A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit with i² = −1. Every real number is complex (take b = 0), so equations like x² = −9 now have solutions. Add and subtract the parts separately, multiply by distributing, and replace i² with −1.
Key facts:
- A complex number is a + bi with real a and b, and i² = −1.
- Every real number is a complex number with b = 0.
- Add or subtract real parts and imaginary parts separately.
- To multiply, distribute and then replace i² by −1.
- The powers of i cycle through i, −1, −i, 1 every four exponents.
What sets of numbers make up the real numbers?
The natural numbers are 1, 2, 3 and so on. The integers add 0 and the negatives. A rational number is a quotient \(\frac{a}{b}\) of integers with \(b\neq 0\); its decimal stops or repeats. An irrational number such as \(\sqrt{2}\) or \(\pi\) has a decimal that never stops or repeats. Together they fill the number line: the real numbers.
What is the imaginary unit i?
No real number squared gives a negative result, so mathematicians define a new number.
\(i^{2}=-1\), so \(i=\sqrt{-1}\). For any \(a > 0\), \(\sqrt{-a}=i\sqrt{a}\). Example: \(\sqrt{-36}=6i\) and \(\sqrt{-20}=2i\sqrt{5}\).
What is a complex number?
A complex number has the form \(a+bi\), where \(a\) and \(b\) are real. The number \(a\) is the real part and \(b\) is the imaginary part. If \(b=0\) the number is real, so every real number is also complex. Two complex numbers are equal only when both parts match. In the complex plane the point \((a,b)\) stands for \(a+bi\).
How do you add, subtract and multiply complex numbers?
To add or subtract, combine real parts and imaginary parts separately: \((a+bi)+(c+di)=(a+c)+(b+d)i\). To multiply, use the distributive property and replace \(i^{2}\) with \(-1\).
Worked example: find \((2+3i)(4-5i)\).
- Distribute: \(2\cdot 4+2\cdot(-5i)+3i\cdot 4+3i\cdot(-5i)\).
- Simplify: \(8-10i+12i-15i^{2}\).
- Replace \(i^{2}\) with \(-1\): \(8-10i+12i+15\).
- Combine like terms: \(23+2i\).
What are the powers of i?
The powers of \(i\) repeat every four steps: \(i^{1}=i\), \(i^{2}=-1\), \(i^{3}=-i\), \(i^{4}=1\). Divide the exponent by 4 and use the remainder: 0 gives 1, 1 gives \(i\), 2 gives \(-1\), 3 gives \(-i\). For \(i^{27}\), the remainder of \(27: 4\) is 3, so \(i^{27}=-i\).
Real and Complex Numbers: frequently asked questions
Is zero a complex number?
Yes. Zero is \(0+0i\), so it is real, rational, an integer, and complex. It is not a natural number in this course.
Why is \(\sqrt{-a}=i\sqrt{a}\) only for positive a?
The rule rewrites the square root of a negative number using the imaginary unit. Writing \(\sqrt{-16}=i\sqrt{16}=4i\) keeps the number organized before you multiply.
What is the difference between an imaginary number and a complex number?
A complex number is any \(a+bi\). It is called imaginary when \(b\neq 0\) and purely imaginary when \(a=0\), such as \(5i\).
How do you find a large power of i?
Divide the exponent by 4 and keep only the remainder. Since \(i^{4}=1\), the multiples of 4 drop out. For \(i^{50}\) the remainder is 2, so the answer is \(-1\).
Can the sum of two complex numbers be real?
Yes. When the imaginary parts are opposites, they cancel. For example \((3+5i)+(2-5i)=5\).
See every chapter on the Algebra 2 math page.