Pure Imaginary

A pure imaginary number is a complex number with a real part of 0, written in the form bi where b is a real number. In Intermediate Algebra, it comes up when you work with square roots of negative numbers and the imaginary unit i.

What is Pure Imaginary?

A pure imaginary number is a number in Intermediate Algebra that has no real part at all, only an imaginary part. It is written as bi, where b is a real number and i is the imaginary unit, with i^2 = -1.

That means numbers like 4i, -7i, and (1/2)i are pure imaginary. The coefficient in front of i can be any real number, including positive, negative, fraction, or decimal. The only thing that makes it pure imaginary is that there is nothing added to it on the real side.

This is different from a general complex number, which can have both a real part and an imaginary part, like 3 + 2i. If you can split a number into a real part plus an imaginary part, it is complex. If the real part is 0, it becomes pure imaginary.

A helpful way to picture it is on the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. Pure imaginary numbers sit on that vertical axis, because their real part is zero. So 5i is five units above 0, and -5i is five units below 0.

In this course, pure imaginary numbers usually appear when you simplify negative square roots. For example, sqrt(-49) becomes 7i, because sqrt(49) is 7 and sqrt(-1) is i. That is one of the main reasons the imaginary unit exists in the first place, to let you rewrite answers that do not stay in the real number system.

Why Pure Imaginary matters in Intermediate Algebra

Pure imaginary numbers show up any time Intermediate Algebra moves beyond the real number system. They let you write answers for expressions like sqrt(-16) instead of stopping at "no real solution." That makes them a bridge between square roots, complex numbers, and later work with polynomial equations.

You also need pure imaginary numbers to keep your notation clean. Writing 6i is much better than leaving an answer in a broken form like sqrt(-36). Once you simplify, you can combine like terms, compare numbers, and check whether an answer is real, imaginary, or fully complex.

This term also helps you avoid a common mistake: treating i like a variable. It is not a placeholder you solve for. It is a defined number with the rule i^2 = -1, so you handle it by algebraic rules, not by isolating it like x. That difference matters when you multiply, divide, or simplify powers of i.

Pure imaginary numbers are also a good checkpoint for understanding the structure of the complex number system. If you can tell the difference between 4, 4i, and 4 + 4i, you are getting the language of the chapter right. That language shows up in problem sets where you simplify radical expressions, classify numbers, and work with complex arithmetic.

How Pure Imaginary connects across the course

Imaginary Unit

The imaginary unit i is the building block of every pure imaginary number. A number like 8i is pure imaginary because i is the special symbol that stands for sqrt(-1), and the coefficient 8 is a real number. If you forget what i means, the whole form bi gets harder to simplify correctly.

Complex Number

A pure imaginary number is a specific kind of complex number. Complex numbers can have a real part and an imaginary part, like 3 + 2i, but pure imaginary numbers have a real part of zero. That distinction matters when you classify answers or decide whether a result is purely imaginary or fully complex.

Real Part

The real part of a pure imaginary number is always 0. For 9i, the real part is 0 and the imaginary part is 9i. When you look at an expression, separating the real part from the imaginary part helps you recognize whether you are dealing with a pure imaginary number or a complex number with both parts.

Real Number

Pure imaginary numbers are not real numbers, even though their coefficients are real. The number 12 is real, while 12i is pure imaginary. That difference matters when you are simplifying radicals or classifying values on a worksheet, because the presence of i moves the number out of the real number system.

Is Pure Imaginary on the Intermediate Algebra exam?

A quiz question might ask you to simplify sqrt(-81) or classify 6i as real, imaginary, or complex. The move is to rewrite the negative square root using i, then simplify the remaining radical if needed. For sqrt(-81), you get 9i. If a problem gives you 3 + 5i, you should identify it as complex, not pure imaginary, because the real part is 3. On problem sets, you may also be asked to tell whether a number lies on the real axis or the imaginary axis in the complex plane.

Pure Imaginary vs Complex Number

These are easy to mix up because every pure imaginary number is complex, but not every complex number is pure imaginary. A complex number can have a real part and an imaginary part, like 2 - 3i. A pure imaginary number has no real part, so its form is just bi.

Key things to remember about Pure Imaginary

• A pure imaginary number has the form bi, where b is a real number and i is the imaginary unit.

• The real part of a pure imaginary number is 0, so it sits on the imaginary axis in the complex plane.

• Numbers like 7i and -3/2 i are pure imaginary, but 4 + 7i is not because it has a real part.

• In Intermediate Algebra, pure imaginary numbers usually appear when you simplify square roots of negative numbers.

• If you see i^2, use the rule i^2 = -1 instead of treating i like an unknown variable.

Frequently asked questions about Pure Imaginary

What is pure imaginary in Intermediate Algebra?

A pure imaginary number is a number written as bi, where b is real and i is the imaginary unit. Its real part is 0, which means it lies on the imaginary axis in the complex plane. You usually meet it when simplifying negative square roots, like sqrt(-25) = 5i.

Is 4i a pure imaginary number?

Yes. 4i has no real part, so it is pure imaginary. The number 3 + 4i is not pure imaginary because it includes a real part of 3.

How do you tell if a number is pure imaginary or complex?

Look at the real part. If the real part is 0, the number is pure imaginary. If it has both a real part and an imaginary part, it is a complex number, but not pure imaginary.

Why do we use pure imaginary numbers?

They let you write answers for square roots of negative numbers instead of stopping at no real solution. They also make it possible to work with the full complex number system, which comes up later in algebra when you simplify expressions and solve equations.