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Adults.
Algebra for Adults:
Part 2, Review, Addition and Subtraction:
December 2, 2002:
We define counting as the simple process of adding one (1)
to a number to get the next number in the sequence.
For example 3 + 1 = 4.
We define the set of counting numbers as the numbers {1,2,3,4,5,...}.
The three dots mean, and so on.
We define 0 (zero) as the condition of not having any of something.
The set of counting digits are {0, 1, 2, 3, 4, 5, 6 ,7, 8, 9}. This is a finite or limit set having ten members.
In the decimal (base 10) number system the digits have weights
based on their column position.
For example, here is a simple number. 37542.
The 7 in this number is really equal to 7000.
The 2 is equal to 2.
The 5 is equal to 500.
We expand numbers by separating them into their equivalent
weights or column values.
For example:
532 = 500 + 30 + 2
4050 = 4000 + 50
2476 = 2000 + 400 + 70 + 6
We define a set as a group of things with common or similar
characteristics like a set of china or a set of encyclopedias.
We discovered that there can be sets within sets and sets that
overlap to some degree or other.
We notice that there can be finite sets and infinite sets.
For example, the set of 10 counting digits is finite because It
has a limited and definable number of members.
The set of counting numbers is infinite because it has no end
or limit.
We define the + sign as being the operator for addition and the = sign as being the sign for equality of elements.
In addition, we define the addends and the sum such that addend
+ addend = sum.
Addends are the numbers we add. Sum is the result.
Thus, in addition we are adding one or more numbers together
to produce a sum.
For example:
3 + 6 = 9
7 + 7 = 14
6 + 3 = 9
8 + 9 = 17
Notice that changing the order of the addends does not change
the sum.
Adding 0 to any number results in the same number.
The order in which we add numbers does not affect the sum.
We can count by numbers other that 1.
For example we can count by twos by adding 2 to a number to get
the next number in the sequence.
2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
1, 3, 5, 7, 9, 11, 13, 15.
We can use an Addition Table to solve simple addition problems.
The sum of the two addends is the number at the intersect of the
addend row and the addend column.
For example 4 + 5 = 9.
We define the set of all even numbers as 0, 2, 4, 6, 8, and all the numbers who's last digit is one of these.
The set of all odd numbers is defined as 1, 3, 5, 7, 9, and all the numbers who's last digit is one of these.
We can use the expansion method to find sums of larger numbers.
For example:
235 = 200 + 30 + 5, or 2 hundreds + 3 tens + 5 ones.
463 = 400 + 60 + 3, or 4 hundreds + 6 tens + 3 ones.
The sum is 6 hundreds + 9 tens + 8 ones or, 600 + 90 + 8 = 698.
We always add our columns from right to left to allow for a carry situation. This occurs when we get a sum of more than 9 in any column. In that case, we record the lowest digit part in the column being added and we carry the rest of the sum into the next column where it becomes another addend.
For example:
Review of Subtraction:
We find the basic rule of subtraction to be minuend - subtrahend
= difference.
Subtrahend is the number we take away. Minuend is the number we
take it away from. Difference is what's left.
We related that to addition, such that, Minuend = Subtrahend
+ Difference.
We see that the minuend is the same as the sum in addition and
the subtrahend and difference are the same as the addends in addition.
We can use the addition table to see that subtraction is implied
in addition.
So subtraction is a way of checking our addition results, because
it is nothing more than addition in reverse.
We defined the - (minus) sign as being the operator for subtraction.
For example:
11 - 2 = 9
17 - 16 = 1
12 - 3 = 9
9 - 5 = 4
We can use the expansion method to find the difference of larger
numbers.
For example:
754 = 700 + 50 + 4, or 7 hundreds + 5 tens + 4 ones.
322 = 300 + 20 + 2, or 3 hundreds + 2 tens + 2 ones.
The difference is 4 hundreds + 3 tens + 2 ones = 400 + 30 + 2
= 432.
We subtract larger numbers in order from right to left. This
allows us to deal with the case where our subtrahend in any column
is larger than our minuend. In this case, we must borrow from
the next higher, leftmost column.
For example:
It is always possible to check results using addition.
For example:
Previously, We defined the set of counting numbers as the numbers
{1,2,3,4,5,...}. The three dots mean, and so on.
We can also call this the set of all positive integers.
We define the set of negative numbers or negative integers {-1,
-2, -3, -4, ...}.
A number line serves to make this concept clear.
Knowing this we can subtract larger numbers from smaller numbers.
Regardless of the size of the numbers we are really just finding
the difference.
When we subtract a larger number from a smaller number, the result
is a negative number.
Back to Algebra for Adults.
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