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Mathematics is especially useful when it helps you predict,and number patterns are all about prediction. What will the50th number of this pattern be? How many cupcakes wouldwe need if we gave a party for the town instead of just ourclass? Working with number patterns leads directly to theconcept of functions in mathematics: a formal description ofthe relationships among different quantities.

Recognizing number patterns is also an important problem-solving skill. If you see a

pattern when you look systematically at specific examples, you can use that patternto generalize what you see into a broader solution to a problem.

The youngest children begin simply by counting. They count by 1s, then by 2s, 5s,and 10s. These patterns give students a natural strategy to understand addition andmultiplication. When considering a number pattern such as 2, 4, 6..., a young studentwill ask herself, By what number can I count (add) to get to the next number in thepattern and the next and the next?

 As the student gets older, her knowledge of patterns advances from sums toproducts. When asked for the 50th number in the pattern, she will know to multiply 2times 50.

High school students can start to understand functions, such as f(x) = 2x + 2, wherex is the numer ical sequence 0, 1, 2, 3,…. They begin with simple in-out machinesand gradually adapt their understanding to the abstractions of algebra.

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Arithmetic Sequences

An Arithmetic Sequence is made by adding some value each time.

Examples:

1, 4, 7, 10, 13, 16, 19, 22, 25, ... 

This sequence has a difference of 3 between each number.

The pattern is continued by adding 3 to the last number each time.

3, 8, 13, 18, 23, 28, 33, 38, ...  

This sequence has a difference of 5 between each number.

The pattern is continued by adding 5 to the last number each time.

The value added each time is called the "common difference"

What is the common difference in this example?

19, 27, 35, 43, ... 

Answer: The common difference is 8 

The common difference could also be negative, like this:

25, 23, 21, 19, 17, 15, ... 

This common difference is -2 

The pattern is continued by subtracting 2 each time.

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Geometric Sequences

A Geometric Sequence is made by multiplying by some value each time.

Examples:

2, 4, 8, 16, 32, 64, 128, 256, ... 

This sequence has a factor of 2 between each number.

The pattern is continued by multiplying the last number by 2 each time.

3, 9, 27, 81, 243, 729, 2187, ... 

This sequence has a factor of 3 between each number.

The pattern is continued by multiplying the last number by 3 each time.

Special Sequences

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, ... 

This sequence is generated from a pattern of dots which form a triangle.

By adding another row of dots and counting all the dots we can find the next number of thesequence:

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Square Numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, ... 

The next number is made by squaring where it is in the pattern.The second number is 2 squared (22 or 2×2)The seventh number is 7 squared (72 or 7×7) etc

Cube Numbers

1, 8, 27, 64, 125, 216, 343, 512, 729, ... 

The next number is made by cubing where it is in the pattern.The second number is 2 cubed (23 or 2×2×2)The seventh number is 7 cubed (73 or 7×7×7) etc

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Fibonacci Numbers 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...  

The next number is found by adding the two numbers before it together.The 2 is found by adding the two numbers in front of it (1+1)The 21 is found by adding the two numbers in front of it (8+13)The next number in the sequence above would be 55 (21+34)

A factor of a number will divide that number exactly, without any remainder.

Example : 2 is a factor of 16 because 2 goes into 16 exactly 8 times, with no remainder.

But note that 16 also has other factors apart from 2. These other factors are

  4 : which goes into 16 exactly 4 times  8 : which goes into 16 exactly 2 times  as well as 16 itself : 16 divided by 16 equals 1  and also 1 : 16 divided by 1 equals 16

So the number 16 has a total of 5 factors.

Example : 24 has the following factors

1,2,3,4,6,8,12,24

Example : 3 is a factor of 9 because 3 divides into 9 exactly, without any remainder.

Example : 3 is also a factor of

27, 18, 30, 15,....and many more

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Highest Common Factor 

The Highest Common Factor (HCF) of two numbers is the highest number that is a

factor of both numbers 

Example : The numbers 36 and 24 have a common factor of 4. But this not the Highest Common Factor, because 6 is also a common factor. The number 6 itself is still not therequired HCF - the HCF is actually 12.

Multiples 

Multiples of a number are obtained just by multiplying that number by a whole number.

Example The multiples of 6 are

6, 12 , 18, 24, 30, etc

 Note that 6 is a multiple of 6 because it is equal to 6 multiplied by one.

Lowest Common Multiple 

The Lowest Common Multiple (LCM) of two numbers is a multiple of both numbers andlower than all other numbers that are also multiples of both the numbers.

Example The Lowest Common Multiple of 6 and 4 is 12.

12 is lower than other common multiples like 24, 36, 48, and there is no common multiple of6 and 4 which is lower than 12.

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Square number

In mathematics, a square number, sometimes also called a perfect square, is an integer  thatis the square of an integer; in other words, it is the product of some integer with itself. So, forexample, 9 is a square number, since it can be written as 3 × 3. Square numbers are  non-negative. Another way of saying that a (non-negative) number is a square number, is that its

square root is again an integer. For example, √9 = 3, so 9 is a square number.

A positive integer that has no perfect square divisors except 1 is called square-free. 

The usual notation for the formula for the square of a number n is not the product n × n, butthe equivalent exponentiation n2, usually pronounced as "n squared". For a non-negativeinteger n, the nth square number is n

2, with 02 = 0 being the zeroth square. The concept ofsquare can be extended to some other number systems. If rational numbers are included, thena square is the ratio of two square integers, and, conversely, the ratio of two square integers isa square (e.g., 4/9 = (2/3)2).

