Solved on Jan 13, 2024

Identify real/imaginary complex numbers and write in standard form $a+bi$ . Examples: $6i$ , $\frac{1+i}{3}$ , $\sqrt{7}$ , $\frac{\pi}{2}$ .

STEP 1

Assumptions
1. A complex number is in the form $a+bi$ , where $a$ is the real part and $b$ is the imaginary part.
2. A complex number is real if $b=0$ .
3. A complex number is imaginary if $a=0$ and $b \neq 0$ .
4. The standard form of a complex number does not include $i$ in the denominator.

STEP 2

Identify the type of the complex number $6i$ and write it in standard form.
Since there is no real part and the imaginary part is $6i$ , we can write it as $0 + 6i$ .

STEP 3

Conclude the type of the complex number $6i$ .
The complex number $6i$ is purely imaginary.

STEP 4

Identify the type of the complex number $\frac{1+i}{3}$ and write it in standard form.
First, we need to divide both the real and imaginary parts by $3$ .

STEP 5

Perform the division to obtain the standard form.
$\frac{1+i}{3} = \frac{1}{3} + \frac{i}{3}$

STEP 6

Conclude the type of the complex number $\frac{1+i}{3}$ .
The complex number $\frac{1+i}{3}$ is neither purely real nor purely imaginary since it has both a real part, $\frac{1}{3}$ , and an imaginary part, $\frac{i}{3}$ .

STEP 7

Identify the type of the complex number $\sqrt{7}$ and write it in standard form.
Since $\sqrt{7}$ is a real number and there is no imaginary part, we can write it as $\sqrt{7} + 0i$ .

STEP 8

Conclude the type of the complex number $\sqrt{7}$ .
The complex number $\sqrt{7}$ is purely real.

STEP 9

Identify the type of the complex number $\frac{\pi}{2}$ and write it in standard form.
Since $\frac{\pi}{2}$ is a real number and there is no imaginary part, we can write it as $\frac{\pi}{2} + 0i$ .

STEP 10

Conclude the type of the complex number $\frac{\pi}{2}$ .
The complex number $\frac{\pi}{2}$ is purely real.

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Identify real/imaginary complex numbers and write in standard form $a+bi$ . Examples: $6i$ , $\frac{1+i}{3}$ , $\sqrt{7}$ , $\frac{\pi}{2}$ .

STEP 1

Assumptions
1. A complex number is in the form $a+bi$ , where $a$ is the real part and $b$ is the imaginary part.
2. A complex number is real if $b=0$ .
3. A complex number is imaginary if $a=0$ and $b \neq 0$ .
4. The standard form of a complex number does not include $i$ in the denominator.

STEP 2

Identify the type of the complex number $6i$ and write it in standard form.
Since there is no real part and the imaginary part is $6i$ , we can write it as $0 + 6i$ .

STEP 3

Conclude the type of the complex number $6i$ .
The complex number $6i$ is purely imaginary.

STEP 4

Identify the type of the complex number $\frac{1+i}{3}$ and write it in standard form.
First, we need to divide both the real and imaginary parts by $3$ .

STEP 5

Perform the division to obtain the standard form.
$\frac{1+i}{3} = \frac{1}{3} + \frac{i}{3}$

STEP 6

Conclude the type of the complex number $\frac{1+i}{3}$ .
The complex number $\frac{1+i}{3}$ is neither purely real nor purely imaginary since it has both a real part, $\frac{1}{3}$ , and an imaginary part, $\frac{i}{3}$ .

STEP 7

Identify the type of the complex number $\sqrt{7}$ and write it in standard form.
Since $\sqrt{7}$ is a real number and there is no imaginary part, we can write it as $\sqrt{7} + 0i$ .

STEP 8

Conclude the type of the complex number $\sqrt{7}$ .
The complex number $\sqrt{7}$ is purely real.

