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Lesson 2 Number, Part II
Rules and Definitions
Rules
No new rules for Lesson 2
Definitions
- square root: If x is greater than 0, then
( )
x is the unique positive real number 2
such that x = x .
- digit: Any of the Hindu-Arabic numerals 1 through 9, and 0.
- counting (natural) numbers: Numbers used to count objects. Ex. 1, 2, 3, .....
- whole numbers: Counting numbers and the number 0.
- integers: Whole numbers, and all negative numbers.
- real numbers: Any number used to describe a positive or negative number. Includes all integers, and all decimal numbers and fractions.
2
3
1.2234
− 17
43.11
3
1
8
- imaginary numbers: The result of taking the square root of a negative number. Normally, the −1 is factored out and exchanged for the symbol “i”. i2 = -1.
2A
Origin of modern numerals and arithmetic symbols
Hindu-Arabic digits: Hindu-Arabic numerals arrived in western Europe at least by the 12th century. They include the digits 0,1,2,3,4,5,6,7,8,9.
Roman numerals: Developed by Romans in the centuries before Christ (B.C.), they are still used occasionally to mark dates, page numbers, and hours on a
clock face. The symbols I, X, C, M stand for 1, 10, 100, and 1000. The symbols
V, L, D stand for 5, 50, and 500. Normally, Roman numerals are added. For
example, CLXII = 100 + 50 + 10 + 2 = 162. If a smaller numeral precedes a larger
one, then subtract. For example, IV = 5-1 = 4. Likewise, IX = 9, XC=90, IC=99. For
example, DXC = 500 + 90 = 590.
Base 10, 2, etc. systems: The “base” of a system relates to its place value. For example, the base 10 system (also called the decimal numeral
system) works in multiples of 10. As you can see in the example below, moving from right to left, each place is 10 times more than the place to its right.
_______ _______ _______ _______ . _______ _______
103=1000 102=100
101=10
100=1
10-1=1/10 10-2=1/100
Use of the base 10 system dates back to at least 3000 B.C. , with Egyptians using a heiroglyphic grouping system. A quick glance at a normal
human’s hands suggests that counting by 10’s was probably standard practice in the pre-Flood world, too. Positional base systems have a number of digits
equal to the base value. For example, the base 10 system has 10 digits, 0
through 9. The base 2 system uses 2 digits, 0 and 1. The base 2 system is
important in computer applications, where the entire system is run by
switching electric current on (1) or off (0) at different times and locations.
A positional base 2 example is formed simply by switching out the 10’s in the
base 10 example with 2’s:
_______ _______ _______ _______ . _______ _______
23=8
22=4
21=2
20=1
2-1=1/2
2-2=1/4
Example 2.1 Convert the binary number 1011 to base 10.
solution: 1(8) + 0(4) + 1(2) + 1(1) = 8 + 2 + 1 = 11.
An example of the binary system in use on a
computer is with the color quality for a computer
monitor. For example, the “16 bit” quality, allows for
216 possible colors to be displayed. You will learn
more about “bits” in Lesson 25.
Arithmetic symbols: Our most common modern arithmetic symbols include “+” for addition, “-” for subtraction, “×” or “·” or “()”for multiplication, and “÷”
for division. Most of these symbols were first used in arithmetic and algebra in
the 15th-17th centuries. “+” and “-” are also used to describe positive and
negative numbers, like +3 and -3.
It is important to know that math is considered the “language of science”. It is a tool used to study God’s creation. And like every other language, it has its
own symbols. Sometimes, the symbols mean just one word, like “+” means “plus”, for example. However, some symbols have a meaning that is a phrase or
even a sentence.
As you learn algebra, you will learn a lot of new symbols. It will be confusing
at first, because you will forget what the new symbols mean. But be patient
and persevere! Just like if you were learning a new language, youwill naturally start memorzing what the symbols mean by using them repeatedly.
You will also see that using the symbols is faster than writing out the words or phrases corresponding to each symbol. If you don’t believe me, then
get a piece of paper and write “29 + 4 = 33”. Now, write the words that express this statement of equality: “Twenty-nine plus four equals thirty-three.” Aren’t
you glad you aren’t like the Greeks, who did not use a symbolic format for mathematics?
2B
Types of numbers, number lines
Example 2.2. True or False. All real numbers are also integers.
solution: False. Real numbers include decimal numbers, and integers don’t. It is true however, that all integers are real numbers.
