Download Whole Numbers

− 1 , etc ..
π , 2 , 3 , etc..
SECTION 1.1
A set is a collection of objects.
The set of natural numbers is {1,2,3,4,5,….}
The set of whole numbers is {0,1,2,3,4,5,…}
Whole numbers are used for counting objects (such as money, but
not cents!) However, they do not include fractions or decimals.
The digits in a whole number have place value.
Place
Value
3 4 5
ThreeDigit
Groups
(separated
by commas)
5 7 6
4 0 2
8 9 7
4 1 5
Ones group
Thousands group
Millions group
Billions group
Trillions group
Verbal Form:
30,542 = Thirty thousand, five hundred forty-two.
(Notice we don’t use the word “and”.)
Standard Notation:
Uses only digits (0 through 9) and commas to state the number.
16 million = 16,000,000
Expanded Form:
States the standard form of each place value.
31,567
3 is the place value for ten thousands so this represents 30,000
1 is the place value for thousands, so this represents
1,000
5 is the place value for hundreds, so this represents
500
6 is the place value for tens, so this represents
60
7 is the place value for ones, so this represents 7 ones or
7
31, 567 in Exanded Form is 30,000 + 1,0000 + 500 + 60 + 7
Rounding Whole Numbers:
Step 1: Locate the “rounding digit”, which is the digit at the place value you are rounding to.
Step 2: Look at the digit directly to the right of the rounding digit. This is the test digit. If the test
digit is < 5 (less than 5), keep the rounding digit the same and change all digits to the right of it
to 0.
If the test digit is ≥5 (greater than or equal to 5), then increase the rounding digit by 1 and
change all the digits to the right of it to 0.
On a number line, numbers are written so that ascending numbers are to the right. 10 is to the left
of 20 on the number line, so 20 is greater than 10 (20 > 10). It can also be stated that
descending numbers are to the left, so 10 is less than 20 (10 < 20)
0
10
20
30
40
50
48
Ask "is 48 closer to 40 or 50?" It is closer to 50, it is about 50.
0
10
20
30
40
50
1888
Ask "is 81 is closer to 80 or 90?" It is closer to 80. It is about 80.
Round off the same way for larger numbers.
0
1000
2000
Is 1888 closer to 1000 or 2000? 1888 is closer to 2000.
Is 43,556 closer to 40,000, or 50,000? Compare the first two numbers.
43 is closer to 40 than 50. It is 40,000.
"Tricky" numbers are those with 5 as the number that decides. Is 650 closer to 600 or 700? On a
number line 650 is half way. Math has a rule for this. When the digit is 5 or greater, round up. When the
digit is less than 5, round down.
Example: Round 24 to the nearest ten. Round 36 to the nearest ten
24
35 Rounds up
Rounds down
0
10
20
30
40
50
You may need to estimate to a certain place. Look at the number in the place to the right. Then round.
Round 684 to the tens place.
The number 4 in the ones place to the right of the tens. It rounds down to 680.
Round 6423:
to the nearest ten is 6420.
to the nearest hundred is 6400.
to the nearest thousand, it is 6000.
Round 589,457:
to the nearest ten, it is ________.
to the nearest hundred, it is _______.
to the nearest thousand, it is ______.
to the nearest ten thousand is _______.
to the nearest hundred thousand is _______.
You can round it to the nearest million, it is 1,000,000.
Adding and Subtracting Whole Numbers
Addition Terms: The numbers being added are called addends.
Subtraction Terms: The number you are taking away is the subtrahend, the number you are subtracting
from is the minuend, and the answer is the difference.
Mathematical properties are often used to simplify computation.
Below are three addition properties stated in words, shown with a numeric example,
and shown with an algebraic example.
The Zero Property of Addition is also called the Identity Property of Addition.
Associative Property of Addition
When numbers or variables are added, for example
(2 + 3) + 4 = 2 + (3 + 4) and (a + b) + c = a + (b + c)
The addends can be grouped in different ways without changing the result.
Commutative Property of Addition
When numbers or variables are added, for example 2 + 3 = 3 + 2 and a + b = b + a,
The order of the addends can be changed without changing the result.
Zero Property of Addition
When 0 is added to a number or variable, for example, 2 + 0 = 2 and a + 0 = a,
the result is the same number or variable.
These properties can be used when adding numbers in your head.
Example:
337 + 18 = (300 + 30 + 7) + (10 + 8) = 300 + (30 + 10) + (7+ 8) = 300 + 40 + 15 = 355
Estimating Sums and Differences
When an exact answer is not necessary, an estimate can be used. The most common method
of estimating sums and differences is called “front-end rounding”, which is to round each number
to its largest place value, so that all but the first digit of the number is 0.
Examples:
Estimate 4,894 + 429
Round 4,894 to the nearest thousand. 4,894 → 5,000
Round 429 to the nearest hundred 429 → 400
Add the rounded numbers. 5,000 + 400 = 5,400
The actual answer is 4,894 + 429 = 5353, which is close to 5,400.
If both addends are rounded up, the estimated sum will be greater than the actual sum,
and if both addends are rounded down, the estimated sum will be less than the actual sum.
Such generalizations are not possible with subtraction.
Estimate 6,209 − 383.
What is the largest place value of 6,209? Round to the nearest _________. 6,209 ->______
What is the largest place value of 383? Round to the nearest _________. 383 -> ________
Subtract the rounded numbers. _____ - ________ = __________
The actual answer is 6,209 – 383 = 5826
How close was your estimate?
Pictograph
A pictograph uses an icon to represent a
quantity of data values in order to decrease the
size of the graph. A key must be used to explain
the icon.
Advantages Easy to read
Visually appealing
Handles large data sets easily using keyed icons
Disadvantages Hard to quantify partial icons
Icons must be of consistent size
Best for only 2-6 categories
Very simplistic
Bar graph
A bar graph displays discrete data in separate
columns. A double bar graph can be used to
compare two data sets. Categories are
considered unordered and can be rearranged
alphabetically, by size, etc.
Advantages Visually strong
Can easily compare two or three data sets
Disadvantages Graph categories can be
reordered to emphasize certain effects
Use only with discrete data
1999 Quarterly Financial Data for NIKE
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
Year
164
69
124
95
200
150
100
50
Line graph
A line graph plots continuous data as points and
then joins them with a line. Multiple data sets
can be graphed together, but a key must be
used.
Advantages Can compare multiple continuous
data sets easily
Interim data can be inferred from graph line
Disadvantages Use only with continuous data
0
1st Qtr
2nd Qtr
3rd Qtr
4th Qtr
Example of “Real World” graphs
The following graph has two sets of data overlaying each
other.
One is a scattered diagram with connected dots. The other is a
bar graph. They both share the same x-values, which in this
case are levels of education. The y-values for the bar graph
(Median Weekly Earnings) are on the left, since earnings is
represented by dollars. The y-values for the scattered diagram
(Unemployment Rates) are on the right, since unemployment
rates are represented as percentages.
A sample data point would as follows:
People with an Associate Degree have a median weekly
earnings of about $700 and about a 3% unemployment rate.