Conjugate in Math
The term conjugate means a pair of things joined together. For example, the two smileys: smiley and sad are exactly the same except for one pair of features that are actually opposite of each other. If you look at these smileys, you will notice that they are the same except that they have opposite facial expressions: one has a smile and the other has a frown. Similarly, conjugate in math refers to either the conjugate of a surd or the conjugate of a complex number where there is just a sign change in the number with respect to a few conditions.
Let us see what is conjugate in math in different cases and what is the use of finding it.
| 1. | What is Conjugate in Math? |
| 2. | Conjugate of a Surd |
| 3. | Conjugate of a Complex Number |
| 4. | Conjugate and Rational Factor |
| 5. | Conjugates and Rationalization |
| 6. | FAQs on Conjugate in Math |
What is Conjugate in Math?
The conjugate in math is formed by changing the sign between two terms in a binomial with respect to the condition that the sum and product of the binomial and its conjugate are rational. Here, the binomial can be either a surd or a complex number. Observe the following binomials and their conjugates.
| Binomial | Conjugate | Sum and Product are Rational Numbers | |
|---|---|---|---|
| Surd | 1 + √3 | 1 - √3 | Sum = (1 + √3) + (1 - √3) = 2 Product = (1 + √3) (1 - √3) = 1 - 3 = -2 |
| Complex Number | 2 + i | 2 - i | Sum = (2 + i) + (2 - i) = 4 Product = (2 + i) (2 - i) = 22 - i2 = 4 - (-1) = 5 |
We study two types of conjugates in math.
- Conjugate of a Surd
- Conjugate of a Complex Number
Let us study each of these in detail in the upcoming sections.
Conjugate of a Surd
The conjugate of a surd x + y√z is always x - y√z and vice versa. This is because the sum is (x + y√z) + (x - y√z) = 2x, and the product = (x + y√z) (x - y√z) = x2 - (y√z)2 = x2 - y2z (by the formula of a2 - b2) are rational numbers. For example, for the surd 3 + √2, the conjugate surd is 3 - √2, this is because:
- Their sum = (3 + √2) + (3 - √2) = 6 and
- Their product = (3 + √2) (3 - √2) = 9 - 2 = 7
Here, both 6 and 7 are rational numbers. Note that to find the conjugate of a surd, just change the sign of the surd (irrational part), but it is not changing the middle sign always. For example, the conjugate of the surd √2 + 3 is -√2 + 3, but NOT √2 - 3. Because, if we assume that √2 - 3 is the conjugate of √2 + 3, then their sum is 2√2, which is NOT a rational number. Here are some more examples of conjugates of surds in math.
| Surd | Conjugate |
|---|---|
| 2√5 + 3 (write it as 3 + 2√5) | 3 - 2√5 |
| -√7 - 3 (write it as -3 - √7) | -3 + √7 |
| 3 - √2 | 3 + √2 |
The easiest way of writing conjugate surd is to find the conjugate is just to write the given number in the order of rational part first and irrational part next and then change the middle sign.
Conjugate of a Complex Number
The complex conjugate of a complex number z = x + iy is x - iy (and vice versa) and it is represented by \(\bar{z}\) as their sum (2x) and the product x2 + y2 both are rational numbers. To write the complex conjugate,
- Write the given complex number in the form of x + iy (real part first and then the imaginary part)
- Change the middle sign. Then the complex conjugate of x + iy is x - iy.
Here are some examples of conjugates of complex numbers.
| Complex Number | Conjugate |
|---|---|
| 2 - i | 2 + i |
| 3i + 5 (write it as 5 + 3i) | 5 - 3i |
| (-1/2) + 5i | (-1/2) - 5i |
Conjugate and Rational Factor
If the product of two surds is a rational number, then each one of them is called the rational factor of the other. For example, the rational factors of 2 + √3 are each of 2 - √3 and -2 + √3. This is because by multiplying 2 + √3 with each of their conjugates result in a rational number as shown below.
