Rational Numbers, Exponents, Perfect Squares, Integer Problems
What is a Rational Number?
The formal definition for a rational number is a number that can be in the form p/q where p and q are integers (whole numbers) and q is not equal to zero. In other words, rational numbers can be written as a ratio of two integers (or a fraction).
Examples:
What is an Irrational Number?
An irrational number is a real number that cannot be written as a simple fraction.
A good way to tell if a number is rational is if it can be written as a ratio of two integers.
Example 1: 1.5 is rational, because it can be written as the ratio 3/2
Example 2: 0.333… (3 repeating) is also rational, because it can be written as the ratio ⅓
Example 3: Pi is irrational, because we cannot write it as a simple fraction
Π (pi) = 3.1415926535897932384626433832795…
What is an Exponent?
The exponent of a number shows how many times to multiply a number by itself.
Example 1: In the image above, the “2” means to multiply 8 twice: 8 x 8 = 64
● can be called “8 to the power 2” or “8 to the second power” or “8 squared”
Example 2: = 5 x 5 x 5 = 125
● can be called “5 to the power 3” or “5 to the third power” or “5 cubed”
Example 3: = 2 x 2 x 2 x 2 = 16
● can be called “2 to the power 4” or “2 to the 4th”
If the exponent is a negative number, that number shows how many times to divide a number by
itself.
Example 4: = 1 / 8 = 0.125
● can be called “8 to the power -1” or “8 to the negative 1”
Example 5: = 1 / (5 x 5 x 5) = 1 / 125 = 0.008
● can be called “5 to the power -3” or “5 to the negative 3”
What if the Exponent is 1 or 0?
1 If the exponent is 1, then you have the number itself (= 9 )
0 If the exponent is 0, then the answer is 1 (= 1 )
What is a Perfect Square?
A perfect square is the product of an integer multiplied by itself.
Example 1: What is 3 squared?
● 3 x 3 = 9
Try to memorize the perfect squares in the tables below!
Order of Operations
1. Parentheses
2. Exponents (Power, Square Roots)
3. Multiply and Division (left-to-right)
4. Add and Subtract (left-to-right)
This can easily be remembered as PEMDAS: Please Excuse My Dear Aunt Sally
○ Multiplication and division are ranked equally
○ Addition and subtraction are ranked equally
After you have done “P” and “E”, go from left to right doing any “M” or “D” as they are found in the problem.
Then, go from left to right doing any “A” or “S” as they are found in the problem.
Example 1: How do you work out 3 + 6 x 2 ?
Multiplication before Addition. M before A.
6 x 2 = 12
3 + 12 = 15
Answer: 15
Example 2: How do you work out ( 3 + 6 ) x 2
Parentheses before Multiplication. P before M.
3 + 6 = 9
9 z 2 = 18
Answer: 18
Example 3: How do you work out: 12 / 6 x 3 / 2
Multiplication and Division are ranked equally, so go left to right. M or D.
12 / 6 = 2
2 x 3 = 6
6 / 2 =3
Answer: 3
Full Example 1: How do you work out 7 + ( 6 x + 3 )
