Operations on Integers
Addition
Like signs – add the absolute values, and prefix the negative sign if the addends are negative
Unlike signs – get the absolute values of the numbers and subtract the smaller absolute value from the
other and prefix the negative sign if the negative addend has the larger absolute value.
E.g. -4 + (-6) = -10
-2 + 7 = 7 - 2 = 5
3 ÷ (-9) = - (9 – 3) = -6
Subtraction – if the minuend and subtrahend are both positive and the minuend is greater than the subtrahend,
proceed to subtract forthwith. Otherwise, change the sign of the supposed subtrahend and
proceed as in addition.
E.g. 8 – 3 = 5
4 – 5 = 4 + (-5) = -1
Multiplication (Division) – to multiply (or divide) two integers with
Like signs – get the product (or quotient) of their absolute values
Unlike signs – get the negative of the product (or quotient) of their absolute values
E. g. -5 x -4 = 20
-3 x (2) = -6
12 / (-3) = -4
Real Numbers
o Rational – numbers that are in rational form
Fractions (n/m, where n & m are whole numbers)
Mixed (1 ½)
Proper (1/2)
Improper (3/2)
Decimals
Repeating
Terminating
Integers
Natural (counting numbers/positive whole numbers)
o Odd (2n + 1)
o Even (2n)
Negative whole numbers
Zero (0)
Prime numbers – a whole number whose divisor are 1 and itself only
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67
10 10 10 10 11 12 13 13 13 14 15 15 16
71 73 79 83 89 97
1 3 7 9 3 7 1 7 9 9 1 7 3
16 17 17 18 19 19 19 19 21 22 22 22 23 23 24 25 25 26 26
7 3 9 1 1 3 7 9 1 3 7 9 3 9 1 1 7 3 9
27 27 28 28 29 30 31 31 31 33 33 34 34 35 35 36 37 37 38
1 7 1 3 3 7 1 3 7 1 7 7 9 3 9 7 3 9 3
38 39 40 40 41 42 43 43 43 44 44 45 46 46 46 47 48 49 49
9 7 1 9 9 1 1 3 9 3 9 7 1 3 7 9 7 1 9
50 50 52 52 54 54 55 56 56 57 57 58 59 59 60 60 61 61 61
3 9 1 3 1 7 7 3 9 1 7 7 3 9 1 7 3 7 9
63 64 64 64 65 65 66 67 67 68 69 70 70 71 72 73 73 74 75
1 1 3 7 3 9 1 3 7 3 1 1 9 9 7 3 9 3 1
75 76 76 77 78 79 80 81 82 82 82 82 83 85 85 85 86 87 88
7 1 9 3 7 7 9 1 1 3 7 9 9 3 7 9 3 7 1
88 88 90 91 91 92 93 94 94 95 96 97 97 98 99 99
3 7 7 1 9 9 7 1 7 3 7 1 7 3 1 7
List of Prime Numbers up to 1000
Composite number – a whole number that has at least one other divisor or factor aside from 1 and itself.
Greatest Common Factor GCF – the GCF of two or more numbers is the product of all the common prime
factors of the numbers under considerations.
Least Common Multiple LCM – the LCM of two or more numbers is the smallest number that is divisible
by all given numbers. It is the product of all unique factors of the given numbers taken and the highest
power in which each occurs.
Divisibility laws - An integer is divisible a number(integer)if it can be divided exactly by the number, that is, the
remainder after diving is zero.
An integer is divisible by 2 if it ends with 0, 2, 4, 6, or 8, which means that the number is even. A
number that is not divisible by 2 is odd.
An integer is divisible by 3 if the sum of the digits is divisible by 3. Example: 21345
An integer is divisible by 4 if the last two digits form a number which is divisible by 4. Example: 11312
An integer is divisible by 5 if it ends with either 0 or 5. Example: 2000010, 345675
An integer is divisible by six if it is divisible by both 2 and 3. Example: 1233408
An integer is divisible by 7 if the number represented without its units’ digit minus twice the units digit
of the original integer, is divisible by 7. Example: 581 because 58 – 2(1) = 56 (56 is divisible by 7)
An integer is divisible by 8 it the last 3 digits from a number which divisible by 8. Example: 4572128
An integer is divisible by 9 if the sum of the digits is divisible by 9. Example 2312343
An integer is divisible by 10 if its last digit is 0.
