ACT Math: Complex Numbers on the ACT

Complex numbers often appear in the final, most challenging questions of the ACT math section, testing your algebraic fluency and careful computation under pressure. Mastering them not only secures these valuable points but also strengthens your overall number sense and problem-solving skills. This guide provides a comprehensive foundation in complex numbers, directly aligned with the types of problems you will encounter on test day.

Imaginary Numbers and the Complex Plane

Our journey begins with the imaginary unit $i$, which is defined by the property $i^2=-1$. This definition allows us to take the square root of negative numbers. For example, $\sqrt{-4}=\sqrt{4}\cdot \sqrt{-1}=2i$. A complex number is any number that can be written in the form $a+bi$, where $a$ and $b$ are real numbers. Here, $a$ is called the real part and $b$ is called the imaginary part.

Complex numbers can be visualized on the complex plane. This is a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part. The number $3+2i$ would be plotted as the point $(3,2)$. This geometric view becomes crucial when discussing the absolute value of a complex number later.

Performing Operations with Complex Numbers

You add, subtract, and multiply complex numbers much like you handle binomials, with the crucial extra step of simplifying any instance of $i^2$ to $-1$.

• Addition/Subtraction: Combine like terms (real with real, imaginary with imaginary).
• Example: $(5+3i)+(2-7i)=(5+2)+(3-7)i=7-4i$.
• Example: $(4+i)-(6-3i)=4+i-6+3i=(4-6)+(1+3)i=-2+4i$.

• Multiplication: Use the distributive property (FOIL) and simplify.
• Example: $(3+2i)(1-4i)=3(1)+3(-4i)+2i(1)+2i(-4i)$
• $=3-12i+2i-8i^2$
• Since $i^2=-1$, this becomes $3-10i-8(-1)=3-10i+8=11-10i$.

Simplifying Powers of i

The powers of $i$ follow a predictable four-step cycle. This cycle is essential for simplifying high powers of $i$ quickly on the ACT. $$i^1=i,i^2=-1,i^3=i^2\cdot i=-i,i^4=i^2\cdot i^2=1$$

After $i^4=1$, the cycle repeats. To simplify a high power like $i^{47}$, find the remainder when the exponent is divided by 4.

• $47÷4=11$ remainder $3$.
• Therefore, $i^{47}=i^3=-i$.

This method is far faster than manually multiplying $i$ by itself 47 times.

Complex Conjugates and Division

The complex conjugate of a complex number $a+bi$ is $a-bi$. Conjugates have a special property: when you multiply a complex number by its conjugate, the result is always a real number. $$(a+bi)(a-bi)=a^2-(bi{)}^2=a^2-b^2{i}^2=a^2+b^2$$

This property is the key to division of complex numbers. To divide by a complex number, you multiply the numerator and denominator by the conjugate of the denominator. This process, called rationalizing the denominator, eliminates $i$ from the denominator.

• Example: Simplify $\frac{2+i}{3-i}$.
• Multiply numerator and denominator by the conjugate of the denominator, $3+i$:

$$\frac{2+i}{3-i}\cdot \frac{3+i}{3+i}=\frac{(2+i)(3+i)}{(3-i)(3+i)}$$

• Calculate the numerator: $(2+i)(3+i)=6+2i+3i+i^2=6+5i-1=5+5i$.
• Calculate the denominator: $(3-i)(3+i)=9-i^2=9+1=10$.
• Therefore, the result is $\frac{5+5i}{10}=\frac{5}{10}+\frac{5}{10}i=\frac{1}{2}+\frac{1}{2}i$.

Finding the Absolute Value (Modulus)

The absolute value (or modulus) of a complex number $a+bi$, denoted $|a+bi|$, represents its distance from the origin $(0,0)$ on the complex plane. Using the Pythagorean Theorem on the point $(a,b)$, the formula is: $$|a+bi|=\sqrt{a^2+b^2}$$

• Example: Find $|3-4i|$.
• $|3-4i|=\sqrt{(3{)}^2+(-4{)}^2}=\sqrt{9+16}=\sqrt{25}=5$.

This connects directly to the multiplication property of conjugates: $(a+bi)(a-bi)=a^2+b^2=|a+bi{|}^2$.

Common Pitfalls

1. Misapplying the Powers of i Cycle: The most common error is miscounting the cycle or confusing $i^3$ with $i^2$. Remember the sequence: $i,-1,-i,1$. Always reduce the exponent modulo 4 (find the remainder when divided by 4) before deciding.
2. Forgetting to Distribute the Negative: When subtracting complex numbers, such as $(a+bi)-(c+di)$, you must subtract both the real and imaginary parts of the second number. The correct process is $a-c+(b-d)i$. A frequent mistake is to write $a-c+(b-d)i$ but forget to distribute the negative to the $d$, writing $b-di$ instead.
3. Incorrect Division: Failing to multiply the numerator and denominator by the conjugate of the denominator will leave you with an $i$ in the denominator, which is not considered simplified form on the ACT. Remember, you must multiply both the top and bottom by the same value.
4. Absolute Value Formula Mix-Up: The absolute value of $a+bi$ is $\sqrt{a^2+b^2}$, not $a^2+b^2$ and not $\sqrt{a^2+(bi{)}^2}$. You square the real coefficient $a$ and the real coefficient $b$—you do not square the $i$.

Summary

• The imaginary unit is defined as $i^2=-1$, and a complex number has the form $a+bi$.
• Add and subtract by combining real and imaginary parts. Multiply using distribution, always simplifying $i^2$ to $-1$.
• The powers of $i$ cycle every four exponents: $i,-1,-i,1$. Simplify high powers by finding the exponent's remainder when divided by 4.
• The complex conjugate of $a+bi$ is $a-bi$. Multiplying a number by its conjugate yields a real number, which is the key technique for division.
• The absolute value $|a+bi|=\sqrt{a^2+b^2}$ represents the number's distance from the origin on the complex plane.
• On the ACT, approach these problems methodically, double-check your arithmetic, and watch for sign errors, especially in the final questions where these concepts typically appear.