Mathematics

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Mathematics

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Mathematics

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Mathematics

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Mathematics

Ordering of Decimals

Ordering Decimals

“Could I have a 3.65 and an 0.8, please … ?”
NO, not THAT type of ordering. I mean putting them in order

Ordering Decimals can be tricky. Because often we look at 0.42 and 0.402 and say that 0.402 must be bigger because there are more digits. But no!

We can use this method to see which decimals are bigger:

  • Set up a table with the decimal point in the same place for each number.
  • Put in each number.
  • Fill in the empty squares with zeros.
  • Compare using the first column on the left
  • until one number If the digits are equal move to the next column to the right wins.

If you want ascending order you always pick the smallest first

If you want descending order you always pick the largest first

Example: Put the following decimals in ascending order:

1.506, 1.56, 0.8

In a table they look like this:

Units Decimal
Point
Tenths Hundredths Thousandths
1 . 5 0 6
1 . 5 6
0 . 8

Fill in the empty squares with zeros:

Units Decimal
Point
Tenths Hundredths Thousandths
1 . 5 0 6
1 . 5 6 0
0 . 8 0 0

Compare using the first column (Units)

Two of them are “1”s and the other is a “0”. Ascending order needs smallest first, and so “0” is the winner:

Answer so far: 0.8

Now we can remove 0.8 from the list:

Units Decimal
Point
Tenths Hundredths Thousandths
1 . 5 0 6
1 . 5 6 0
.

Compare the Tenths

Now there are two numbers with the same “Tenths” value of 5, so move along to the “Hundredths” for the tie-breaker

Compare the Hundredths

One of those has a 6 in the hundredths, and the other has a 0, so the 0 wins (remember we are looking for the smallest each time). In other words 1.506 is less than 1.56:

Answer so far: 0.8, 1.506

Remove 1.506 from the list:

Units Decimal
Point
Tenths Hundredths Thousandths
.
1 . 5 6 0
.

Only one number left, it must be the largest:

Answer: 0.8, 1.506, 1.56

Done!

Example: Put the following decimals in DESCENDING order:

0.402, 0.42, 0.375, 1.2, 0.85

In a table they look like this:

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 4 0 2
0 . 4 2  
0 . 3 7 5
1 . 2    
0 . 8 5  

And we want to go from highest to lowest (descending).

Fill in the empty squares with zeros:

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 4 0 2
0 . 4 2 0
0 . 3 7 5
1 . 2 0 0
0 . 8 5 0

Compare using the first column (Units):

There is a 1, all the rest are 0. Descending order needs largest first, so 1.2 must be the highest. (Write it down in your answer and cross it off the table).

Answer so far: 1.2

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 4 0 2
0 . 4 2 0
0 . 3 7 5
0 . 8 5 0

Compare the Tenths.

The 8 is highest, so 0.85 is next in value.

Answer so far: 1.2, 0.85

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 4 0 2
0 . 4 2 0
0 . 3 7 5

Now there are two numbers with the same “Tenths” value of 4, so move along to the “Hundredths” for the tie-breaker

One number has a 2 in the hundredths, and the other has a 0, so the 2 wins. So 0.42 is bigger than 0.402:

Answer so far: 1.2, 0.85, 0.42, 0.402

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 3 7 5

Only 0.375 left, so the answer is:

Answer: 1.2, 0.85, 0.42, 0.402, 0.375

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Mathematics

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Mathematics

What is Function?

What is a Function?

A function relates an input to an output.

It is like a machine that has an input and an output.

And the output is related somehow to the input.

 

                                  f(x)   f(x) = … ” is the classic way of writing a function.
And there are other ways, as you will see!

Input, Relationship, Output

We will see many ways to think about functions, but there are always three main parts:

  • The input
  • The relationship
  • The output

Example: “Multiply by 2” is a very simple function.

Here are the three parts:

Input Relationship Output
0 × 2 0
1 × 2 2
7 × 2 14
10 × 2 20

For an input of 50, what is the output?

Some Examples of Functions

  • x2 (squaring) is a function
  • x3+1 is also a function
  • Sine, Cosine  and Tangent are functions used in trigonometry
  • and there are lots more!

Names

First, it is useful to give a function a name.

The most common name is “f“, but we can have other names like “g” … or even “marmalade” if we want.

But let’s use “f”:

f(x)

We say “f of x equals x squared”

what goes into the function is put inside parentheses () after the name of the function:

So f(x) shows us the function is called “f“, and “x” goes in

And we usually see what a function does with the input:

f(x) = x2 shows us that function “f” takes “x” and squares it.

Example: with f(x) = x2:

  • an input of 4
  • becomes an output of 16.

In fact we can write f(4) = 16.

 

The “x” is Just a Place-Holder!

Don’t get too concerned about “x”, it is just there to show us where the input goes and what happens to it.

It could be anything!

So this function:

f(x) = 1 – x + x2

Is the same function as:

  • f(q) = 1 – q + q2
  • h(A) = 1 – A + A2
  • w(θ) = 1 – θ + θ2

The variable (x, q, A, etc.) is just there so we know where to put the values:

f(2) = 1 – 2 + 22 = 3

 

Sometimes There is No Function Name

Sometimes a function has no name, and we see something like:

y = x2

But there is still:

  • an input (x)
  • a relationship (squaring)
  • and an output (y)

Relating

At the top we said that a function was like a machine. But a function doesn’t really have belts or cogs or any moving parts – and it doesn’t actually destroy what we put into it!

A function relates an input to an output.

Saying “f(4) = 16” is like saying 4 is somehow related to 16. Or 4 → 16

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Arithmatics, Mathematics

Arithmatics

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an elementary part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry, and analysis. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory

Totally dependent on the operation of Addition,Subtraction, Multiplication and Division.

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Mathematics, Trigonometry

Trigonometry/Function

If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

  • Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

    245px-TrigonometryTriangle.svg

\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.

  • Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.
  • Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{a}{\,c\,}*\frac{c}{\,b\,}=\frac{a}{\,c\,} / \frac{b}{\,c\,}=\frac{\sin A}{\cos A}\,.

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).

The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively:

\csc A=\frac{1}{\sin A}=\frac{\textrm{hypotenuse}}{\textrm{opposite}}=\frac{c}{a} ,
\sec A=\frac{1}{\cos A}=\frac{\textrm{hypotenuse}}{\textrm{adjacent}}=\frac{c}{b} ,
\cot A=\frac{1}{\tan A}=\frac{\textrm{adjacent}}{\textrm{opposite}}=\frac{\cos A}{\sin A}=\frac{b}{a} .
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Mathematics

Algebra

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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics.