noviembre 22, 2015

By Ana

POWER AND ROOTS

Introduction

Powers are used when we want to multiply a number by itself repeatedly.

1. Powers

When we wish to multiply a number by itself we use powers.

For example, the quantity 7⁴= 7 × 7 × 7 × 7 is usually written as 7⁴. The number 4 tells us the number of sevens to be multiplied together. In this example, the exponent, or index, is 4. The number 7 is called the base.

Example

6² = 6 · 6 = 36. We say that ‘6 squared is 36’, or ‘6 to the power of 2 is 36’.

4³ = 4 · 4 · 4 = 64. We say that “4 cubed is 64”, or “4 to the power of 3 is 64”

2⁵ = 2× 2 × 2 × 2 × 2 = 32. We say that ‘2 to the power of 5 is 32’.

Most calculators have a button marked x^y, or simply ˆ to calculate a power. Ensure that you are using the calculator correctly by verifying that

7⁵ = 16807.

OPERATIONS AND RULES

Square roots

When 5 is squared we obtain 25. That is 5² = 25.

The reverse of this process is called finding a square root. The square root of 25 is 5. This is written as $\sqrt{25}=$ 5

Note also that when −5 is squared we again obtain 25, that is (-5)²= 25. This means that 25 has another square root, −5.

In general, a square root of a number is a number which when squared gives the original number.

There are always two square roots of any positive number, one positive and one negative.

Therefore, negative numbers do not possess any square roots.

Perfect squares

A square number, also called a perfect square, is an integer that is the square of another integer.
In other words, it is the product of some integer with itself.
So, for example, 9 is a square number, since it can be written as 3²

Cube roots and higher roots

The cube root of a number, is the number which when cubed gives the original number.

For example, because 4³= 64 we know that the cube root of 64 is 4, written

. All numbers,

both positive and negative, possess a single cube root.

Higher roots are defined in a similar way: because 2⁵ = 32, the fifth root of 32 is 2.

EJERCICIOS
Potencias
Repaso potencias y raices
Operaciones combinadas

noviembre 16, 2015

By Ana

FRACTIONS

When an object is divided into a number of equal parts, then each part is called a fraction.

There are different ways of writing a fraction.

For example, two fifths of an object can be written as

a common fraction ® 2/5
a decimal ® 0.4
a percentage ® 40%

2 numerator says how many parts in the fraction
5 denominator says how many equal parts in the whole object

Always remember: denominator can NEVER be 0.
Why? Because you cannot divide by 0.

Equivalent fractions

· Two fractions are said to be equivalent when simplifying, both of them produces the same fraction written in its simplest terms.

· Equivalent fractions are fractions with identical values.

· To create a pair of equivalent fractions, you multiply (or divide, cancelling down) the top (numerator) and bottom (denominator) of a given fraction by the same number.

· Two fractions are equivalent if the cross-products are the same.

OPERATIONS WITH FRACTIONS
Fraction operations

EJERCICIOS

Ordenar fracciones

Problems

Interactivo. Operaciones

Problemas interactivos 1

Problemas interactivos 2

Operaciones combinadas

Problemas con fracciones

Más ejercicios...

octubre 25, 2015

By Ana

Prime Numbers

A prime number can be divided, without a remainder, only by itself and by 1. For example, 17 can be divided only by 17 and by 1.

Some facts:

The only even prime number is 2. All other even numbers can be divided by 2.
If the sum of a number's digits is a multiple of 3, that number can be divided by 3.
No prime number greater than 5 ends in a 5. Any number greater than 5 that ends in a 5 can be divided by 5.
Zero and 1 are not considered prime numbers.
Except for 0 and 1, a number is either a prime number or a composite number. A composite number is defined as any number, greater than 1, that is not prime.

To prove whether a number is a prime number, first try dividing it by 2, and see if you get a whole number. If you do, it can't be a prime number. If you don't get a whole number, next try dividing it by prime numbers: 3, 5, 7, 11 (9 is divisible by 3) and so on, always dividing by a prime number (see table below).

octubre 11, 2015

By Ana

DIVISIBILITY

Resultado de imagen de dividend, divisor, quotient and remainder.

In a division of natural numbers, we can find four elements: D=dividend, d=divisor, q=quotient and r=remainder.

D=d x q +r

A division is exact, if its remainder is zero. In this case, D=d·q is verified.

When the division between two numbers is exact, we say there is a relation of divisibility between them. D is divisible by d.

Multiples of a number: A number b is a multiple of another number a if the division b:a is exact. The first multiples of a number are obtained by mulyipling the number by each of the natural numbers: 1, 2, 3, 4, …

Factors of a number: A number a is a factor of another number b if the division b:a is exact. A factors is any number that will divide into another number exactly (with no part left over): 8 can be divide by 2 (the factor in this example) 4 times. However, in the total number 8 has several factors: 1, 2, 4 and 8

Prime and composite numbers: A prime number (or a prime) is a natural number wich has exactly two distint natural numbers divisors: 1 and itself. If a numbers has more than two divisors, it is called composite number. The number 1 is by definition not a prime number

Test of divisibility:

Here are some quick and easy checks to see if one number will divide exactly:

Divisible by 2_ A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8. Example: 2 346 is divisible by 2 because the last digit is 6.

Divisible by 3_ A number is divisible by 3 if the sum of the digits is divisible by 3. Example: 23 457 is divisible by 3 because the sum of the digits is 21 (2+3+4+5+7=21), and 21 is divisible by 3.