Refer to the Lesson 2 definitions for different types of numbers. Be prepared to answer questions like the following:
Example 2.3. True or False. All counting numbers are whole numbers.
solution: True. Counting numbers are members of the set of numbers called whole numbers.
Real numbers include all positive and negative numbers, with and
without decimal places. This includes all numbers we can write as ratios
of integers (rational numbers), along with all numbers that cannot be written that way (irrational numbers). Irrational numbers are unique because they have no recognizable repeating pattern of decimals, no matter how many decimal places you try to show. For example, the irrational number π, pronounced “pie”, rounded to 2 decimal places, equals 3.14. Rounded to 20
decimal places, it equals 3.14159265358979323846. People have attempted to
measure pi out to quadrillions of decimal places, so as far as we know, it has an
infinite and non-repeating decimal pattern!
You may also be thinking, “if there are real numbers, are there imaginary
numbers too?” Well, actually, there are! They are also called complex numbers, and we’ll discuss them later. First, let’s continue with real numbers and discuss
ways to organize them.
Number lines are a useful tool for understanding signed numbers. A number line is just a horizontal line, with or without arrowheads at the ends, divided into equal sections, like you see on a ruler. The number line is usually drawn
with zero at the center, positive values to the right and negative values to the left.
One way to think of negative numbers is that they are the opposite of a posi-
tive number. For example, -2 is the opposite of 2. If you look at the number line, it is easy to see that they are both 2 “tick marks” from 0, you just move in
opposite directions from 0 to reach each value.
Example 2.4 Organize the following numbers by graphing them on a number line:
3
4.1,2,6 ,−2
4
solution: Place a dot on the number line to identify the location of each number. This procedure is called “graphing”. Some points are more difficult to locate exactly, like 4.1. Just do your best to estimate its location.
Squares and square roots - The Greeks did much to develop mathematics in
the centuries before Christ, but their cyclical view of time and their timidity towards accepting difficult ideas like infinity helped stifle progress in math and science until Christians rejected their ideas some 2,000 years later. Because
God is eternal, Christians have no problem with the reality of infinity. Many
scholars believe the Greek’s inability to accept irrational numbers (Lesson 3) is what led them to focus more on geometry than arithmetic. For example,
consider a square with sides 1 unit long. According to the Pythagorean
theorem (Lesson 8), the diagonal of this square will have a value equal to 2 . Like π, 2 is another irrational number. There is simply no way to write
it as a rational number. However, it was easy
enough for the Greeks to draw the diagonal
representing 2 , while covering up the fact that its
exact value could not be found! For the Greeks,
religious worship of human reason caused them to
2
ignore certain aspects of reality that defy human
1
intellect.
Squares and square roots will be covered more in
1
Lessons 31 and 34, but for now, work on memorizing
A square with sides 1 unit long
the following table.
has a diagonal equal to 2 .
2
n
n
1
1
2
4
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
11
121
12
144
13
169
Example 2.5 Find the value of the following numbers:
36
9
121
solution: Using the table to the left, think “what times
what equals 36?”
6x6 equals 36, so the square root of 36 = 6. Likewise:
9=3
121 = 11
Imaginary numbers: Review the definitions listed at the beginning of this
2
lesson for square roots and complex numbers. If x = x , then ( −1) = −1 ,
and since −1 = i, then i2 = -1. The symbol “i” is used in place of −1 , because the mathematician Leonhard Euler found it easier and faster to complete
lengthy handwritten calculations this way. Leonhard Euler, a Christian man, is often considered the greatest mathematician ever, and many of the symbols he
developed are still used today.
( )
2
Think about Example 1.5 above. To solve it, we thought “what times what equals 36?” But how do you simplify −1 ? “what times what” equals -1? If you
said “1 times -1”, that is incorrect, because both factors have to be the
same number! The truth is, there is no real number that you can multiply by itself to get -1. That is why −1 is called an imaginary number.
Imaginary numbers may arise in science and engineering courses where
negative values may be measured, like in a system with alternating, or “AC” electrical current. Sometimes, these systems have complex numbers (Lesson
95), which have both a real and imaginary part. For now, just know that
“imaginary” is just a word to describe these numbers, that they are a little weird
because you can’t plot them on a number line, and that you can factor
them out. For example, if you had −36 , factor out the −1 like this:
−1
Next, replace the
−1
36
with i, and the
36
with 6, to get 6i.
Example 2.6 Simplify the following:
solution: For the first two, factor out −1 first:
−19 = −1 19 = 19i
−16 = −1 16 = 4i
−19
−16
5i 2
For 5i2, remember that i2 = -1, which simplifies the problem to
5 (-1) = -5.
