MATHS SYMBOLS - OTHER OPERATIONS (1) - ABSOLUTE VALUE - ROUNDING to INTEGER - PLUS or MINUS - RECIPROCAL - RATIO - PROPORTIONS and FIRST PROPERTIES - BRACKETS - EQUALITY SIGN - APPROXIMATELY EQUAL - NOT EQUAL - LESS - MUCH LESS - LESS THAN or EQUAL TO - GREATER - MUCH GREATER THAN - GREATER THAN or EQUAL TO - PROPORTIONALITY - DEFINITION

1. ENZO EXPOSYTO
MATHS
SYMBOLS
OTHER OPERATIONS - 1
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2. |-2| ‖1.6‖ ⌊4.3⌋ ⌈4.3⌉
± /
OTHER
OPERATIONS - 1
() [ ] { } ⟨ ⟩
= ≠ ≤ ≥ :=
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3.
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4. 1 - Absolute Value 5
2 - Rounding to Integer 9
3 - Other Simple Operations 12
4 - Proportions - Definition 21
5 - Proportions - First 4 Properties 25
6 - Brackets 32
7 - Relations - Relationships 36
8 - SitoGraphy 54
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5. ABSOLUTE
VALUE
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6. Absolute Value - 1
| vertical line
|n| Say: modulus of n
Say: absolute value of n
|n| is the positive value of n if n is negative
|n| is the null value of n if n is null
|n| is the positive value of n if n is positive
It is the size of a number,
ignoring the sign.
For example:
|-3| = 3
|0| = 0
|3| = 3
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7. Absolute Value - 2
|x| Say: modulus of x
Say: absolute value of x
|x| is the positive value of x if x is negative
|x| is the null value of x if x is null
|x| is the positive value of x if x is positive
It is the size of a variable,
ignoring the sign.
Briefly:
|x| = - x (if x < 0)
|x| = x (if x = 0)
|x| = x (if x > 0)
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8. Absolute Value - 3
|x| Say: modulus of x
Say: absolute value of x
y = x and y = |x|
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9. ROUNDING
to
INTEGER
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10. Nearest Integer
‖...‖ the nearest integer to ...
‖x‖ means the nearest integer to x
This may also be written:
[x]
nint(x)
round(x)
...
For example:
‖1.2‖ = 1
‖2.6‖ = 3
‖−2.4‖ = −2
‖4.49‖ = 4
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11. Lower or Upper Integer
⌊x⌋ floor brackets;
it rounds a number to lower integer;
For example:
⌊5.4⌋ = 5
⌈x⌉ ceiling brackets;
it rounds a number to upper integer;
For example:
⌈5.4⌉ = 6
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12. OTHER
SIMPLE
OPERATIONS
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13. Plus or Minus
± plus or minus, positive or negative;
the equation x = 4 ± 2√9, with 2√9 = 3,
has two solutions:
x = 4 + 3 = 7
and
x = 4 - 3 = 1
± plus or minus, positive or negative;
used to indicate a range;
10 ± 2
or, equivalently,
10 ± 20%
means the range from (10 − 2) to (10 + 2);
for example: 8, 9, 10, 11, 12
minus or plus, negative or positive
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14. Reciprocal - 1
1 reciprocal of n;
n
We get the reciprocal of a number n
when we write the fraction 1
n
It is so, because n multiplied by its reciprocal gives us 1:
n x 1 = n = 1
n n
Therefore, the definition is:
the reciprocal of a number ’n’ is the number which
multiplied by ’n’ gives 1.
Note that does not exist the reciprocal of 0,
because the product of any number multiplied by 0
is always 0 and, then, it’s never equal to 1.