Divisible by 4_ A number is divisible by 4 if the number formed by the last two digits is either 00 or divisible by 4. Example: 24 516 is divisible by 4 because 16 is divisible by Divisible by 5_ A number is divisible by 5 if the last digit is either 0 or 5. Example: 9 876 345 is divisible by 5 because the last digit is 5.

Divisible by 6 A number is divisible by 6 if it is divisible by 2 (the last digit is 0, 2, 4, 6 or 8) and it is also divisible by 3 (the sum of the digits is divisible by 3) Example: 534 is divisible by 6 because is divisible by 2 (the last digit is 4) and it is divisible by 3 (the sum of the digits 5+3+4=12 is divisible by 3)

Divisible by 10_ A number is divisible by 10 if the last digit is 0. Example: 12 345 890 is divisible by 10 because the last digit is 0.

Divisible by 11_ To check if a number is divisible by 11, sum the digits in the odd positions counting from the left (the first, the third, …) and then sum the remainder digits. If the difference between the sums is either 0 or divisible by 11, then so is the original number. Examples: 145 879 635 918 291 Digits in odd positions: 1+5+7+6+5=24 Digits in odd positions: 9+8+9=26 Digits in even positions: 4+8+9+3=24 Digits in even positions: 1+2+1=4 The diference is 24-24=0 The difference: 26-4=22 So 145 879 635 is divisible by 11. So 918 291 is divisible by 11.

Factoring numbers: Prime factoring is to factor and then continue factoring a number untill you can no longer reduce the factors into constituent factors any further. Any number can be written as a product of prime numbers in a unique way (except for the order)

Greatest common factor (GCF): There are different ways to find the GCF of numbers.

Least common multiple (LCM): There are different ways to find the LCM of numbers.

Example:

My prime factorizations are:

2940 = 2 × 2 × 3 × 5 × 7 × 7

3150 = 2 × 3 × 3 × 5 × 5 × 7

I will write these factors out, all nice and neat, with the factors lined up according to occurrance:

This orderly listing, with each factor having its own column, will do most of the work for me

The Greatest Common Factor, the GCF, is the biggest number that will divide into (is a factor of) both

2940

and

3150

. In other words, it's the number that contains all the factors common to both numbers. In this case, the GCF is the product of all the factors that

2940

and

3150

have in common.

Looking at the nice neat listing, I can see that the numbers both have a factor of

2; 2940

has a second copy of the factor

, but

3150

does not, so I can only count the one copy toward my GCF. The numbers also share one copy of

, one copy of

, and one copy of

Then

the GCF is 2 × 3 × 5 × 7 = 210.

On the other hand, the Least Common Multiple, the LCM, is the smallest number that both

2940

and

3150

will divide into. That is, it is the smallest number that contains both

2940

and

3150

as factors, the smallest number that is a multiple of both these values. Then it will be the smallest number that contains one of every factor in these two numbers.

Looking back at the listing, I see that

3150

has one copy of the factor of

;

2940

has two copies. Since the LCM must contain all factors of each number, the LCM must contain both copies of

. However, to avoid overduplication, the LCM does not need three copies, because neither

2940

nor

3150

contains three copies.

This fact often causes confusion, so let's spend a little extra time on this. Consider two smaller numbers, 4 and 8, and their LCM. The number 4 factors as 2 × 2; 8 factors as 2 × 2 × 2. The LCM needs only have three copies of 2, in order to be divisible by both 4 and 8. That is, the LCM is 8. You do not need to take the three copies of 2 from the 8, and then throw in two extra copies from the 4. This would give you 32. While 32 is a common multiple, because 4 and 8 both divide evenly into 32,32 is not the LEAST (smallest) common multiple, because you over-duplicated the 2s when you threw in the extra copies from the 4. Again, let the nice neat listing keep track of things when the numbers get big.

So, my LCM of

2940

and

3150

must contain both copies of the factor

. By the same reasoning, the LCM must contain both copies of

, both copies of

, and both copies of

Then

the LCM is 2 × 2 × 3 × 3 × 5 × 5 × 7 × 7 = 44,100.

By using this "factor" method of listing the prime factors neatly in a table, you can always easily find the LCM and GCF. Completely factor the numbers you are given, list the factors neatly with only one factor for each column (you can have 2s columns, 3s columns, etc, but a 3 would never go in a 2s column), and then carry the needed factors down to the bottom row.

For the GCF, you carry down only those factors that all the listings share; for the LCM, you carry downall the factors, regardless of how many or few values contained that factor in their listings.

PROBLEMAS PARA PRACTICAR
LCM, para practicar
HCF para practicar

septiembre 20, 2015

By Ana

INTEGERS

Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on)

"Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.

Integers

The set of integers consists of zero (0), the natural numbers (1, 2, 3, …), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e. −1, −2, −3, …).

The Zahlen symbol, often used to denote the set of all integers

Order

An integer is positive if it is greater than zero and negative if it is less than zero

Properties of addition and multiplication on integers
	Addition	Multiplication
Closure:	a + b is an integer	a × b is an integer
Associativity:	a + (b + c) = (a + b) + c	a × (b × c) = (a × b) × c
Commutativity:	a + b = b + a	a × b = b × a
Existence of an identity element:	a + 0 = a	a × 1 = a
Existence of inverse elements:	a + (−a) = 0	An inverse element usually does not exist at all.
Distributivity:	a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)
No zero divisors: (*)		If a × b = 0, then a = 0 or b = 0 (or both)

EJERCICIOS

Operaciones combinadas

Crucigrama