2C Arithmetic Review
Arithmetic basics: We discussed place value in 2A. In basic arithmetic, it helps to consider place value in each of the four operations. Also, be
thinking of “fact families” and how addition “undoes” subtraction, multiplication “undoes” division, and vice versa.
Addition. Think about how Jesus, after the Resurrection, led the disciples to
a place where they caught exactly 153 fish (John 21:11). Considering place
value, we would say the 1 is in the 100s place, the 5 in the tens place, and the 3
in the ones place. In other words, 153 is composed of 1 “hundred”, 5 “tens”, and
3 “ones”. Using addition, we can write it as follows:
100 + 50 + 3 = 153, or
100
50
+ 3
153
In addition, each of the numbers added together is called an addend, and the result is called the sum. “+” means add, like 3+2, or a positive number, like +3.
357
51 b) -4 + 7 and c) $27.95 + $14.15
Example 2.7 Find the sum of a)
+ 16
solution: a) start by adding all the digits in the ones place together. In the example about 153 fish, the ones place only summed to 3, but with 1.7a, the sum is 14.
Since it is the ones column, you can only write a ones place down, so put the
4 down, and then “carry” the 1 to the top of the tens column. Now add the tens,
which should be 1 + 5 + 5 + 1 = 12. The 2 is actually in the tens place and is written
below the line, but the 1 is in the hundreds place, and must be “carried” to the
hundreds column. Lastly, add 1+3 = 4, and your sum should be 424.
357
51
+ 16
424
b) In this problem, you need to add a negative and positive integer. A number
line helps with solving these. Starting at 0, first move left 4 spaces, then right 7. You should end on +3. Another way to think about it is -4+7 =7 - 4. It’s easier to “see” that 7-4 equals 3, but not as easy to see that -4 + 7 = 3.
c) First, line up the two values, relative to their decimal points, and add, using
the same method you used in a).
27.95
+ 14.15
$42.10
Subtraction. How do you know if Example 2.7c is correct? The simplest way to check is to remember subtraction “undoes” addition. It’s good to remember too that, while in addition we often need to “carry” from one place value to the next higher, in subtraction we often need to “borrow” from a higher place value and give to a lower one.
Example 2.8 Check the result of Example 2.7c by subtracting $14.15 from $42.10. Does it equal $27.95?
42.10
- 14.15
$27.95
solution: In subtraction, you always need a larger value “on top” of a smaller
one. For example, in the first column, you cannot subtract 0-5, so you need to “borrow” 1 tenth from the tenths place to the left. Now you have 10-5 = 5. Moving to the tenths place, the top digit changed from 1 to 0, so now you have 0-1, which you can’t
do. You need to borrow 1 one from the ones place, and get 10-1 = 9. Now, in the ones
place you have 1-4, so you need to borrow 1 ten from the tens place to get 11-4=7.
Finally, subtract 1 from 3 to get 2, and you get $27.95. By “undoing” the addition
with subtraction, we were able to check our work and confirm 1.6c was worked correctly.
The result of subtraction is called the difference. The larger number, or minuend, is always written first, or above, the subtrahend. Example 2.9 Find the difference between 108 and 255. Check your answer by adding.
solution: Line them up, minuend above subtrahend, and subtract.
255
- 108 and check
147
147
+ 108
255
Multiplication. Let’s say you had 3 cartons of eggs, with one dozen (12) eggs in each carton. There are two ways to determine how many eggs you have. The first is to add 12+12+12 = 36. The other way is to multiply. There are three
groups of 12, so 3 x 12 = 36.
Also, note that 3·12, 3(12), and (3)(12) all mean “three times twelve”.
Multiplication is basically a fast way to add when you have equal
groups of things. The result of multiplication is called the product,
and the numbers that are multiplied together are called factors.
Example 2.10 Find the product of 24 and 315.
solution: For place-value purposes, the larger number is written above the smaller number. The 4 is multiplied by 315, paying careful attention to “carry” to the
appropriate place. For example, 4x5=20, but you can’t just write 20 down since there
are more places to multiply by. The 2 is “carried” to the tens place, and added on after
4 is multiplied by 1. When it is time to multiply the 2 by 315, a new row is started and
0 is written first to “hold” the place of the 2, which is in the tens place, not the ones place. The last step involves adding the two rows together. Using the multiplication procedure , or algorithm, we’ve broken the 24 into 20 and 4, and multiplied 315 by
each, so that 20(315) + 4(315) = 6300 + 1260 = 7560.