Fractions
Meanings
a. Part of a whole or group
b. Indicated division
c. Ratio
Kinds of Fractions
As to relation between the numerator and the denominator
a. Proper – the numerator is less than the denominator. E.g. ¾
b. Improper – the numerator is equal to or greater than the denominator. E.g. 5/3
As to relation of the denominators of two or more fractions
a. Similar – the denominators are equal. E.g. 2/5 and 4/5
b. Dissimilar – the denominators are not equal. E.g. 3/7 and 4/9
Other classes
a. Equivalent – fractions having the same value. E.g. 3/7 and 9/21
b. Mixed – composed of a whole number and a proper fraction. E.g. 5 ¼
Rules involving Zero
a. Zero numerator and non-zero denominator – the value is zero
b. Zero denominator – no value, undefined
c. Zero value – the numerator is zero
Operations on Fractions
1. Multiplication – multiply numerator by numerator and denominator by denominator to get the numerator
and denominator respectively of the product.
e.g. 3x1=3
5 4 20
2. Division – multiply the supposed dividend by the reciprocal of the supposed divisor.
e.g. 4 ÷ 3 = 4 x 7 = 28
5 7 5 3 15
3. Addition (Subtraction)
a. Similar Fractions – add (subtract) the numerators and copy the common denominator
e.g. 3 +2=5
11 11 11
b. Dissimilar fractions – use a common denominator (preferably the least) to make the addends
(minuend and subtrahend) similar and do as in the preceding rule.
e.g. 1 + 2 = 7 + 10 = 17
5 7 35 35 35
Simplifying Fractions
A fraction is in simplest form if the numerator and the denominator are relatively prime (their GCF is 1). Thus,
to simplify fractions, express both the numerator and the denominator as products of a number and their GCF.
The fraction is then decomposed into two fractions one of which has the GCF both as its numerator and its
denominator. This fraction reduces to 1. The other fraction thus is the desired simplest form.
e.g. Simplify
16 = 4 x 4 = 4 x 4 = 4
28 4 x 7 4 7 7
Ordering fractions
Two fractions are equivalent if their cross products are equal. Otherwise, that fraction the numerator of which
was used to get the greater of the two cross products is the larger fraction.
e.g. 3 = 15
5 20
Ratio and Proportion
Ratio – comparison of two numbers a and b, where b ≠ 0, and expressed as “a to b”, or a:b, or “a/b”.
Proportion – a statement of equality between two ratios
Given two equal ratios, one comparing a to b and another comparing c to d, the proportion may be
expressed thus:
a : b :: c : d, or alternatively, a/b = c/d.
In either of the above forms, a and d are referred to as extremes, while b and c are referred to as the
means.
Partitive Proportion
If a quantity q is to be partitioned into p1, p2, p3, … pn, so that the partitions are in the ratio a1 : a2 : a3: ….: an, then
the size of the kth partition may be computed as follows:
e.g. If 24 hours is to be partition into 3 parts so that the parts are in the ration 1 : 2: 5, how many hours would
the third part be?
Percentage, Base and Rate
Percentage – The result obtained when a number is multiplied by a percent. P=R x B
P
Rate – The ratio of amount to the base. It is written in percent. R=
B
P
Base – The whole in a problem. The amount you are taking a percent of. B=
R
Example.
1. What is 5% of 20?
2. 24 is 16% of what number?
3. What percent of 25 is 9?
Simplifying Rational Expressions
The following principle can be used to simplify rational expressions, where A, B, and C are
polynomials.
A∙C A
= , B and C are nonzero
B ∙C B
Product and Quotient of Rational Expressions
1. To multiply two rational expressions, multiply the numerations and multiply the denominators.
A C AC
2. x = , B and D are nonzero.
B D BD
3. To divide two rational expressions, multiply by the reciprocal of the divisor.
A C A D
4. ÷ = x , B, C and D are nonzero.
B D B C
Solving Equations Involving Rational Expressions
1. Multiply both sides of the equation by the LCD.
2. Solve the resulting equation.
3. Check the solution to determine whether it is an excluded value and therefore extraneous.
3 8
1. Solve + 5= .
x−7 x−7
