​​​

• Types of numbers: N, W, Z, Q, I, R, C, P                                                                                                           Back to Menu

 

 

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

• Natural Numbers (N) – counting numbers: 1, 2, 3, 4, ….

 

• Integers (Z) – positive & negative whole numbers: 1, -2, 3, 0, -5, -15, etc

​

• Rational Numbers (Q) - A number can be written as a Ratio of two integers (i.e. a fraction).

Fractions:

3/2  

3 = numerator

2 = denominator

-  = fraction line

 

                                                 Example of Rational Numbers

0.5 can be written as 1/2

-0.5 can be written as -1/2

3.5 can be written as 7/2

-3.5 can be written as -7/2

0.33 can be written as 1/3

-0.33 can be written as -1/3

 

• Irrational Numbers (I) – Numbers that cannot be written as a ratio of two integers.

 

Pi - no pattern 3.1415926535897932384626433832795 ...

Euler's Number (e) e=2.7182818284590452353602874713527

The Golden Ratio 1.61803398874989484820...

Several Square roots

The square root of 2 (√2) =1.4142135623730950...

The square root of 4 is not an irrational number (√4=2, (√9=3)

 

• Real Numbers (R) – The class of Rational & Irrational numbers

 

• Complex Numbers (C) – Numbers in the form of a + bi, where a, b = real numbers, i= square root of -1.

​

​

Other numbers

​

​

• Whole numbers (W) – counting numbers including 0 (zero): 0, 1, 2, 3 …

 

• Prime numbers (P) - A natural number greater than 1 which is a product of 1 and itself (i.e. cannot be obtained as a product of two smaller natural numbers). E.g.

 

2 is a prime number because can be obtained only by multiplying itself by 1:

1 x 2 = 2

5 is a prime number because can be obtained only by multiplying itself by 1:

1 x 5 = 5

7 is a prime number because can be obtained only by multiplying itself by 1:

1 x 7 = 7

17 is a prime number because can be obtained only by multiplying itself by 1:

1 x 17 = 17

19 is a prime number because can be obtained only by multiplying itself by 1:

1 x 19 = 19

 

• Composite numbers - A natural number that is not prime, and can be written as the product of two smaller natural numbers.

 

Examples:

4 is a composite number because it is a product (2 × 2), both numbers (2 and 2) are smaller than 4

6 is a composite number because it is a product of (3x2), both numbers (3 and 2) are smaller than 6

8 is a composite number because it is a product of (4x2), both numbers (4 and 2) are smaller than 8

 

• Even numbers – numbers that are divisible with 2, and with zero remainder (r).

For example, 4 is an even number because 4:2=2 (r=0)

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, …

 

• Odd numbers: numbers that are divisible with 2, but have remainders.

For example, 5 is an odd number because 5:2=2 (r=5) or 5:2=2.5

 

 

 

 

 

 

• The Place Value of Natural Numbers                                                                                                                Back to Menu

 

 

 

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

 

 

​

​

​

​

​

• Comparing numbers                                                                                                                                          Back to Menu

 

• Comparing signs - Same, Less & More ​ = Greater or smaller than

​

Same/Equal to:

=

Less/Smaller than:

< 

Less/Smaller & Equal:

≤

More/Greater than:

> 

More/Greater & Equal:

≥

​

 

• Comparing Natural numbers 1-10-100

0 = 0

0 < 1

5 > 2

10 > 8

103 < 1003

10,000 < 10,001

1 < 100 > 50 < 64 > 8 = 8 < 94 > 5 < 7 >10

 

 

• Comparing Integers

0 < 1

0 > -1

2 > -4

-10 > -20

-5 > -8

2 > -15

​

 

• Comparing decimal numbers

0.5 >0.25

0.05 > 0.02

-0.75 < 0.60

1 > 0.2

-0.2 > -0.5

 

• Comparing fractions

​

Comparing fractions with the same denominator

In the case of 2 fractions with the same denominator - the greatest fraction will have the greatest numerator.

3/2 > 2/2

10/2 < 20/2

3/4 > 2/4

-3/4 < 2/4

1/3 < 2/3

0/5 < 1/5

0/5 > -1/5

5/2 > 3/2

Comparing fractions with different denominators

In this case, we have to use fraction simplification or multiplication to obtaining of fractions with the same denominator.

Example: 7/2 and 3/4

To obtain the same denominator, 7/2 must be multiplied by 2: 

7x2 / 2x2   = 14 / 4.

Therefore, the comparison will be between 14/4 and 3/4.

14/4 > 3/4

consequently

7/2 > 3/4 

Verification: 7/2 = 14/4 =3.5, while 3/4 = 0.75.