315
x 24
1260
+ 6300
7560
Example 2.11 Find the product of 1.3 and 22.4.
solution: For multiplication problems that include decimals, the simplest way to solve them is to ignore the decimals until the end. Then, count the number of total
decimal places, and add them back in. For this example, both factors have one
decimal place, so we will add back in 1+1 = 2 decimal places. Start at the right, move two places to the left, and add the decimal point:
224
x 13
672
+ 2240
2912
=29.12
Division. Multiplication can be “undone” by division. Like subtraction is used to check the results of addition (or vice versa), division is used to check the results of multiplication. For example, we can divide the result of Example 2.10 by 24,
and see if we get 315 for the answer:
315
24 7560
-72
36
-24
120
-120
0
The result of division is called the quotient. In the example above, we divided 7560 by 24. The first number is the dividend, and the second number is the
divisor. In the three operations covered so far, step-by-step procedures, or
algorithms, were used to find results. The example above used the
long-division algorithm. To use the algorithm, we think “how many times can the divisor ‘go into’ “ one or more places in the dividend. For example, 24
cannot go into 7, it is too big. But it can go into 75 at most 3 times, so 3 is the first place we write down in the quotient. Notice it is written directly above the dividend’s hundreds place (5). Next, we subtract 72 from 75 to get 3, and then
“bring down” the next place, which is 6. Now, we notice 24 goes into 36 one time, so we write a 1 next to the 3 in the quotient, subtract to get 12, and
bring down the next (and last) place, which is 0. 24 goes into 120 exactly 5 times, leaving us with a remainder of zero. Sometimes though, a problem has a non-zero remainder.
Example 2.12 Find the quotient of 450 ÷ 7.
64
64
7 450 R2
x
7
solution: Use long division to solve -42
and check
448
30
-28
448+2=450
2
The quotient is 64 R2, meaning 64 and a remainder of 2. We check the solution by
multiplying, which “undoes” the division. We multiply 64 x 7, then add 2, which gets us
back to 450.
2
. Alternatively, we can put a 7
decimal place after the 0 in 450, and solve in decimal form. Let’s solve it to 4
decimal places.
We can also write the answer as the mixed number, 64
64.2857
7 450.0000
-42
30
-28
20
-14
60
-56
40
-35
50
-49
1
This is a very long, long division problem, and we could keep
going more because the remainder is not zero! However, since we are only solving it to 4 decimal places, we will stop
with the remainder of 1 shown. We will cover division and
decimal answers more in Lesson 5.
Example 2.13 The farmer had $50.00 to spend on chickens. If the feed store sold chicks for $2.50 each, how many could the farmer purchase? Assume tax is included in the
price.
solution: When solving word problems, first ask yourself “What am I solving for?” In this problem, you need to find out how many chickens the farmer can purchase
for $50.00. This would be a division problem, because you need to divide the $50.00
into $2.50 portions for each chicken. Notice that both values are multiplied by
100 first, which removes the decimals and simplifies the long division process.
20
250 5000
-500
00
- 0
0
Answer: 20 chicks
Practice Set 2
A calculator is not necessary for any problems in Practice Set 2.
1. Natural numbers are also called ___________ numbers.
2. i2 = _____.
3. True or False. All integers are real numbers. ______________.
4. Convert the binary number 1111 to base 10
5. Convert LXIV to the Hindu-Arabic numeral system.
6. Write Roman numeral values for a) 35
7. Simplify:
a) 8i2
8. Add. -2 + 5 + 17
b) 45
c) 14
b) −14 9. Add. 4,704,219 10. Add. 4.0311 + 82.15
+ 387,565
11. Subtract. 411.501 - 82.333
12. Multiply. 7,421
x 830
13. The supplies required to make the cakes cost $20.45. If the cakes were sold for $39.95, how
much profit was made? Hint: Find the difference.
14. Find the product of 671.4 and 1.65.
15. Find the product of 16.2 and 0.006.
16. The family started a pecan orchard by tilling 10 rows and planting 17 trees in each row.
How many trees would they plant? If they ordered 2-3’ tall trees that cost $18.95 each, what
was the total cost for all the trees?
17. Divide. 7 252 18. Find the quotient of 491÷8.
19. Find the quotient of 297.5 ÷ 7.
20. The half-gallon container holds about 16 servings of ice cream. In a family of 7, how many
whole servings does each person get? How many servings are left over?