Download 5. operations with natural numbers.

5. OPERATIONS WITH NATURAL NUMBERS.
5.1.- Addition
Adding is the same as putting together or joining two values into one.
We read 3 + 5 = 8 as: “Three plus five is equal to eight” or “Three plus five equals eight” or
“Three plus five is eight”.
The terms in the addition are called addends and the result is called the sum.
Solved Example:
The library lent 45 books last Monday, 50 books on Tuesday and 73 books on Wednesday.
How many books have they lent?
45 + 50 + 73 = 168 books.
Answer: They have lent 168 books
5.2.- The properties of addition
The properties of addition are: closure, commutative, associative, and additive identity.
Closure property: Addition of two natural numbers is always another natural number.
For example: 6 + 7 = 13 (13 is another natural number)
Commutative property: When we add two natural numbers, the sum is the same no matter
what the order of the addends is. a + b = b + a.
For example 4 + 2 = 2 + 4.
Associative Property: When we add three or more natural numbers, the sum is the same no
matter how we group the addends. a + b + c = (a + b) + c = a + (b + c).
For example: (2 + 3) + 4 = 2 + (3 + 4)
Additive Identity Property: The sum of any natural number and zero is the original number.
For example 5 + 0 = 5.
5.3.- Subtraction of natural numbers
Subtracting is removing or taking away some objects from a group.
We read 13 – 7 = 6 as: “Thirteen minus seven equals six” or “Thirteen subtract seven equals
six”.
The terms of subtraction are called minuend and subtrahend, the result is called the
difference.
The minuend is the first number, it is the number from which you take something and it must
be the larger number.
The subtrahend is the number that is subtracted and it must be the smaller number.
The difference is the result of the subtraction.
To check if the subtraction is correct we add up the subtrahend and the difference. The result
must be the minuend.
Solved example:
We have saved 3,520 euros but we have spent € 745 on a computer. How much money do we
have left?
3,520 – 745 = 2,775. Answer: We have 2,775 euros left.
Addition and subtraction are opposite operations.
This means that one can be used to ‘undo’ the other. If we start with any number and add any
number and then subtract the same number we added, we return to the number we started with.
The same will happen if we first subtract and then add the same number.
5.4.- Multiplication
Multiplying is doing an addition of equal addends.
3 + 3 + 3 + 3 + 3 = 3 · 5 = 15
We read 3 · 5 = 15 as: “Three times five equals fifteen” or “Three times five is fifteen”.
The factors are the numbers that are multiplied together. The product is the result of
multiplying.
Solved Example:
In my living-room I have a bookcase with three shelves. If there are five books on each shelf,
how many books are there?
5 · 3 = 15 Answer: There are 15 books in my bookcase.
5.5.- The properties of multiplication
The properties of multiplication are: closure, commutative, associative, and additive identity.
Closure property: Multiplication of two natural numbers is always another natural number.
For example 6 · 7 = 42
Commutative property: When two numbers are multiplied together, the product is the same
no matter what the order of the factors is.
For example 4 · 2 = 2 · 4
Associative Property: When three or more numbers are multiplied, the product is the same no
matter how we group the factors.
For example: (2 · 3) · 4 = 2 · (3 · 4).
Multiplicative Identity Property: The product of any number and one is that number.
For example 5 · 1 = 5.
Distributive law for addition: Calling a, b and c to three different natural numbers, it is a law
that: a · (b + c) = a · b + a · c
For exemple: 2·(5 + 7) = 2· 5 + 2· 7
This property is also true for the subtraction: a · (b – c) = a · b – a · c
5.6. Division
Dividing is to share a quantity into equal groups. It is the inverse of multiplication. In Spanish
we write 6 : 2 , but in English it is always 6 ÷ 2, we never use the colon (:).
We read 15 ÷ 5 = 3 as: “Fifteen divided by five equals three”.
There are four terms in a division: dividend, divisor, quotient and remainder.
The dividend is the number that is divided.
The divisor is the number that divides the dividend.
The quotient is the number of times the divisor goes into the dividend.
The remainder is a number that is too small to be divided by the divisor.
Solved Example:
There are 72 sweets in a bag. If we want to distribute them among 12 children, how many
sweets are there for each child?
72 : 12 = 6. Answer: There are six sweets for each child.
The division can be: Exact: the remainder is 0, or non exact: the remainder isn’t 0.
To check if the division is correct we do the division algorithm:
Dividend = Divisor · Quotient + Remainder;
Solved example:
Find out the result of the division 237 : 13 and then check the result with the division
algorithm:
237 : 13 = 18 remainder = 3
Dividend = Divisor · Quotient + Remainder; 13 · 18 + 3 = 237, so it is correct.
Multiplication and division are opposite operations.
Multiplying any number by another number and then dividing the result by the same number
we multiplied by, will give us as a result the number we started with. Similarly, if we first
divide and then multiply by the same number, we will get the number we started with (we need
to remember that division by zero is not defined).
Working with the Properties of Mathematics
1 ) Which of the following does not show the Commutative Property of Addition ?
A. a + b = b + a
B. 5 + x = x + 5
C. ab = ba
D. 3x + 4y = 4y + 3x
2 ) Which property is used in the following ?
9 x (7 + 3) = 9 x 7 + 9 x 3
A. Distributive Property
B. Associative Property
C. Commutative Property
D. None of the above
3 ) Which is an example of Associative Property of Addition ?
A. 8 + 0 = 8
B. 6 + 9 = 9 + 6
C. 2 + (-2) = 0
D. (9 + 5) + 7 = 9 + (5 + 7)
4 ) Which is an example of Identity Property of Addition ?
A. 8 x 1 = 8
B. (5 + 6) + 2 = 5 + (6 + 2)
C. 4 + 5 = 5 + 4
D. 3 + 0 = 3
5 ) Which property of addition is used in the following ?
(2 + 4) + 5 = 2 + (4 + 5)
A. Distributive Property
B. Commutative Property
C. Associative Property
D. Identity Property
6 ) Which property is used in the following expression ?
7(9 + 2) = 63 + 14
A. Associative Property of Multiplication
B. Associative Property of Addition
C. Commutative Property of Addition
D. Distributive Property
7 ) Which property is used in the following expression ?
(8 x 6) x 4 = 6 x (4 x 8)
A. Distributive Property of Multiplication
B. Associative Property of Multiplication
C. Associative Property of Addition
D. Commutative Property of Addition
8 ) Which Property of Multiplication is shown ?
(7 + 6) x 8 = 7 x 8 + 6 x 8
A. Identity Property
B. Distributive Property
C. Commutative Property
D. Associative Property
9 ) Which equation shows the Identity Property of Multiplication ?
A. a(b + c) = ab + ac
B. (a + b) + 7 = a + (7 + b)
C. a + a + a = 3 x a
D. a x 1 = a
10 ) Which of the following is an example of Commutative Property of Addition ?
A. 4 x 1 = 4
B. 8 + 5 = 5 + 8
C. (5 + 3) + 9 = 5 + (3 + 9)
D. 6 + 7 = 2 + 6