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15. Reciprocal - 2
1 reciprocal of n;
n
Examples (1):
the reciprocal of 3 is 1 because 3 x 1 = 3 = 1
3 3 3
the reciprocal of 4 is 1 because 4 x 1 = 4 = 1
4 4 4
the reciprocal of -2 is 1 because -2 x 1 = -2 = 1
-2 -2 -2
the reciprocal of -7 is 1 because -7 x 1 = -7 = 1
-7 -7 -7
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16. Reciprocal - 3
1 reciprocal of n;
n
Examples (2):
the reciprocal of 1 is 6 because 1 x 6 = 6 = 1
6 6 6
Indeed, if we write the fraction 1 of n = 1, we have:
n 6
1 = 1 x 6 = 6
1 1
6
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17. Reciprocal - 4
1 reciprocal of n;
n
Examples (3):
the reciprocal of 1 is 5 because 1 x 5 = 5 = 1
5 5 5
the reciprocal of - 1 is -8 because - 1 x (-8) = + 8 = 1
8 8 8
the reciprocal of - 1 is -9 because - 1 x (-9) = + 9 = 1
9 9 9
the reciprocal of 1 is 2√2 because 1 x (2√2) = 2√2 = 1
2√2 2√2 2√2
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18. Ratio - 1
: ratio;
/ a ratio of 7:4 or 7/4 (seven to four)
between pears and apples
means that,
for each seven pears,
there are four apples.
The ratio of two natural numbers is the relationship
which considers the size of the first
with respect to the size of the second.
The two numbers in a ratio are called terms:
the first term
and
the second term.
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19. Ratio - 2
: Example 1 - Multiple
/
What is the ratio of 20 to 4?
Answer: 20 is five times 4 and, then
5 is the ratio - the relationship - of 20 to 4.
Example 2 - Part
What is the ratio of 4 to 20?
Answer: 4 is the fifth part of 20, then
1 is the ratio - the relationship - of 4 to 20;
5
it’s the reciprocal ratio of 20 to 4.
Note that the terms are exchanged.
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20. Ratio - 3
: Example 3 - Parts
/
What is the ratio of 4 to 6?
Answer: 4 is two thirds of 6, then
2 is the ratio - the relationship - of 4 to 6
3
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21. PROPORTIONS
-
Definition
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22. Proportions - Definition - 1
a:b=c:d proportion
Since the meaning is the same, we can write:
a:b=c:d
or
a/b=c/d
or
a = c
b d
where b ≠ 0 and d ≠ 0
Say: a is to b as c is to d
a and d are named extremes
b and c are called means
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23. Proportions - Definition - 2
a:b=c:d A proportion is a statement that two ratios are equal.
For example:
2:4 = 3:6
or
2/4 = 3/6
or
2 = 3
4 6
2 = 3 is equivalent to 1 = 1
4 6 2 2
Because the two ratios are equal,
these equalities represent the same proportion.
2 is to 4 as 3 is to 6
2 and 6 are named extremes
4 and 3 are called means
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24. Proportions - Definition - 3
a:b=c:d A proportion is a statement that two ratios are equal.
In other words, if, and only if, two ratios are equal,
the equality with the two ratios is a proportion.
For example, are the next equalities true?
Are they a proportion?
4:2 = 6:2
or
4/2 = 6/2
or
4 = 6
2 2
They aren't true, because
4 = 2 and 6 = 3
2 2
These equalities don't exist and don't represent a proportion
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25. PROPORTIONS
-
First 4 Properties
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26. Proportions - First 4 Properties - (1)
a:b=c:d proportion - Property (1) - Cross Products
a x d = b x c
Say: the product of the extremes
is equal to the product of the means
1) From the proportion 2:4 = 3:6 we get
2 x 6 = 4 x 3
12 = 12
2) From the proportion 15:5 = 9:3
or
15 = 9 that's equivalent to 3 = 3 and is true
5 3
we have
15 x 3 = 5 x 9
45 = 45
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27. Proportions - First 4 Properties - (2a)
a:b=c:d proportion - Property (2a) - Extremes Switching Property
d : b = c : a with b ≠ 0 and a ≠ 0
If we switch the extremes each other,
we get two equal ratios
and, then, the new equality is a proportion
