How Does i Work With Real Numbers in Math?

In math, the imaginary unit ( i ) is very important because it helps us work with numbers that can't be explained by regular numbers alone. It makes the number system bigger and allows us to solve equations that don’t have normal solutions. The key thing to know about ( i ) is:

$$i^2 = -1$$

This shows the main difference between real numbers and imaginary numbers. For Year 9 students, it’s really important to understand how ( i ) interacts with real numbers in different math operations.

What You Need to Know About i

1. Definition:

   • The imaginary unit ( i ) is defined by the equation ( i^2 = -1 ).

2. Powers of i:

   • The powers of ( i ) follow a repeating pattern:
     • ( i^1 = i )
     • ( i^2 = -1 )
     • ( i^3 = i \times i^2 = i \times -1 = -i )
     • ( i^4 = (i^2)^2 = (-1)^2 = 1 )
     • ( i^5 = i^1 = i ) (and this pattern keeps going every four powers).

Here’s a simple way to remember the pattern:

• If ( n \equiv 0 ,(\text{mod } 4) ), then ( i^n = 1 )
• If ( n \equiv 1 ,(\text{mod } 4) ), then ( i^n = i )
• If ( n \equiv 2 ,(\text{mod } 4) ), then ( i^n = -1 )
• If ( n \equiv 3 ,(\text{mod } 4) ), then ( i^n = -i )

How i Works with Real Numbers

When you do math operations with ( i ) and real numbers, some rules and patterns show up:

Addition

When you add real numbers to imaginary numbers, you make a complex number:

$$a + bi \quad \text{(where } a \text{ is a real number and } b \text{ is also a real number)}$$

Multiplication

1. Multiplying by i: When you multiply a real number by ( i ), you get an imaginary number:

   • For example, multiplying ( 3 ) by ( i ) gives you ( 3i ).

2. Distributive Property: If you multiply imaginary numbers by real numbers, use the distributive property:

   • Example: ( (2 + 3i)(4) = 8 + 12i ).

3. Combining with Real Numbers: When multiplying complex numbers, you also use the distributive property:

   • Example: ( (2 + 3i)(1 + 4i) = 2 \cdot 1 + 2 \cdot 4i + 3i \cdot 1 + 3i \cdot 4i = 2 + 8i + 3i + 12(-1) = -10 + 11i ).

Division

Dividing by real numbers is easy as long as you don’t divide by zero. But when dividing with ( i ), you usually need to multiply by the conjugate:

$$\frac{a + bi}{c + di} \quad (c \neq 0)$$

Conclusion

The imaginary unit ( i ) helps connect real numbers and complex numbers. It allows us to solve problems that are too hard for just real numbers. By learning about the properties of ( i ) and how it works with real numbers, students can better explore complicated math concepts in Year 9.