The absolute value function strips a real number of its sign. For a complex number z = x + yi, we define the absolute value |z| as being the distance from z to 0 in the complex plane C.

The bar over a set is another notation for the complement of a set. For example: Where U is the universal set. Remember that the complement of C is the set of all elements not in C, in other words, everything in the Universe that isn’t in C.

Conjugate of a Complex Number It is denoted by z and is read as z bar. Thus, z bar means the conjugative of the complex number z. We can write the conjugate of complex numbers just by changing the sign before the imaginary part.

A complex number is a number of the form a + bi, where i = and a and b are real numbers. For example, 5 + 3i, – + 4i, 4.2 – 12i, and – – i are all complex numbers. a is called the real part of the complex number and bi is called the imaginary part of the complex number.

In this complex number, 3 is the real number and 5i is the imaginary number. Complex numbers are numbers that consist of two parts — a real number and an imaginary number. Complex numbers are the building blocks of more intricate math, such as algebra.

A simple shortcut to simplify an imaginary unit raised to a power is to divide the power by 4 and then raise the imaginary unit to the power of the reminder. For example: to simplify j23, first divide 23 by 4. 23/4 = 5 remainder 3. So j23 = j3 = -j …… as already shown above.

To add or subtract two complex numbers, just add or subtract the corresponding real and imaginary parts. For instance, the sum of 5 + 3i and 4 + 2i is 9 + 5i. For another, the sum of 3 + i and –1 + 2i is 2 + 3i. Addition can be represented graphically on the complex plane C.

Start by solving all of the terms in parentheses. For instance, within parentheses, you should multiply before you add, subtract, etc. As an example, let’s try to simplify the expression 2x + 4(5 + 2) + 32 – (3 + 4/2). In this expression, we would solve the terms in parentheses, 5 + 2 and 3 + 4/2, first. 5 + 2 = 7.

To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, (23)5 = 215. For any positive number x and integers a and b: (xa)b= xa· b. Simplify.

To simplify any algebraic expression, the following are the basic rules and steps:

1. Remove any grouping symbol such as brackets and parentheses by multiplying factors.
2. Use the exponent rule to remove grouping if the terms are containing exponents.
3. Combine the like terms by addition or subtraction.
4. Combine the constants.

Key Points to Remember while Solving Simplification Question

1. Replace ‘of’ by ‘Multiplication’
2. Replace ‘/’ by ‘Division’
3. Always do the operations in priority according to ‘BODMAS’
4. Division & Multiplication have the same priority (Start from left)
5. Addition & Subtraction have the same priority.

To simplify an expression containing a square root, we find the factors of the number and group them into pairs. For example, a number 16 has 4 copies of factors, so we take a number two from each pair and put it in front of the radical, finally dropped, i.e., √16 = √(2 x 2 x 2 x 2) = 4.

7/4=(4+3)/4=4/4+3/4=1+3/4=1 3/4. Answer: 7/4 is as a mixed number 1 3/4.

How to Convert an Improper Fraction to a Mixed Number

1. Divide the numerator by the denominator.
2. Write down the whole number result.
3. Use the remainder as the new numerator over the denominator. This is the fraction part of the mixed number.

A fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). Example: 5/3 (five thirds) and 9/8 (nine eighths) are improper fractions.