1) From the proportion 2:4 = 3:6
if we switch the extremes each other, we get
6 : 4 = 3 : 2 Since it means
3 : 2 = 3 : 2 the new equality is a proportion
2) From the proportion 15:5 = 9:3
if we switch the extremes each other, we have
3 : 5 = 9 : 15 Since its meaning is
3 : 5 = 3 : 5 the new equality is a proportion
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28. Proportions - First 4 Properties - (2b)
a:b=c:d proportion - Property (2b) - Means Switching Property
a : c = b : d with c ≠ 0 and d ≠ 0
If we switch the means each other,
we get two equal ratios
and, then, the new equality is a proportion
1) From the proportion 2:4 = 3:6
if we switch the means each other, we get
2 : 3 = 4 : 6 Since it means
2 : 3 = 2 : 3 the new equality is a proportion
2) From the proportion 15:5 = 9:3
if we switch the means each other, we have
15 : 9 = 5 : 3 Since its meaning is
5 : 3 = 5 : 3 the new equality is a proportion
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29. Proportions - First 4 Properties - (3)
a:b=c:d proportion - Property (3) - Upside-Down Property
b : a = d : c with a ≠ 0 and c ≠ 0
If we switch extreme for mean to both sides,
we get two equal ratios
and, then, the new equality is a proportion
1) From the proportion 2:4 = 3:6 we get
4 : 2 = 6 : 3 Since it means
2 = 2 the new equality is a proportion
2) From the proportion 15:5 = 9:3 we have
5 : 15 = 3. : 9 Since its meaning is
1 : 3 = 1 : 3 the new equality is a proportion
We can write as well:
5 = 3 that's equivalent to 1 = 1 and is true
15 9 3 3
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30. Proportions - First 4 Properties - (4a)
a:b=c:d proportion - Property (4a) - Denominator Addition Property
a + b = c + d with b ≠ 0 and d ≠ 0
b d
1) From the proportion 2:4 = 3:6 we get
2 + 4 = 3 + 6 If we simplify, it means
4 6
6 = 9 that's equivalent to 3 = 3 and is true
4 6 2 2
2) From the proportion 15:5 = 9:3 we have
15 + 5 = 9 + 3 If we simplify, it means
5 3
20 = 12 which is equivalent to 4 = 4 and is true
5 3
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31. Proportions - First 4 Properties - (4b)
a:b=c:d proportion - Property (4b) - Denominator Subtraction Property
a - b = c - d with b ≠ 0 and d ≠ 0
b d
1) From the proportion 2:4 = 3:6 we get
2 - 4 = 3 - 6 If we simplify, it means
4 6
- 2 = - 3 that's equivalent to - 1 = - 1 and is true
4 6 2 2
2) From the proportion 15:5 = 9:3 we have
15 - 5 = 9 - 3 If we simplify, it means
5 3
10 = 6 which is equivalent to 2 = 2 and is true
5 3
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32. BRACKETS
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33. Simple Brackets
( left parenthesis
) right parenthesis
() parentheses
[ ] left and right square brackets
{ } curly brackets or braces
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34. Angle Brackets
⟨ ⟩ angle brackets, chevrons, diamond brackets
Premise:
an n-tuple is a sequence (or ordered list)
of n elements, where n is a non-negative integer.
We can use the angle brackets:
⟨⟩ is the empty tuple (or 0-tuple)
⟨a, b⟩ is an ordered pair (or 2-tuple)
⟨a, b, c⟩ is an ordered triple (or 3-tuple)
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35. Integer Brackets
‖ ‖ double vertical line brackets
⌊x⌋ floor brackets
⌈x⌉ ceiling brackets
Examples:
see pages 8-10
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36. RELATIONS
RELATIONSHIPS
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37. Equality - 1
= the equals sign or equality sign
Say: ... is equal to ...
Example 1:
4 = 4
means that
4, and only 4, is equal to 4;
4.00000000000000000…00001 is not equal to 4
Example 2:
3 + 4 = 7
means that
the sum of 3 plus 4 is 7
and, then, because 7 is equal to 7,
3 + 4 is equal to 7
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38. Equality - 2
= the equals sign or equality sign
Example 3:
2√9 = 3
(the square root of 9 is equal to 3)
means that
the square root of 9 is 3
and, then, because 3 is equal to 3,
2√9 is equal to 3.
