Addition of Whole numbers Subtraction of Whole numbers Properties of Addition

1. Chapter 1
Addition and Subtraction of
Whole Numbers
FUNDAMENTAL OF MATHEMATICS

2. Chapter 1: Addition and Subtraction of
Whole Numbers
• 1.3 Addition of Whole Numbers
• 1.4 Subtraction of Whole Numbers
• 1.5 Properties of Addition
Source:
Fundamental of Mathematics Denny Burzynski & Wade Ellis
https://cnx.org/contents/XeVIW7Iw@4.6:l0Q7pQNZ@5/Addition-of-Whole-Numbers

3. 1.3 Addition of Whole Numbers
• Objectives:
• understand the addition process
• be able to add whole numbers
• be able to use the calculator to add one whole number to
another

4. 1.3 Addition of Whole Numbers
• Suppose we have two collections of objects that we combine
together to form a third collection. For example,
is combine with to yield
We are combining a collection of four objects with a
collection of three objects to obtain a collection of seven
objects.

5. 1.3 Addition of Whole Numbers
• Addition
The process of combining two or more objects (real or intuitive)
to form a third, the total, is called addition.
In addition, the numbers being added are
called addends or terms, and the total is called the sum.
The plus symbol (+) is used to indicate addition, and the equal
symbol (=) is used to represent the word "equal." For example, 4
+ 3 = 7 means "four added to three equals seven."

6. Addition Visualized on the Number Line
• Addition is easily visualized on the number line. Let's visualize
the addition of 4 and 3 using the number line.
• To find 4 + 3
• Start at 0.
• Move to the right 4 units. We are now located at 4.
• From 4, move to the right 3 units. We are now located at 7
• Thus, 4 + 3 = 7

7. The Addition Process
We'll study the process of addition by considering the sum of 25
and 43
We write this as 68.

8. The Process of Adding Whole Numbers
To add whole numbers,
The process:
1) Write the numbers vertically, placing corresponding
positions in the same column.
2) Add the digits in each column. Start at the right (in the
ones position) and move to the left, placing the sum at the
bottom.

9. Addition Involving Carrying
• It often happens in addition that the sum of the digits in a
column will exceed 9. This happens when we add 18 and 34. We
show this in expanded form as follows.

10. Addition Involving Carrying
• This same example is shown in a shorter form as follows:
8 + 4 = 12 Write 2, carry 1 ten to the top of the next
column to the left.

11. Addition Involving Carrying
• Sample Set B
• Perform the following additions. Use the process of carrying
when needed.
1) Find the sum 2648, 1359, and 861.
2) The number of students enrolled at NCL College in the years
1984, 1985, 1986, and 1987 was 10,406, 9,289, 10,108, and
11,412, respectively. What was the total number of students
enrolled at NCL College in the years 1985, 1986, and 1987?

12. Calculators
Calculators provide a very simple and quick way to find sums of
whole numbers.
Sample Set C
Use a calculator to find each sum.
a) 9,261 + 8, 543 + 884 + 1,062
b) 10,221 + 9,016 + 11, 445

13. Exercises
For the following problems, perform the additions. If you can,
check each sum with a calculator.
1) 43,156,219 + 2,013,520 2)

14. Exercises
3) Perform the additions
and round to the nearest
hundred.
4) replace the letter m
with the whole number
that will make the addition
true.
5) The enrollment in public and nonpublic schools in the years
1965, 1970, 1975, and 1984 was 54,394,000, 59,899,000,
61,063,000, and 55,122,000, respectively. What was the total en-
rollment for those years?

15. Exercises
6) Find the total number of scientists employed in 1974.

16. 1.4 Subtraction of Whole Numbers
• Objectives
• understand the subtraction process
• be able to subtract whole numbers
• be able to use a calculator to subtract one whole number from
another whole number

17. 1.4 Subtraction of Whole Numbers
• Subtraction is the process of determining the remainder when
part of the total is removed.
• Suppose the sum of two whole numbers is 11, and from 11 we
remove 4. Using the number line to help our visualization, we
see that if we are located at 11 and move 4 units to the left,
and thus remove 4 units, we will be located at 7. Thus, 7 units
remain when we remove 4 units from 11 units.

18. 1.4 Subtraction of Whole Numbers
• The Minus Symbol
The minus symbol (-) is used to indicate subtraction. For example,
11 – 4 indicates that 4 is to be subtracted from 11.
• Minuend
The number immediately in front of or the minus symbol is called the minuend,
and it represents the original number of units.
• Subtrahend
The number immediately following or below the minus symbol is called
the subtrahend, and it represents the number of units to be removed.
• Difference
The result of the subtraction is called the difference of the two numbers. For
example, in 11 - 4 = 7, 11 is the minuend, 4 is the subtrahend, and 7 is the
difference.

19. Subtraction as the Opposite of Addition
• Subtraction can be thought of as the opposite of addition. We
show this in the problems in Sample Set A.
• Sample Set A
8 – 5 = 3 since 3 + 5 = 8
9 – 3 = 6 since 6 + 3 = 9
• Practice Set A: Complete the following statements.
7 – 5 = ______ + 5 = 7
17 – 8 = ______ + 8 = 17

20. The Subtraction Process
We'll study the process of the subtraction of two whole numbers
by considering the difference between 48 and 35.
which we write as 13.