STEP 9

Identify the type of the complex number $\frac{\pi}{2}$ and write it in standard form.
Since $\frac{\pi}{2}$ is a real number and there is no imaginary part, we can write it as $\frac{\pi}{2} + 0i$ .

STEP 10

Conclude the type of the complex number $\frac{\pi}{2}$ .
The complex number $\frac{\pi}{2}$ is purely real.

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Identify real/imaginary complex numbers and write in standard form a+bia+bia+bi. Examples: 6i6i6i, 1+i3\frac{1+i}{3}31+i​, 7\sqrt{7}7​, π2\frac{\pi}{2}2π​.

STEP 1

STEP 2

Identify the type of the complex number 6i6i6i and write it in standard form.Since there is no real part and the imaginary part is 6i6i6i, we can write it as 0+6i0 + 6i0+6i.

STEP 3

Conclude the type of the complex number 6i6i6i.The complex number 6i6i6i is purely imaginary.

STEP 4

Identify the type of the complex number 1+i3\frac{1+i}{3}31+i​ and write it in standard form.First, we need to divide both the real and imaginary parts by 333.

STEP 5

Perform the division to obtain the standard form.1+i3=13+i3\frac{1+i}{3} = \frac{1}{3} + \frac{i}{3}31+i​=31​+3i​

STEP 6

Conclude the type of the complex number 1+i3\frac{1+i}{3}31+i​.The complex number 1+i3\frac{1+i}{3}31+i​ is neither purely real nor purely imaginary since it has both a real part, 13\frac{1}{3}31​, and an imaginary part, i3\frac{i}{3}3i​.

STEP 7

Identify the type of the complex number 7\sqrt{7}7​ and write it in standard form.Since 7\sqrt{7}7​ is a real number and there is no imaginary part, we can write it as 7+0i\sqrt{7} + 0i7​+0i.

STEP 8

Conclude the type of the complex number 7\sqrt{7}7​.The complex number 7\sqrt{7}7​ is purely real.

STEP 9

Identify the type of the complex number π2\frac{\pi}{2}2π​ and write it in standard form.Since π2\frac{\pi}{2}2π​ is a real number and there is no imaginary part, we can write it as π2+0i\frac{\pi}{2} + 0i2π​+0i.

STEP 10

Conclude the type of the complex number π2\frac{\pi}{2}2π​.The complex number π2\frac{\pi}{2}2π​ is purely real.

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Identify real/imaginary complex numbers and write in standard form $a+bi$ . Examples: $6i$ , $\frac{1+i}{3}$ , $\sqrt{7}$ , $\frac{\pi}{2}$ .

Identify the type of the complex number $6i$ and write it in standard form.
Since there is no real part and the imaginary part is $6i$ , we can write it as $0 + 6i$ .

Conclude the type of the complex number $6i$ .
The complex number $6i$ is purely imaginary.

Identify the type of the complex number $\frac{1+i}{3}$ and write it in standard form.
First, we need to divide both the real and imaginary parts by $3$ .

Perform the division to obtain the standard form.
$\frac{1+i}{3} = \frac{1}{3} + \frac{i}{3}$

Conclude the type of the complex number $\frac{1+i}{3}$ .
The complex number $\frac{1+i}{3}$ is neither purely real nor purely imaginary since it has both a real part, $\frac{1}{3}$ , and an imaginary part, $\frac{i}{3}$ .

Identify the type of the complex number $\sqrt{7}$ and write it in standard form.
Since $\sqrt{7}$ is a real number and there is no imaginary part, we can write it as $\sqrt{7} + 0i$ .

Conclude the type of the complex number $\sqrt{7}$ .
The complex number $\sqrt{7}$ is purely real.

Identify the type of the complex number $\frac{\pi}{2}$ and write it in standard form.
Since $\frac{\pi}{2}$ is a real number and there is no imaginary part, we can write it as $\frac{\pi}{2} + 0i$ .

Conclude the type of the complex number $\frac{\pi}{2}$ .
The complex number $\frac{\pi}{2}$ is purely real.