3.5 > 0.75

 

Other examples:

5/2 > 6/5

8/5 < 7/2

7/2 > 9/6

3/5 < 5/8

 

 

 

​

​

• Basic Arithmetic Operations                                                                                                                           Back to Menu

​

First Order Operations: Addition & Subtraction

​

• Addition

a + b = c

a, b = terms, summand, addend,

c = sum

Example:

1 + 2 = 3

• Subtraction

a – b = c

a = term, minuend

b = term, subtrahend

c = difference

​

                               Example: 

5 - 4 = 1

​

​

​

Second Order Operations: Multiplication & Division

​

• Multiplication

a x b = c

a = factor, multiplier

 b = factor, multiplicand

c = product

Example:

                             5 x 2 = 10

 

 

• Division (Ratio, Fraction)

a : b = c (d)

a = dividend, numerator 

b = divisor, denominator

c = quotient

d = remainder

 

a : b or  a/b

 

Example:

6 : 2 = 3

Or

6/3 = 2

​

​

​

​

​

• The Order of Operations

​

​

Types of Brackets

• Small Brackets ( )

• Big Brackets    [ ]

• Curly Braces  { }

​

​

Rules:

A. Always start to calculate the operation from brackets, in the following order: small brackets, big brackets, curly braces;

A1. Inside the brackets calculate first the 2nd order operations (i.e., multiplication & division) in the order they appear;

A2. Inside the brackets calculate the 1st order operations after finishing to calculate the 2nd order operations;

A3. After calculating the operations inside the brackets, continue to calculate the operations outside the brackets;

B. If no brackets, apply the rule B & C (i.e., always calculate first the 2nd order operations and afterward the 1st order operations in the order they appear;

C. If there are only 1st order operations – calculate in the order they appear;

D. If there are only 2nd-order operations – calculate in the order they appear.

​

​

 

Example: 

10 x (8-3x5+7) – 10:2-3x{5x5+10:7-2[2x(6-5+6x4:3)-10+3]-100x2:50} +5

​

​

​

Example Observation 1:

The operations outside the brackets are calculated, where possible, by applying the rule B (calculate the 2nd order operations first, followed by 1st order operations);

 

Example Observation 2:

In the exercise below, the order of calculations will be:

RED (the 2 small brackets),

YELLOW [the big brackets],

ORANGE {the curly braces}

GREEN – the operations outside the brackets

​

​

Example - Calculation: 

​

10 x (8-3x5+7) – 10:2-3x{5x5+10:7-2[2x(6-5+6x4:3)-10+3]-100x2:50} +5

 

= 10 x (8-3x5+7) – 10:2-3x{5x5+14:7-2[2x(6-5+6x4:3)-10+3]-100x2:50} +5

= 10 x (8-15+7) – 5-3x{25+2-2[2x(6-5+24:3)-10+3]-200:50}+5

=10x (-7+7)-5-3x{27-2[2x(6-5+8)-10+3]-4}+5

=10x0-5-3x{27-2[2x(1+8)-10+3]-4}+5

= 0-5-3x{27-2[2x9-10+3]-4}+5

=-5-3x{27-2[18-10+3]-4}+5

=-5-3x{27-2[8+3]-4}+5

=-5-3x{27-2x11-4}+5

= -5-3x {27-22-4}+5

= -5-3x{5-4}+5

= -5-3x1+5

= -5-3+5

= -8+5

= -3

 Back to Menu

​

• Multiplication & Division Table

​​

• Observations​​

• Multiplication is a repeated addition.

3 x 4 = 12

3 x 4 = 3 + 3 + 3 + 3 = 12

3 x 4 = 4 + 4 + 4 =12

​

• Any number multiplied by 0 (zero) is 0 (zero).

3 x 0 = 0

0 x 3 = 0

​

• Zero divided by any number is 0 (zero).

0 : 2 = 0

0 : 7 = 0

0 : 100 = 0

​

• Any number divided by 0 (zero) is an impossible operation – an ERROR.

2 : 0  – impossible operation – error

7 : 0  – impossible operation – error

100 : 0  – impossible operation – error

​

• The division is the reverse operation of multiplication

6 x 2 = 12 

12 : 2 = 6   or  12 : 6 = 2

  

​

​

• Multiplication Table

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

• Division Table

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​ Back to Menu

​

​

​

• Operations with Integers (Z) - Positive & Negative whole numbers

​

Rules for Addition & Subtraction with Integers (Z):

​

+a +b = +c

Sign: +

Operation: Addition

+1+2 = +3

​

+a -b = +c

Sign + if a>b

Operation: Subtraction

+5-2 = +3

​

+a -b = -c

Sign - if a<b

Operation: Subtraction

+5-7 = -2

​

-a +b = +c

Sign + if a<b

Operation: Subtraction

-7+9=+2

​

-a +b = -c

Sign - if a>b

Operation: Subtraction

-3+1=-2

​

-a-b = -c

Sign: -

Operation: Addition

-5-3=-8

 

​

Resuming the Operations & Sign of Addition & Subtraction with Integers (Z):

​

​

​

​

​

​

​

​

​

​

​

​

​

​

 

 

 

 

Rules for Multiplication & Division with Integers (Z):

​

+a x (+b) = +c Operation: Multiplication

+5 x (+3) = +15

+a x (-b) = -c Operation: Multiplication

+4 x (-5) = -20

-a x (+b) = -c Operation: Multiplication

-2 x (+5) = - 10

-a x (-b) = +c Operation: Multiplication

-5 x (-3) = +15

​

Idem for Division

​

​

Resuming the Sign of Multiplication with Integers (Z):

 

​

​

​

​

​

​

​

​

​

 

 

 

 

 

​

​

 Back to Menu