Focus

Our goal is to review two basic type of numbers the integers and the decimals and then we will learn a new one, the fractions.

Explore

From simple to complex:

Is zero the most basic number? Or 1? We use these to manipulate quantities, zero is when there is nothing left, 1 , 2 , 3 etc. can help us  counting how many of a thing we have.  It is easy to count up to 10 using the fingers of our both hands and then continue on with groups of ten – this system is the most commonly used aka as the decimal system (base 10).

Machines prefer the binary system made of only 0 and 1 (base 2). Why so? Electronic components can only make sense of 0 (no current going through, off) and 1 (the current is going through, on). In this system the numbers are written columns associated with the powers of  2 (1-2-8-16-32-64-128-256-etc): for example, the number 2 is written “10”. The first column is 0 and the second column is 1. The number 3 is written: 11.  (first colum is 1+ 2 in the second column). In base 2, the number 11011 is equal to 1+2+0+8+16=27 in base 10.

Similar column system is used in base 10: this time we use powers of 10 (10-100-1000-10000). The number 27 is 2 times 10 plus 7 = 27.

These numbers are part of the family of “integers”. They are made of one of the 10 numbers organized in columns. There is whole with no decimal or fraction involved. They can be as large as needed: What other integers do you know? For example 9 000 000 (9 million) is an integer. There is no limit of how big they can be, there is an infinite number of integers.

We can also have negative integers, adding a negative sign in front.

For example, minus 18. When is this number used? For example to describe the optimal temperature in a freezer: -18oC. Negative numbers are also used to measure financial loss. They are also useful when evaluating the depth below the surface of  an ocean.

We can use a number line to show positive, negative integers, and zero. The further away a positive number from zero the larger they are. On the contrary, negative numbers are getting smaller when they are further away from zero.

——————- drawing with -15 and -10 and 0

Some integers are very special they have unique properties:

Let’s talk briefly about “prime” numbers, they have been discovered by mathematicians. The multiples of the simple prime numbers are important to recognize and know.

There is also Even and Odd numbers that have special properties.

We can perform all sort of calculation with numbers. Addition, subtraction, multiplication and division. Calculation are more challenging with large numbers and most people use a calculator to perform these. Practicing to become fast and accurate with these calculations is one of the activities learned in math classes. Such abilities are used for tests and games.

Some mathematicians specialized in the study of integers and discover new properties that are surprising and non obvious. For example, a direct application is encrypting technologies are based on the multiplication of two large numbers, a private and a public key. The public key is visible but the private key is confidential. Computers are unable to multiple two large numbers (or that takes a very long time) and that is the basis of secret and encryption as they can not easily guess the secret key. For example, eCommerce is using 256-encryption key to communicate sensitive information (such as credit card info), it is an integer with TO DO

 

Even though integers are fascinating numbers and super useful, in the modern times humans have found they are  not sufficient to perform some more sophisticated calculation when dealing with parts. Indeed, integers can only be used to count whole things. So how do we manage to add, multiply, and performing calculations when dealing with the parts of an object.

For example, let’s take a pizza and cut three equal parts. Each part is equal to 1 third:

1/3

Where is this fraction located in the number line: in between 0 and 1. Less than a whole part and more than nothing.

Let’s take the same pizza and cut it in 4 and 6. We ccan form two more fractions:

1/3 and 1/4

Where are they located? Again, between 0 and 1.

In fact there is an infinite number of fractions located between 0 and 1.

How about between 1 and 2, what fractions exist in between these 2 numbers:

If I take six pizzas that were all cut in 3 each, what if I take several parts, how do I add up these fractions, what is the final result, is it an integer or is it a fraction?

Let’s take three pieces:

1/3+ 1/3+1/3= 3/3 = 1

the result is an integer it is exactly 1

What about taking 6 pieces:

1/3+ 1/3+1/3 +1/3+ 1/3+1/3= 6/3=2

the result is an integer it is exactly 2

Can you guess the result for 4 pieces?

1/3+ 1/3+1/3+1/3=4/3=1+1/3

This is a fractional result, where is this number located? Between 1 and 2

How do we express this result? Do we write 4/3? or do we write 1+1/3?

It all depends. Mathematicians prefer to use only one fraction (and we will see why later) but in real life it is very convenient to isolate a whole part and a smaller remaining wrote as a fraction like this: 1+ 1/3

Let’s introduce some vocabulary:

numerator

denominator

when a numerator  is larger than the denominator, the fraction is said to be improper, such as 4/3. (numerator is 4 larger than the denominator  3)

If you are asked to write a result as a proper form: 1+1/3

1/3 is a proper form because the numerator 1 is not larger than its denominator.

Already, we have some abstraction to comprehend. Knowing that 4/3 and 1+1/3 represent the exact same quantity. How can we make sense of this?There is  lot of knowledge regarding fractions that can help and we will memorize and learn to be more agile at calculating with fractions.

Not sure why there is no TV game for calculating fractions fats and accurately the same way there is one for integers. Fractions are for sure more complex and more sophisticated. They are essential to perform accurate operations on parts.

Fractions are tricky because some of them are equal to Integers and even Decimals.

For example:

3/3=1 is an integer

1/2=0.5 is a decimal

1/3 is not an integer and is not a decimal, it can only be represented accurately using the fraction symbol

When we calculate with fractions it can be tempting to mix integers, decimals and fractions, there is nothing wring with doing that:

1 + 1/2 + 1/3= 1.5 + 1/3 = 4/3 + 1/2 = 11 /6

if we mix all these types of numbers, there are many correct ways to write the result, but if we use only one fraction as the result, it is unique. This is why in math class, we prefer to express all our results and numbers in fractions to make sure we are goign to find the same exact result!