6.1 IN-DEPTH PATH squares

Theory

A squared number is a number multiplied by itself. 

An example of a squared number is 52  = 25  this is the same as 5 . 5 = 25 .

Another example is 32 = 9 this is the same as 3 . 3 = 9.  

A square root is the opposite of a squared number. 

An example of this is √25 = 5 because 52 = 5 . 5 = 25 

Another example is √9 = 3 because 32 = 3 . 3 = 9 

 

The  square of a square root (√...)2 

( √5 )2 = √5 . √5 = √25 = 5

( √7 )2 = √7 . √7 = √49 = 7

 

You can also split square roots with multiplication:

√500 = √(5 . 100) = √5 . √100 = √5 .10 = 10√5 

Test yourself exercise

You really need to understand how to calculate with squares in order to understand the complex theory and the imaginary numbers. So, you are going to test yourself with this exercise. 

 

Test yourself exercise: 

Calculate

(2--8) + (-9)2 - 7 + (√3)2 + √(10+6) 

 

Complex numbers & the imaginary number

Extra learning objectives

At the end of the lesson you are able to... 

  • ... explain others what a complex number is. 
  • ... explain others what a imaginary number is.
  • ...calculate the square root of a negative number using the imaginary number (i).

Theory

A complex number is a number that consist of two parts a real part (just a normal number) and an imaginary part. A complex number always looks like this:  z = a + bi. 

a = the real number 

bi = the imaginary part 

i = the imaginary number. 

Jobs 

Imaginary numbers are also used in some jobs. Think of electrical engineers who uses it to calculate AC circuits. An alternating currents is an electric current which is used for electricity. 

electrical engineers

Sales analysts also use complex numbers in order to make predictions and to understand the sales process better. 

Sales analysts

Economists also use imaginary numbers to make profit predictions.

economist

For now, you are only going to learn how to calculate the square root of a negative number using the imaginary number (i).

The rule

The most important rule of the complex theory is:

imaginairy number, i kwadraat is min een. i squared equals minus one. complex theory, complex numbers, real numbers

This means that √-1 is the same as √(i 2) = i 

Examples 

  • √-3 =  √( 3 . -1 ) = √( 3 . i 2 ) = √3 . √i 2 = i√3 
  • √-5 =  √( 5 . -1 ) = √( 5 . i 2 ) = √5 . √i 2 = i√5 
  • √-16 =  √( 16 . -1 ) = √( 16 . i 2 ) = √16 . √i 2 = 4i  
electrical

Conclusion

You can calculate the square root of negative numbers by splitting the negative number underneath the square root into ... . -1. When you have done the splitting, you need to replace the -1 by i 2 

Exercises

  • Exercise 1: 

Calculate 

a) 42 + 3 - 1 

b) 62 - 32

c) (7 - 6)2 

d) (1/4)2 + (2/3)2 

e) 7 - 62 : 3 

f) √49 + 42 - 1 

g) 3 . √121 + 7 . √625

  • Exercise 2

Calculate 

a) (-8)2

b) √81

c) √49 + 42 - 1 

d) 3 . √121 + 7 . √625

  • Exercise 3: 

Calculate 

a)√-7

b) √( 9 - 10 ) 

c) 3√-13 + 5 . 4 2