21. The Process of Subtracting Whole Numbers
To subtract two whole numbers,
The process:
1) Write the numbers vertically, placing corresponding
positions in the same column.
2) Subtract the digits in each column. Start at the right, in the
ones position, and move to the left, placing the difference
at the bottom.

22. 1.4 Subtraction of Whole Numbers
Sample Set B
Perform the following subtractions.
1) Find the difference between 977 and 235.
Write the numbers vertically, placing the larger
number on top. Line up the columns properly.
The difference between 977 and 235 is 742.

23. 1.4 Subtraction of Whole Numbers
2) In Keys County in 1987, there were 809 cable television
installations. In Flags County in 1987, there were 1,159 cable
television installations. How many more cable television
installations were there in Flags County than in Keys County in
1987?

24. Subtraction Involving Borrowing
Minuend and Subtrahend
It often happens in the subtraction of two whole numbers that a
digit in the minuend (top number) will be less than the digit in
the same position in the subtrahend (bottom number). This
happens when we subtract 27 from 84.
We do not have a name for 4 – 7. We need to rename
84 in order to continue. We'll do so as follows:

25. Subtraction Involving Borrowing
We do not have a name for 4 – 7. We need to rename 84 in order
to continue. We'll do so as follows:
Our new name for 84 is 7 tens + 14 ones.
= 57

26. Subtraction Involving Borrowing
Borrowing
The process of borrowing (converting) is illustrated in the
problems of Sample Set C.
• Sample Set C
1. Borrow 1 ten from the 8 tens. This leaves 7 tens.
2. Convert the 1 ten to 10 ones.
3. Add 10 ones to 4 ones to get 14 ones.

27. Borrowing from Zero
Borrowing from a Single Zero
To borrow from a single zero,
• Decrease the digit to the immediate left of zero by one.
• Draw a line through the zero and make it a 10.
• Proceed to subtract as usual.

28. Borrowing from Zero
Consider the problem
Since we do not have a name for 3 – 7, we must borrow from 0.
Since there are no tens to borrow, we must borrow 1 hundred.
One hundred = 10 tens.
503
-37
-----
We can now borrow 1 ten from 10
tens (leaving 9 tens). One ten = 10
ones and 10 ones + 3 ones = 13
ones.

29. Borrowing from a group of zeros
To borrow from a group of zeros,
• Decrease the digit to the immediate left of the group of zeros by one.
• Draw a line through each zero in the group and make it a 9, except the
rightmost zero, make it 10.
• Proceed to subtract as usual.

30. Borrowing from a group of zeros
Sample Set F
Perform each subtraction.

31. Calculators
In practice, calculators are used to find the difference between
two whole numbers.
Practice Set G
1) Use a calculator to find the difference between 7338 and 2809.
2) Use a calculator to find the difference between 31,060,001 and
8,591,774.

32. Exercises
For the following problems, perform each subtraction.
1) Subtract 26,082 from 35,040.
2) How much bigger is 3,080,020 than 1,814,161?
3) The 1980 population of Singapore was 2,414,000 and the 1980
population of Sri Lanka was 14,850,000. How many more
people lived in Sri Lanka than in Singapore in 1980?
4) Add the difference between 815 and 298 to the difference
between 2,204 and 1,016.

33. Exercises
5) How many more social, psychological, mathematical, and
environmental scientists were there than life, physical, and
computer scientists?

34. 1.5 Properties of Addition
• Objectives
• understand the commutative and associative properties of
addition
• understand why 0 is the additive identity

35. The Commutative Property of Addition
Sample Set A
Add the whole numbers
8 + 5 = 13
5 + 8 = 13
The numbers 8 and 5 can be added in any order. Regardless of
the order they are added, the sum is 13.
If two whole numbers are added in any order, the sum will not
change.

36. The Commutative Property of Addition
Practice Set A
1) Use the commutative property of addition to find the sum of
837 and 1,958 in two different ways.
2) Use the commutative property of addition to find the sum of
265,094 and 32,508 in two different ways.

37. The Associative Property of Addition
Using Parentheses
It is a common mathematical practice to use
parentheses to show which pair of numbers we wish to combine
first.
Sample Set B: Add the whole numbers.
If three whole numbers are to be added, the sum will be the same if
the first two are added first, then that sum is added to the third, or,
the second two are added first, and that sum is added to the first.

38. The Associative Property of Addition
Practice Set B
Use the associative property of addition to add the following
whole numbers two different ways.
1) 17, 32 and 25.
2) 1,629; 806 and 429.

39. The Additive Identity
• Sample Set C
• Add the whole numbers.
29 + 0 = 29
0 + 29 = 29
Zero added to 29 does not change the identity of 29.
0 Is the Additive Identity
The whole number 0 is called the additive identity, since when
it is added to any whole number, the sum is identical to that
whole number.

40. Exercises
1) The fact that (a first number + a second number) + third
number = a first number + (a second number + a third
number) is an example of the property of addition.
2) The fact that 0 + any number = that particular number is an
example of the property of addition.
3) The fact that a first number + a second number = a second
number + a first number is an example of the property of
addition.
4) Use the numbers 15 and 8 to illustrate the commutative
property of addition.
5) Use the numbers 6, 5, and 11 to illustrate the associative
property of addition.

41.  end 
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