- (2 + √3) (2 - √3) = 4 - 3 = 1
- (2 + √3) (-2 + √3) = -4 + 3 = -1
Note that -2 + √3 is NOT a conjugate of 2 + √3, its only a rational factor.
Sometimes, rational factor and conjugate of a number are referred to as the same but there is one minor difference between them. The sum of a binomial and its rational factor does NOT need to be a rational number, but the sum of a binomial and its conjugate SHOULD be a rational number. On the other hand, the product of a binomial with each of its rational factor and conjugate should be a rational number.
Conjugates and Rationalization
Rationalization of the denominator is the process of multiplying a fraction (with an irrational or complex denominator) by its rational factor or conjugate to make the denominator a rational number. This is because it is very convenient to have rational numbers in the denominators to compare or make some calculations in math. Here are two examples to understand how to rationalize the denominators by using conjugates.
Rationalization of Surds Example: Rationalize the denominator of 1 / (3 + √2).
Solution: The conjugate of 3 + √2 is 3 - √2. Multiplying and dividing the given fraction by 3 - √2,
1 / (3 + √2) × (3 - √2) / (3 - √2)
= (3 - √2) / (32 - (√2)2)
= (3 - √2) / (9 - 4)
= (3 - √2) / 5
Rationalization of Complex Numbers Example: Rationalize the denominator of 1 / (1 + i).
Solution: The complex conjugate of 1 + i is 1 - i.
1 / (1 + i) × (1 - i) / (1 - i)
= (1 - i) / (12 - i2)
= (1 - i) / (1 + 1) (Because by powers of iota, i2 = -1)
= (1 - i) / 2
Important Notes on Conjugate in Math:
- If z = x + √y and its conjugate is \(\bar{z}\) = x - √y, then z + \(\bar{z}\) = 2x and x - \(\bar{z}\) = 2√y.
- If z = x + iy and its conjugate is \(\bar{z}\) = x - iy, then z + \(\bar{z}\) = 2x and x - \(\bar{z}\) = 2iy.
- For a complex number z, if z + \(\bar{z}\) = 0, then z is purely imaginary.
- The modulus of a complex number and its conjugate are always the same. i.e., |z| = |\(\bar{z}\)|, for any complex number z.
- The conjugates are very useful in rationalization.
☛ Related Topics:
FAQs on Conjugate in Math
What are the Math Conjugates?
The math conjugate of a number is a number that when multiplied or added to the given number results in a rational number. For example,
- The conjugate of a surd 6 + √2 is 6 - √2.
- The conjugate of a complex number 5 - 3i is 5 + 3i.
Is Finding Conjugate Means Changing the Middle Sign Always?
No, finding the conjugate does NOT mean changing the middle sign all the time. For example, the conjugate of √5 - 1 is NOT √5 + 1 as their sum is NOT a rational number. After finding the conjugate of a number, check whether both the sum and product of the number and its conjugate are rational or not always.
How to Find Conjugate in Math?
To find the conjugate of a number,
- Step 1: Write the number such that its real part comes first.
- Step 2: Change the middle sign.
What is an Example of Conjugate?
The sum and product of a number and its conjugate are always rational. For example, the complex conjugate of the complex number 3 - 5i is 3 + 5i as (3 - 5i) + (3 + 5i) = 6 and (3 - 5i) (3 + 5i) = 9 + 25 = 34. Here, both 6 and 34 are rational numbers.
Why it is Called Conjugate in Math?
"Conjugate" means two things that are joined together (or) two things that have common features except for one difference. The conjugate of a number is a mathematical value having a reciprocal relation with the given number.
Does Conjugate Need to Have a Minus Symbol Always?
No, the conjugate doesn't need to have the middle sign to be minus always. For example, the conjugate of 4 - √3 is 4 + √3 and it doesn't have a minus symbol.
Is Rational Factor Same as Conjugate?
Sometimes both are considered as the same but there is one difference. For any number,
- the product of the number and its rational factor must be rational.
- both the sum and the product of the number and its conjugate must be rational.