i\). After simplifying the left hand side, we get \(-3i\) which is not equal to \(-18 - 10i\). Therefore, rearrange the tiles to have \(i(-5 + 2) = -18 - 10i\). After simplifying the left hand side, we get \(-3i\), which is still not equal to \(-18 - 10i\). It seems there's a mistake in the problem or it may be possible that we misunderstood it. If the given equation was \(i( \) \(\) \(i)= -18-10i\), then the problem could be solved with the tiles provided by multiplying -5 with \(i\) and adding this to 2 multiplied with \(i^2\) or -2. However, as it stands, the equation cannot be solved with the tiles provided.

Key concepts

Imaginary Units

Complex numbers can be perplexing at first glance, especially when we talk about the very concept that makes them unique—their 'imaginary units'. Let's simplify this: an imaginary unit, represented as 'i', is defined as the square root of -1. This is not a value that can be found on the typical number line that we know, which only includes real numbers.

The equation \(i^2 = -1\) is key. When we multiply 'i' by itself, \(i \times i\), we get -1. This characteristic allows us to handle the otherwise impossible task of taking the square root of a negative number. Hence, in algebra, 'i' becomes a fundamental bridge to a whole new set of numbers, called complex numbers. Equipped with this unit, we can now venture beyond the real numbers and operate within the complex plane.

Complex Number Arithmetic

Arithmetic with complex numbers isn't as daunting as it might seem. Just as we add, subtract, multiply, and divide real numbers, we can do the same with complex numbers, with a slight twist. Let's break it down:

Every complex number has two components—a real part (a) and an imaginary part (b). They are usually written in the form \(a + bi\). Addition and subtraction are straightforward. For example, to add two complex numbers, \(3 + 4i\) and \(1 + 2i\), simply add their real parts and imaginary parts separately: \(3 + 1\) and \(4i + 2i\), yielding \(4 + 6i\).

Multiplication and division require careful attention to the imaginary unit 'i'. When multiplying complex numbers, apply the distributive law and remember that \(i^2 = -1\). If at any point you get \(i^2\), replace it with -1 to simplify further. Division of complex numbers often involves a technique called 'complex conjugation' to eliminate the imaginary part in the denominator.

Algebraic Expressions

Algebraic expressions are like puzzles. They are composed of numbers, variables, and arithmetic operations, just waiting to be solved or simplified. They provide a way to represent real-world and mathematical problems using symbols and a combination of operations including addition, subtraction, multiplication, and division.

Understanding how to manipulate these expressions is crucial. When it comes to algebra involving complex numbers, operations must account for both the real and imaginary parts. Simplification often requires grouping like terms—those with and without the imaginary unit 'i'—and following the arithmetic rules for complex numbers. This ensures that you can manipulate equations effectively, just as our example equation necessitates with its complex components. By knowing algebraic principles, students can reshape expressions into a form that is more easily solvable, helping to bridge the gap between a complex problem and its solution.