2√2 = 1.4
is absolutely wrong
because 1.4 is not equal to 2√2.
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39. Equality - 3
= the equals sign or equality sign
Example 4:
log2(8) = 3
(the logarithm in base 2 of 8 is equal to 3)
means that
the logarithm in base 2 of 8 is 3
and, then, because 3 is equal to 3,
log2(8) is equal to 3
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40. Equality - 4
= the equals sign or equality sign
Example 5:
x - 2 = 4
(x minus 2 is equal to 4)
It's an equation and
this equality sign means that
x represents the only one number by which
the left side is equal to the right side;
x = 6 because only 6 is such that 6 - 2 = 4
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41. Equality - 5
= the equals sign or equality sign
Example 6:
x2 - 1 = 0
(x squared minus 1 is equal to 0)
It's an equation and this equality sign means that
x represents the only two numbers by which
the left side is equal to the right side;
x = ± 1 because only x = +1 and x = -1
are such that
(+1)2 - 1 = 0
and
(-1)2 - 1 = 0
In other words, by x = ± 1,
the left side is equal to the right side.
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42. Equality - 6
= the equals sign or equality sign
Example 7:
y = x
(y is equal to x)
It's a function and this equality sign means that
y and x represent the same math object
(both symbols have the same value
and, if x varies, y varies exactly as x)
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43. Approximately Equal
... is approximately equal to …
Examples:
2.0001 2
3.9 + 2.101 6
2√2 1.4142
e 2.718281828
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44. Not Equal - 1
≠ ... is not equal to …
< >
Examples:
4.00000000000000000…00001 ≠ 4
2.0001 ≠ 2
3.9 + 2.101 ≠ 6
2√2 ≠ 1.4142
e ≠ 2.718281828
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45. Not Equal - 2
≠ ... is not equal to …
< >
Example 6:
1 with x < > 0
x
means that 1 can exist if x < 0 or if x > 0
x
Example 7:
1 where x < > 1
x-1
means that 1 can be if x < 1 or if x > 1
x-1
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46. Less
< ... is less than ...
Examples:
4 < 4.00000000000000000…00001
2 < 2.0001
6 < 3.9 + 2.101
1.4142 < 2√2
2.718281828 < e
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47. Much Less
<< ... is much less than …
Examples:
4 << 4,000,000,000
2 << 2 . 108
1010 << 10100
2√2 << 2√2100
2.718281828 << e20
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48. Less Than or Equal To
≤ ... is less than or equal to …
Example 1:
y = 2√(-x) with x ≤ 0
means that 2√(-x) is a real number if x < 0 or if x = 0
Example 2:
y = 2√(1-x) where x ≤ 1
means that 2√(1-x) is an element of R if x < 1 or if x = 1
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49. Greater
> ... is greater than ...
Examples:
4.00000000000000000…00001 > 4
2.0001 > 2
3.9 + 2.101 > 6
2√2 > 1.4142
e > 2.718281828
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50. Much Greater Than
>> ... is much greater than …
Examples:
4,000,000,000 >> 2
5 . 108 >> 3
1010000 >> 10100
2√3100 >> 2√3
π200 >> π2
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51. Greater Than or Equal To
≥ ... is greater than or equal to ...
Example 1:
y = 2√x with x ≥ 0
means that 2√x is a real number by x > 0 or by x = 0
Example 2:
y = 2√(x-1) where x ≥ 1
means that 2√(x-1) is an element of R if x > 1 or if x = 1
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52. Proportionality
proportionality sign
... is proportional to ...
... varies as ...
y x means that y = k . x (k is some constant);
if y = 2 . x, then y x;
the variable y is directly proportional to the variable x
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53. Definition
:= definition
It means:
... is defined as ...
or
... is equal by definition to ...
For example:
y := x2
means
y is defined to be another name for x2,
under certain assumptions taken in context
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54. SitoGraphy
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55. https://en.m.wikipedia.org/wiki/List_of_mathematical_symbols
http://psychstat3.missouristate.edu/Documents/IntroBook3/sbk10.htm
https://www.cliffsnotes.com/study-guides/geometry/similarity/properties-of-proportions
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