Starting with 1, there are square numbers up to and including m.

When you first learned your numbers, way back in elementary school, you started with the counting

numbers: 1, 2, 3, 4, 5, 6, and so on. Your number line looked something like this:

Later on, you learned about zero, fractions, decimals, square roots, and other types of numbers, soyour number line started looking something like this:

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 Addition, multiplication, and division always made sense — as long as you didn't try to divide by zero

— but sometimes subtraction didn't work. If you had "9 –  5", you got 4:

...but what if you had "5 –  9"? You just couldn't do this subtraction, because there wasn't enough

"space" on the number line to go back nine units:

You can solve this "space" problem by using negative numbers. The "whole" numbers start at zeroand count off to the right; these are the positive integers. The negative integers start at zero and countoff to the left:

Note the arrowhead on the far right end of the number line above. That arrow tells you the direction inwhich the numbers are getting bigger. In particular, that arrow also tells you that the negatives are

getting smaller  as they move off to the left. That is, – 5 is smaller than – 4.

This might seem a bit weird at first, but that's okay; negatives take some getting used to. Let's look ata few inequalities, to practice your understanding. Refer to the number line above, as necessary. 

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The importance of negative number in daily live

I've never had to use (-3x/-4y)+(-5n X -3p) on a daily basis. In real life, if you think about it,tax is a negative number.

More

Every debit entry in your checkbook is a negative number. It must be subtracted from ahigher (hopefully) number. If you write checks for more than what is in the bank, youraccount will have a negative number balance, meaning that you owe the bank money.

Also, anyone working with digital electronics will always be working with negative numbers(voltages, etc.)

Fraction

A fraction (from the Latin  fractus, broken) is a number  that can represent part of a whole. The earliest fractions were reciprocals of  integers: ancient symbols representing one part oftwo, one part of three, one part of four, and so on.[1] A much later development were thecommon or "vulgar" fractions which are still used today (½, ⅝, ¾, etc.) and which consist ofa numerator and a denominator, the numerator representing a number of equal parts and thedenominator telling how many of those parts make up a whole. An example is 3/4, in whichthe numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4,tells us that 4 parts make up a whole.

Decimals

A still later development was the fraction, now called simply a decimal, in which thedenominator is a  power of ten, determined by the number of  digits to the right of a decimalseparator , the appearance of which (e.g., a period, a raised period (•), a comma) depends onthe locale (for examples, see decimal separator ). Thus for 0.75 the numerator is 75 and thedenominator is 10 to the second power, viz. 100, because there are two digits to the right ofthe decimal.

A third kind of fraction still in common use is the  percentage, in which the denominator isalways 100. Thus 75% means 75/100.

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Other uses for fractions are to represent ratios, and to represent division. Thus the fraction 3/4is also used to represent the ratio 3:4 (three to four) and the division 3 ÷ 4 (three divided byfour).

In mathematics, the set of all (vulgar) fractions is called the set of  rational numbers, and is

represented by the symbol Q.

Percentage

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In mathematics, a percentage is a way of expressing a number as a fraction of 100 ( per cent  meaning "per hundred" in French). It is often denoted using the  percent sign, "%", or theabbreviation "pct". For example, 45% (read as "forty-five percent") is equal to 45 / 100, or0.45.

Percentages are used to express how large/small one quantity is, relative to another quantity.The first quantity usually represents a part of, or a change in, the second quantity, whichshould be greater than zero. For example, an increase of $ 0.15 on a price of $ 2.50 is anincrease by a fraction of 0.15 / 2.50 = 0.06. Expressed as a percentage, this is therefore a 6%increase.

Although percentages are usually used to express numbers between zero and one, anydimensionless  proportionality can be expressed as a percentage. For instance, 111% is 1.11and -0.35% is -0.0035.

This article aims to be an accessible introduction. For the mathematical definition, see

Decimal representation. 

Numeral systems by culture 

Hindu-Arabic numerals

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Eastern Arabic 

Indian family 

Khmer 

Mongolian 

Thai 

Western Arabic 

East Asian numerals 

Chinese 

Japanese 

Suzhou 

Korean 

Vietnamese 

Counting rods 

Alphabetic numerals 

Abjad 

Armenian 

Āryabhaṭa 

Cyrillic 

Ge'ez 

Greek (Ionian)

Hebrew 

Other systems 

Aegean 

Attic 

Babylonian 

Brahmi Egyptian 

Etruscan 

Inuit 

Mayan 

Quipu 

Roman Sumerian 

Urnfield 

List of numeral system topics  

Positional systems by base 

Decimal (10) 

1, 2, 3, 4, 5, 8, 12, 16, 20, 60 more… 

v • d • e 

The decimal numeral system (also called base ten or occasionally denary) has ten as its  base. It is the numerical base most widely used by modern civilizations.[1][2] 

Decimal notation often refers to the base-10  positional notation such as the Hindu-Arabicnumeral system, however it can also be used more generally to refer to non-positionalsystems such as Roman or  Chinese numerals which are also based on powers of ten.

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In some contexts, especially mathematics education, the term decimal  can refer specifically todecimal fractions, described  below. In such cases, a single decimal fraction is called a"decimal", and non-fractional numbers, even when written in base 10, are not considered"decimals".