Numerical expressions - order of operations and solved examples

Numerical expressions are essential for knowledge of maths, and very important in its use in every day. To help you study maths, here you will learn what are numerical expressions and how to calculate numerical expressions. 

Numerical expressions

A numerical expression is a way of expressing reasoning in a problem and another way to represent a number. Any numerical expression represents a value. But you perceive better checking the following example: 

John went to the supermarket with 10 dollars and bought two cans of mushrooms and a bag of flour. Knowing each can of mushrooms cost $1.30 and the package flour $1.10, indicate: 

a) how much money was spent. 

To solve this problem, you must use the numerical expression: 2 x 1.30 + 1.10 

2 x 1.30 + 1.10 expresses the amount he spent, which is equal to $3.70. 

b) how much money is left. 

To solve this problem, you must use the numerical expression: 10 - (2 x 1.30 + 1.10) 

10 - (2 x 1.30 + 1.10) expresses the value that is left, which is equal to $6.30. 

How to calculate numerical expressions 

To learn how to calculate numerical expressions, is critical to realize that exists a series of operations, and so, there must be rules to implement them. This rules are called order of operations. So, to calculate numerical expressions, you have to know the order of operations of numerical expressions. Below, you can find an example with the rules explained to you understand more easily. 

• Order of operations

In a numerical expression, the operations are not made simply by the order in which they appear. Each type of operation has a priority, which indicates which operations must be done in the first place. Then you can check the order of operations to calculate numerical expressions. 

1st Priority - Firstly, make all calculations within parentheses (these calculations must follow the rules described below). 

2nd Priority - For operations, we must first make the exponents or roots. 

3rd Priority – Calculate the multiplications and divisions in the order they appear. 

4th Priority - Finally, calculate the additions and subtractions in the order they appear. 

• Solved examples of numerical expressions 

Let's start with a simple numerical expression: 

    23-4 x 5 =    --- 3rd P - Make first the multiplication 

= 23-20 =         --- 4th P – Make the subtraction 

= 3 

Now a solved example a little more complicated:

How to find gcd (greatest common divisor)

The divisor of a number is a natural number that divides another, and which results in an integer. For example, the dividers of 12 are 1, 2 , 3, 4, 6 and 12. Set of divisors of a number is always a finite set of numbers.

12: 1 = 12

12: 2 = 6

12: 4 = 3

12: 4 = 3

12 6 = 2

12: 12 = 1

D12 = { 1,2,3,4,6,12 }

The gcd (greatest common divisor), as the name implies, is the greatest common divisor of two or more numbers integers. For example, if D12 = { 1,2,3,4,6,12 } and D18 = { 1,2,3,6,9,18 }, the greatest common divisor between 12 and 18 is 6. Yet, if instead of making the gcd of 12 and 18, we do the gcd of two much larger numbers, it will take too long. Thus, instead of making the dividers of each number, there is a faster method.

Now we'll explain how to do the gcd of two or more numbers integers.

Step 1

By making the decomposition in prime factors of each number. If you don't know how to decompose a number into prime factors, click here.

Step 2

After you've decompose down each number into its prime factors, you will choose the smallest common factors. The non common factores doesnt interest.

Step 3

Calculates the product of the selected factors . The result is the greatest common divisor of the two given numbers integers.

Easy way to divide fractions

Here you can learn a esay way to divide fractions. Unlike addition and subtraction of fractions, division is a simple, easy to learn method. But to learn this easy way to divide fractions, you have to know before how to multiply fractions, and what is the inverse of a number. If you need to study maths, here you can find all you need.

Inverse of a number

The inverse of a number is the number that, multiplied by the first number, gives the result 1.

example:

3 is the inverse of 1/3 and vice versa.
2/5 is the inverse of 5/2 and vice versa,

Now we will explain to you a easy way to divide fractions

• Easy way to divide two fractions:

To divide two fractions, replace the signal division by the multiplication sign, and the second number by its inverse. Then simply multiply both fractions.

• Easy way to divide fractions by a whole number:

If you want to know how to divide fractions by a whole number, you only have to do what was done above, and in the end multiply the numerators and denominators.

Easy way to multiply fractions

Here you can learn a easy way to multiply fractions. Unlike addition and subtraction of fractions, multiplication of fractions is a very simple, easy to learn, method. So, to help you study maths, here we will explain a easy way to multiply fractions.

• How to multiply two fractions:

To multiply two fractions, simply multiply the numerators and denominators.

examples:

• How to multiply a whole number by a fraction.

Here you have a easy way to multiply fractions by a whole number. You only have to multiply the numerator by the whole number, keeping the denominator.

examples:

Do you know what's the magic number?

The number 1089 is known as the magic number. Do you know why? 

To better understand why the number 1089 is the magic number, I'll ask you to choice any number that is composed of 3 different digits. For example, 286. 

Considering the number 286, you begin by writing 286 backwards. Then subtract the smaller number to the larger number: 

682-286 = 396 

Then you write the result backwards, and add up the two numbers: 

396 +693 = 1089 ---- magic number 

Fantastic, isn't it? 1089 is the magic number, and it works with all numbers composed by 3 distinct digits. But beware, if in the first step, the difference is, for example 099, when reverse that number you must count with the zero. In this case, 

453-354 = 099 

099 + 990 = 1089

What is the Pi number?

The Pi number is one of the most famous numbers in maths, and is used in many formulas, always related to circumferences or circles. But what is the Pi number?

