Lesson #1 Integers and Symbols
The following table lists words and symbols that are commonly used in Arithmetic and Algebra. Arithmetic is the mathematics of numbers and operations. Algebra is the mathematics of applying arithmetic rules in situations where the numbers may be unknown (see Lesson 2).
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Keywords |
Description |
Examples |
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Counting Numbers |
natural numbers for counting |
1, 2, 3, 4, ... |
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Whole Numbers |
Counting numbers and 0 |
0, 1, 2, 3, ... |
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Integers |
Whole numbers with signs |
... -2, -1, 0, 1, 2, ... |
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Positive number |
number counted in favor |
+8, or just 8 (income) |
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Negative number |
number counted against |
-8 (expense, debt) |
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Signed number |
any positive or negative number |
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Parentheses |
Parentheses hold a single number value. Parentheses can also indicate priority: do any operations inside parentheses first. |
( ) or [ ] or { } |
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Operations |
instructions for computing with two or more numbers |
+, - , . , / |
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+ addition x + y |
an operation between numbers to find the total combined value |
6 + 3 = 9 Receive $6 and $3 |
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- subtraction x - y |
an operation to find the distance or difference between two numbers (order matters) |
8 - 5 = 3 You receive $8, but owe $3 |
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. multiplicationx× y |
an operation to find the total number in x groups of y items |
4.3 = 12 You need 4 notebooks, which cost $3 each |
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/ division (also fractions) x/y or x ¸ y |
an operation to find the number of items in each group if x items are split into y groups(order matters) |
12/2 = 6 You paid $12 for 2 lunches, or $6 each |
Every real number can be written with just one sign (+ or -). Zero (0) is an exception to this: it is really neither + nor -, but can be considered as either. We will prefer to write 0 with no sign, as if it were +0.
In all work on arithmetic skills, the most important thing is to know what operation you are working with. You should first and foremost look for the operation. The numbers tell you what to use, but the operation tells you what to do with the numbers.
For any of the operations, you can first reduce each number to a single sign:
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++ means + (two incomes are good) |
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+- means - (adding expense is not good) |
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-+ means - (subtracting income is not good) |
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-- means + (subtracting expense is good) |
In general, any even number of negatives will end up positive, and any odd number of negatives will end up negative. Think of positive numbers as "incomes" and negative numbers as "expenses".
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Addition: |
Add numbers with the same sign, and keep that sign. |
5 + 3 = 8 -2 + (-4) = -6 |
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Subtraction: |
Subtract numbers with different signs. Keep the sign from the larger quantity. |
5 - 3 = 2 4 - 7 = -3 |
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Multiplication: |
Multiply all numbers and reduce to one sign. |
(-4)(-3)(-2) = ---24 = -24 |
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Division: |
Divide and reduce to one sign. |
-16/(-2) = --8 = +8 |
Note that the same numbers can be used in different problems. The operation (+, -, ., or /) provides the instruction to perform. In all cases, pay more attention to the operation between numbers than the numbers themselves.
Exercises
Write each number with a single + or - sign.
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1) -2 |
2) 9 |
3) -(-5) |
4) -(+4) |
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5) +(-14) |
6) +(+6) |
7) -(3) |
8) -(-7) |
Write each sentence as an arithmetic statement and then provide the answer. Think carefully about your choice of operation to use.
9) The temperature was 8 degrees, but dropped 15 degrees.
10) David gave me $7 and Jon gave me $12.
11) David gave me $10 and I spent $6.
12) I bought 6 books that each cost $5.
Add or subtract as necessary. Be sure to indicate whether the answer is + or -.
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13) 2 + 4 |
14) -6 + 9 |
15) 7 + 3 |
16) 5 - 2 |
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17) 7 - 9 |
18) -5 - 2 |
19) -8 + 3 |
20) 12 + 17 |
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21) -8 - 10 |
22) -3 + 4 |
23) 9 - 9 |
24) 6 - 5 |
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25) -11 + 11 |
26) - 15 - 4 |
27) 7 - 3 |
28) -4 - 5 |
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29) 6 - (-4) |
30) 8 + (-2) |
31) -1 + (-3) |
32) 15 - (-2) |
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33) -9 - (-3) |
34) -13 + (-6) |
35) 4 - (-3) |
36) -1 - 1 |
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37) 5 - (-2) - 4 |
38) -3 + 4 + 10 |
39) -(-6) - 2 |
40) -7 - (-1) + (-2) |
Multiply or divide as necessary. Be sure to indicate whether the answer is + or -.
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41) 3.4 |
42) 2.5 |
43) 5.2 |
44) -2.6 |
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45) -10(2) |
46) 4(-4) |
47) 8/2 |
48) 20/5 |
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49) -16/8 |
50) -3(-1) |
51) -40/5 |
52) (-2)(-9) |
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53) -3(2)(-2) |
54) (-4)(-5)(2) |
55) (-1)(-6)(-3) |
56) 2(7)(-2) |
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57) -32/(-4) |
58) (-20)/(-10) |
59) -42/6 |
60) 2(-8)/4 |
Mixed problems. Perform the stated operations in each case.
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61) (-2)(-4) |
62) -7 - 8 |
63) 4 + (-3) |
64) +3(4) |
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65) 21/3 |
66) 14 - (-2) |
67) 6(-5) |
68) 8 + (-9) |
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69) 4 - 4 |
70) -3(3) |
71) -8/8 |
72) 16 - 5 |
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73) 3 + 8 |
74) 21 - 17 |
75) 4 - 12 |
76) -(-3)(-2) |
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77) 0/14 |
78) -2(0) |
79) 14 - (-3) |
80) (-2)(4)(-3) |
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81) -3 + (-4) - (-6) |
82) -2 - (-2) |
83) 1(18) |
84) 3 - 9 |
If more than one operation appears in a single problem, we perform the operations according to the following order:
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1 Do anything inside parentheses first. 2 Carry out all multiplications and divisions (left to right) 3 Finish with additions and subtractions (left to right) |
This order of operations is essential! Any other system will produce incorrect results. Start every problem by looking for operations within parentheses. Double-check all of your work for correct signs and operations.
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85) 3(8 - 5) |
86) (2 + 3)(1 - 4) |
87) 2(4) - 7 |
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88) 6 + (7 - 2) |
89) 2 + 3(-4) |
90) 9 + (-2)(-3) |
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91) 5 - (4 - 7) |
92) 2 - 4(5) + 1 |
93) 6(2) - (-3) |
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94) 8(-3)/4 |
95) 2(2)(5)/10 + 6 |
96) 3 - 2(5) + (-2) |
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97) 2(3) - 4(5) |
98) -(5 - 3) + (7 - 10) |
99) (3 - 7)(2 - 5) |
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100) 2(6 + 3) |
101) (5 + 13)/(4 - 6) |
102) 3(-1) + 2(-3) - 4(-5) |
Check Lesson #1 Answers.
Go on to Lesson #2 Variable Expressions.