Operations with Numbers and Properties of Operations

In these revision notes for Operations with Numbers and Properties of Operations, we cover the following key points:

• What is a mathematical operation? What are the four basic operations used in math?
• What is addition? What are the properties of addition?
• Why we sometimes split an addend in two parts.
• How to add two numbers in a column and in a number line.
• What is subtraction? What is/are the property (ies) of subtraction?
• How is subtraction related to addition?
• How to subtract two numbers in a column and in a number line.
• What is multiplication? What does it have in common with addition?
• What are the properties of multiplication?
• How to do a multiplication in a column?
• What is division? How is it related to multiplication?
• What is/are the property (ies) of division?
• How to make a division operation easier?
• What is the remainder of a division?
• What is "power of numbers"? How is it expressed in exercises?

Operations with Numbers and Properties of Operations Revision Notes

A mathematical operation is a process in which a number or quantity is changed in order to give another number or quantity. There are four basic mathematical operations: addition, subtraction, multiplication and division.

Addition is a mathematical operation that shows two or more numbers added together. The result of addition is called the sum and each element (number) participating in an addition operation is called an addend.

The operation of addition is expressed through the plus (+) symbol.

An addition expressed on a number line means a shift due right.

Addition as a mathematical operation has the following properties:

1. Closure property - If two (or more) addends belong to a given number set, the sum also belongs to this set.
2. Commutative property - If we switch the place of addends the sum does not change.
3. Associative property - if we dont start the operations from the two leftmost addends but from two other addends more in the right, the sum does not change.
4. Additive identity property - Any number added with zero gives the same number as the sum. Zero is called the identity element of addition.

Sometimes, it is more appropriate to split an addend in two parts to make the calculations easier. Then we can apply any of the aforementioned properties of addition to find the sum.

We can add numbers in columns for simplicity. In this method, we place the like placeholders in the same column (units below units, tens below tens and so on).

Subtraction is a mathematical operation in which we remove a quantity of items from a collection. Subtraction is represented mathematically through the symbol minus (-).

The participants in a subtraction are as follows: the first number (usually the biggest) from which the quantity is subtracted is called the minuend; the quantity subtracted is called the subtrahend, while the result of subtraction is known as the difference.

Subtraction on a number line is done by moving due left.

Subtraction has only one property in common with addition - the Subtractive Identity Property of Zero, where zero is the subtractive identity element. Based on this property, when we subtract zero from a number, the value of the number remains the same.

Subtraction is commonly considered as the inverse operation of addition. Hence, instead of writing a - b = c, we can write a + (-b) = c as the result is the same in both cases.

We can subtract numbers in a column in the same way as we do in addition. In this case, when the upper digit is smaller than the lower digit, we borrow one "ten" from the next placeholder on its left. We say a "ten" is borrowed from the left placeholder to make the subtraction Possible.

Multiplication is a shorter representation of the repeated addition of equal numbers. It is represented in expressions through the symbols ( × ) or ( · ).

The numbers participating in a multiplication operation are known as factors and the result of multiplication is called the product.

Properties of multiplication include:

1. Closure property of multiplication - If two (or more) factors belong to a given number set, the product also belongs to this set.
2. Commutative property of multiplication - When we switch the place of factors the product does not change.
3. Associative property of multiplication - We can start the operations from anywhere for convenience as the result does no change.
4. Distributive property - When an expression inside brackets containing addition or subtraction is multiplied by a number, the expression can be written without brackets where the given number multiplies every element of the expression separately.
5. Multiplicative identity property - In multiplication, this identity element is the number 1. This means that if we multiply a number by 1, the product is the same as the number itself.

We can multiply two numbers in column for an easier solution. We multiply each number of the upper factor to each number of the lower factor and the products obtained are written in separate rows below each other by starting (due right) from the position of the digit of the lower number involved in the process.

Division is the inverse operation of multiplication. We apply division when we want to cut a quantity into equal parts.

Division is the inverse operation of multiplication. We use the symbol (÷) to represent division in a mathematical expression. The original quantity is known as the dividend, the number that divides the original quantity in equal parts is called the divisor while the result of division is called the quotient.

Division has only one property in common with multiplication: the divisive identity property. According to this property, the number 1 is the identity element for division because any number divided by 1 gives the same number as a result.

When we divide two natural numbers, the result is not always a natural number. However, if we want to keep writing the result as a natural number, we write the closest value as quotient and also the remainder of division inside brackets, i.e. the number left from the division operation.

If we have to multiply the same factor a number of times by itself, we use a shorter notation known as power to represent this recurring multiplication by the same number. In other words, the power of a number says how many times to use that number in a multiplication. The recurring factor is called the base, the number that shows how many times this factor appears in a recurring multiplication is called the exponent and the result of this operation is called the power.