If you measure the perimeter of a circle, and divide it by its diameter, the value of this ratio is equal to approximately 3,14. This happens with the ratio of the length of the circumference of a circle and its diameter in all all circles. For this reason it is said that Pi is a constant, cause it exists direct proportionality between the ratio of the length of the circumference of a circle and its diameter.

P - perimeter of the circumference of the circle

d - diameter

Pi is an irrational number with a non-periodic infinite tithe, which is the ratio between the length of the circumference of a circle and its diameter. Next, we present the Pi number to its 300ª decimal place.

In daily maths, it is usually used the rounding to two decimal places :

How to do prime factorization

The Fundamental Theorem of Arithmetic states that any composite number (number divisible by more than two dividers) , is a product of prime numbers. So, in this article you will learn how to do prime factorization. This decomposition will allow then calculate the lcm (least common multiple) and gcd (greatest common divisor).

Prime Factorization method

With this two examples we will explain how to make the prime factorization (decomposition into prime factors).

Taking one of the examples :

- Check if the number is divisible by the first prime number, 2. If so, perform the division (12: 2 = 6). If not, see if it is divisible by the following prime number, and so on, until you find a prime number that is a divisor of the given number. To do this step is important to know the divisibility rules.

- Repeat the same procedure until you have the result 1.

- The number you want to decompose is the product of all prime numbers placed to the right of the line.

The tree method

In this method of prime factorization, you will decompose the given number, until you have only prime numbers. Look at the example. 

We hope that this article was helpfull to you learn how to do prime factorization and study maths.

Prime numbers up to 100

Do you know what is a prime number? To help you understand what is this special number and to use them to study maths, below you can find the definition of prime numbers and the first prime numbers up to 100.

The simplest definition is that a prime number is a number that is only divisible by itself and 1. That is, it has two dividers. For this reason, it is not considered the number 1 to be a prime number. All the numbers that have more than two dividers are called composite numbers. 

For example, the set of dividers of the number 5 is {1,5}, then 5 is a prime number. 

There are an infinite number of primes. Below are the prime numbers up to 100. 

List of the prime numbers up to 100: 

2, 3, 5, 7, 

11, 13, 17, 19, 

23, 29, 

31, 37, 

41, 43, 47, 

53, 59, 

61, 67, 

71, 73, 79, 

83, 89, 

97

Number sets

In mathematics, to easily organize the study of numbers, it was necessary to make groups. In each group were united the numbers having the same characteristics. Throughout history, number sets have evolved, until we get to the current organization of numbers. In the diagram below you can see the currently existing number sets. We left out purposely complex numbers and other even more complicated. Hope you use this to study maths.

Natural numbers - N

The natural numbers are those that can represent something in nature. Thus, we consider natural numbers all integers greater than zero. N = {1, 2 , 3, 4, 5, ...}

There is a group of natural numbers, but now integrating zero.

Integers - Z

The integers cover all natural numbers, zero, and also the opposite of the natural numbers. Thus, Z = {... -5, -4 , -3, -2, -1 , 0, 1 , 2, 3 , 4, 5 ...}

Rational numbers - Q

The rational numbers include the integers (example: -6) and decimal, finite (example: 2,493) and with infinite repeating decimal (example: 0.456456456 ...) The rational numbers can be represented by a fraction. 

Irrational Numbers - I

This set consists of infinite decimal numbers with a non repeating decimal. As examples, we have 0,428465242 ..., or pi (3.1415926 ...).

Real numbers - R

The real numbers is the set of all number sets referred aboved.

Divisibility rules for 2, 3, 4, 5, 6, 8, 9 and 10

Divisibility rules, also known as divisibility criteria, are rules that let you know if a number can divide another in an exact manner. Thus, it's possible to know whether a number is a divisor of another, without having to actually do the calculations. In this blog you can learn the divisibility rules for 2, 3, 4, 5, 6, 7, 8, 9, 10, and other numbers.

- divisibility rules for 2

The number two divides all pair numbers, that is, all numbers that ends with the digits 0, 2, 4, 6 or 8. 

example: 

68 is pair, so, it's divisible by 2.

912 is pair, so it's also divisible by 2.

- divisibility rules for 3 

A number is divisible by 3, if the sum of its digits is a multiple of 3. 

example: 

The sum of the digits of 531 is 5 +3 +1 = 9. 

9 is a multiple of 3, so 531 is divisible by 3

- divisibility rules for 4 

A number is divisible by 4 if the sum of its units digit with the twice the tens digit, is a multiple of 4. 

example: 

In the number 7236, the sum of the units digit (6) with twice the number of tens (3) is 6+2x3 = 12 

12 is multiple of 4, then 7236 is divisible by 4.

- divisibility rules for 5 

A number is divisible by 5 if it ends in 0 or 5. 

example: 

345 ends in 5, so it is divisible by 5


- divisibility Rules for 6

A number is divisible by 6 if it is divisible by 2 and 3. So, just use the divisibility rules for 2 and 3 to verify.

example, consider the number 972 .
972 is a pair number, so it is divisible by 2 .
9 +7 +2 = 18, 18 is a multiple of 3, so 972 is divisible by 3.
Since 972 is divisible by 2 and 3, so it is also divisible by 6.


- divisibility rules for 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8 .

Example: In the number 4240, the number formed by the last three digits is 240. As 240:8 = 30, then 4240 is divisible by 8.


- divisibility rules for 9

A number is divisible by 9 if the sum of its digits is a multiple of 9 .

example : The sum of the digits of 693 is 6 +9 +3 = 18 .
18 is a multiple of 9, so 693 is divisible by 9


- divisibility by 10

A number is divisible by 10 if the last digit is a zero.

example: 83610

The number ends in zero, so it is divisible by 10 .