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1 Basic Math Review
Numbers
NATURAL NUMBERS
{1, 2, 3, 4, 5, …}
WHOLE NUMBERS
{0, 1, 2, 3, 4, …}
INTEGERS
{…, 3, 2, 1, 0, 1, 2, …}
RATIONAL NUMBERS
All numbers that can be written in the form , where a
and b are integers and .
IRRATIONAL NUMBERS
Real numbers that cannot be written as the quotient of two
integers but can be represented on the number line.
REAL NUMBERS
Include all numbers that can be represented on the number
line, that is, all rational and irrational numbers.
PRIME NUMBERS
A prime number is a number greater than 1 that has only
itself and 1 as factors.
Some examples:
2, 3, and 7 are prime numbers.
COMPOSITE NUMBERS
A composite number is a number that is not prime. For
example, 8 is a composite number since
8 = 2 . # 2 # 2 = 23
Rational Numbers
Real Numbers
23, 22.4, 21 , 0, 0.6, 1, etc. 4_ 2 5
25VN
Irrational
Numbers
Integers p 23, 22, 21, 0, 1, 2, 3, p
Whole Numbers 0, 1, 2, 3, p
Natural Numbers 1, 2, 3, p
3,
VN2, p, etc.
b Z 0
a>b
–5 –4 –3
Negative integers Positive integers
The Number Line
Zero
–2 –1 0 1 2 3 4 5
ISBN-13:
ISBN-10:
978-0-321-39476-7
0-321-39476-3
9 780321 394767
90000
Int Properties
Identity Property of Zero:
Inverse Property:
Commutative Property:
Associative Property:
PROPERTIES OF MULTIPLICATION
Property of Zero:
Identity Property of One: , when .
Inverse Property: , when .
Commutative Property:
Associative Property:
PROPERTIES OF DIVISION
Property of Zero: , when .
Property of One: , when .
Identity Property of One:
Absolute Value
The absolute value of a number is always ≥ 0.
If , .
If , .
For example, and . In each case, the
ƒ -5 ƒ = 5 ƒ 5 ƒ = 5
a 6 0 ƒ a ƒ = a
a 7 0 ƒ a ƒ = a
a
1 = a # 1
a Z 0 a
a = 1
a Z 0
0
a = 0
a # 1b # c2 = 1a # b2 # c
a # b = b # a
a # a Z 0
1
a = 1
a # 1 = a a Z 0
a # 0 = 0
a + 1b + c2 = 1a + b2 + c
a + b = b + a
a + 1-a2 = 0
a + 0 = a
Key Words
Text
More Text
5. Equivalent Fractions Equivalence is a concept that is easy to understand when a fraction wall is used. As you can see, each row has been split into different fractions: top row into 2 halves, bottom row 12 twelfths. An equivalent fraction splits the row at the same place. Therefore: 1 2 = 2 4 = 3 6 = 4 8 = 5 10 = 6 12 The more pieces I split the row into (denominator), the more pieces I will need (numerator). Mathematically, whatever I do to the numerator (multiply or divide), I must also do to the denominator and vice versa, whatever I do to the denominator I must do to the numerator. Take 2 3 as an example. If I multiply the numerator by 4, then I must multiply the denominator by 4 to create an equivalent fraction: 2 × 4 3 × 4 = 8 12 Example problems: Use what you have understood about equivalent fractions to find the missing values in these fraction pairs. 1. 3 5 = 20 Answer: The denominator was multiplied by 4. (20 ÷5 =4) So the numerator must by multiplied by 4. ∴ 3×4 5×4 = 12 20 2. 27 81 = 9 Answer: The numerator was divided by 3. (27 ÷9 =3) So the denominator must be divided by 3. ∴ 27÷3 81÷3 = 9 27 5. Your Turn: a) 2 3 = 9 b) 5 7 = 45 c) 9 10 = 30 d) 52 = 4 13 Return to Your Turn Activity 4. e) What fraction of the large square has dots? f) What fraction of the large square has horizontal lines? Watch this short Khan Academy video for further explanation: “Equivalent fractions” https://www.khanacademy.org/math/arithmetic/fractions/Equivalent_fractions/v/equivalent-fractions Page 9 of 40 6. Converting Mixed Numbers to Improper Fractions A mixed number is a way of expressing quantities greater than 1. A mixed number represents the number of wholes and remaining parts of a whole that you have, while an improper fraction represents how many parts you have. The diagram below illustrates the difference between a mixed number and an improper fraction, using a quantity of car oil as an example. On the left, we use a mixed number to represent 3 whole litres and 1 half litre. We write this mixed number as 3 ½. On the right, we use an improper fraction to represent 7 half litres. We write this improper fraction as 7 2 . Is the same as 3 1 2 = 7 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 You are more likely to encounter mixed numbers than improper fractions in everyday language. For example, you are more likely to say, ‘my car requires 3 ½ litres of oil,’ rather than, ‘my car requires 7 2 litres of oil.’ It is much easier to multiply or divide fractions when they are in improper form. As such, mixed numbers are usually converted to improper fractions before they are used in calculations. To convert from a mixed number to an improper fraction, multiply the whole number by the denominator then add the numerator. This total then becomes the new numerator which is placed over the original denominator. For example: Convert 3 1 2 into an improper fraction. working: 3(𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) × 2(𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) + 1(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛) = 7 Therefore,𝑡𝑡ℎ𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖𝑠𝑠 7 2 Example problems: 1. 2 2 3 = 8 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 3 + 2 = 8) 2. 2 3 7 = 17 7 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (2 × 7 + 3 = 17) 6. Your Turn: Convert these mixed numbers to improper fractions. a) 4 1 2 = c) 7 3 5 = b) 5 1 3 = d) 2 1 8 = Page 10 of 40 7. Converting Improper Fractions to Mixed Numbers While improper fractions are good for calculations, they are rarely used in everyday situations. For example, people do not wear a size 23 2 shoe; instead they wear a size 11 1 2 shoe. = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 7 2 To convert to an improper fraction we need to work out how many whole numbers we have. Here we reverse the procedure from the previous section. We can see that 6 of the halves combine to form 3 wholes; with a half left over. = 31 2 So to work this symbolically as a mathematical calculation we simply divide the numerator by the denominator. Whatever the remainder is becomes the new numerator. Using a worked example of the diagram above: Convert 7 2 7 ÷ 2 = 3 1 2 If I have three whole numbers, then I also have six halves and we have one half remaining. ∴ 7 2 = 3 1 2 That was an easy one. Another example: Convert 17 5 into a 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 . working: 17 ÷ 5 = the whole number is 3 with some remaining. If I have 3 whole numbers that is 15 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠. (3 × 5) I must now have 2 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓ℎ𝑠𝑠 remaining. (17 − 15) Therefore, I have 3 2 5 Example problems: Convert the improper fractions to mixed numbers: 1. 27 6 = 4 3 6 = 4 1 2 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (27 ÷ 6 = 4.5) 𝑠𝑠𝑠𝑠 (4 × 6 = 24) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (27 − 24 = 3) remaining. Don’t forget equivalent fractions. 2. 8 3 = 2 2 3 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁: (8 ÷ 3 = 2.67) 𝑠𝑠𝑠𝑠 (2 × 3 = 6) 𝑤𝑤𝑤𝑤𝑤𝑤ℎ (8 − 6 = 2) remaining. 7. Your Turn: Convert the following improper fractions to mixed numbers: a) 7 5 = c) 53 9 = b) 12 9 = d) 27 7 = Watch this short Khan Academy video for further explanation: “Mixed numbers and improper fractions” (converting both ways) https://www.khanacademy.org/math/cc-fourth-grade-math/imp-fractions-2/imp-mixed-numbers/v/changing-a-mixed-number-to-animproper-fraction Page 11 of 40 8. Converting Decimals into fractions Decimals are an almost universal method of displaying data, particularly given that it is easier to enter decimals, rather than fractions, into computers. But fractions can be more accurate. For example, 1 3 is not 0.33 it is 0.33̇ The method used to convert decimals into fractions is based on the notion of place value. The place value of the last digit in the decimal determines the denominator: tenths, hundredths, thousandths, and so on… Example problems: 1. 0.5 has 5 in the tenths column. Therefore, 0.5 is 5 10 = 1 2 (simplified to an equivalent fraction). 2. 0.375 has the 5 in the thousandth column. Therefore, 0.375 is 375 1000 = 3 8 3. 1.25 has 5 in the hundredths column and you have 1 25 100 = 1 1 4 The hardest part is converting to the lowest equivalent fraction. If you have a scientific calculator, you can use the fraction button. This button looks different on different calculators so read your manual if unsure. If we take 375 1000 from example 2 above: Enter 375 then followed by 1000 press = and answer shows as 3 8 . NOTE: The calculator does not work for rounded decimals; especially thirds. For example, 0.333 ≈ 1 3 The table below lists some commonly encountered fractions expressed in their decimal form: Decimal Fraction 0.125 1 8 0.25 1 4 0.33333 1 3 0.375 3 8 