DIVISIBILITY. BASIC CONCEPTS.The factors of a whole number are the numbers that divide into it exactly. By the way , factors are also called divisor For example, the factors of 18 are 1, 2, 3, 6, 9, and 18. A number is called a prime number if it has exactly two factors (i.e. 1 and itself). The first few prime numbers are 2, 3, 5, 7, 11, 13, … . Note that 1 is not a prime number. The multiples of a number are all the numbers that it will go into exactly. For example, the multiples of 6 are 6, 12, 18, 24, 30, … .A composite number is any number that has more than two factors. Here's a list of composite numbers up to 20. You can see that they can all be factored further. For example, 4 equals 2 times 2, 6 equals 3 times 2, 8 equals 4 times 2, and so forth.By the way, zero and one are considered neither prime nor composite numbers-they're in a class by themselves!You can write any composite number as a product of primes factors. This is called prime factorization.FACTOR TREEFactor trees are used to break a number up into its prime factors. Example: Draw the factor tree for 60. We begin by finding two whole numbers that multiply to make 60, for example 6 × 10: We keep splitting up each number in this way until we reach a prime number. Once a prime number is reached, we circle it and then that part of the diagram is finished. The completed factor tree looks as follows: The circled numbers multiply together to make 60, i.e. 60 = 2 × 2 × 3 × 5. This product is called the prime factorisation of 60. Divisibility Tests
Divisibility by 11: Divisibility by 11 is the most interesting of the above tests (7 will be studied below). We do two sums (the odd numbered digits and the even numbered digits), subtract one sum from the other, and see if this is divisible by 11. By the way, if we end up with zero, then that is divisible by 11. We can repeat that process, just as we did with 3. Let's look at an example: 34871903 3+8+1+0=12 4+7+9+3=23 23-12=11 Is divisible by 11 We can, of course, do the summing in different orders. In fact we can just go from left to right adding and subtracting alternate digits: 3-4+8-7+1-9+0-3=-11 (divisible by 11). LCM (Least Common Multiple) and GCF (Greatest Common Factor)To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of two numbers, you always start out the same way: you find the prime factorizations of the two numbers. Then you put the factors into a nice neat grid of rows and columns, and compare and contrast and take what you need.Find the GCF and LCM of 84 and 140 :
My prime factorizations are: - So: 84 = 2 × 2 × 3× 7
- Also, 140 = 2 × 2 × 5 × 7
The Greatest Common Factor, the GCF(Mcd), is the biggest number that will divide into (is a factor of) both 84 and 140. 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 84 and 140 have in common. This orderly listing, with each factor having its own column, will do most of the work for me.
Then the GCF is 2 × 2 × 7 = 28. On the other hand, the Least Common Multiple, the LCM, it is the smallest number that contains both 84 and 140 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.
Then the LCM is 2 × 2 × 3 × 5 × 7 = 420. 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 down all the factors, regardless of how many or few values contained that factor in their listings. INTEGERS DEFINITION- The number line goes on forever in both directions. This is indicated by the arroes.
- Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line.
- Whole numbers less than zero are called negative integers. thes numbers are to the left of zero on the number line.
- The integer zero is neutral. It is neither positive nor negative.
- The sign of an integer is positive (+) or negative (-), except zero, wich has no sign.
- Two integers are opposites if they are each the same distance away from zero, but opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above, +3 and -3 are labelled as opposites.
ABSOLUTE VALUE OF AN INTEGERThe number of units a number is from zero on the number line. The absolute value of a number is always a positive number (or zero). We specify the absolute value of a numbere n by writing n in between two vertical bars: | n | Examples: | 6 | = 6; | -12 | = 12; | 0 | = 0; | 1234 | = 1234; | -1234 | = 1234 INTEGERS: OPERATIONS WITH SIGNED NUMBERSBefore you do ANY computation, determine the OPERATION! Then follow the instructions for THAT operation. ADDITIONDo the numbers have the SAME SIGN?
Either way Keep the sign of the "LARGE number". "LARGER" is used here as a quick (but mathematically imprecise) way to describe the integer with the greater Absolute Value (ie. distance from zero). In each of the examples above, the SECOND integer has a greater Absolute Value. SUBTRACTIONFirst, change the SUBTRACTION problem to an ADDITION problem: First, copy the problem exactly ( -6 ) - ( +2 ) = 1. The first number stays the same ( -6 ) 2. change the operation ( -6 ) + 3. Switch the NEXT SIGN ( -6 ) + ( -2 ) 4. Follow the rules for addition ( -6 ) + ( -2 ) = ( -8 ) Subtract means: ( +2 ) - ( -6 ) Add the opposite: ( +2 ) + ( +6 ) = ( +8 ) Subtract means: ( -7 ) - ( -3 ) Add the opposite: ( -7 ) + ( +3 ) = ( -4 ) Subtract means: ( +4 ) - ( +9 ) Add the opposite: ( +4 ) + ( -9 ) = ( -5 ) MULTIPLICATION OR DIVISIONFirst, DO the multiplication or division. Then determine the sign: count the number of negative signs ... Are there an EVEN number of negative signs? - YES (an EVEN number of negative signs) the answer is POSITIVE
- NO (an ODD number of negative signs) the answer is NEGATIVE
First, copy the problem exactly: ( -2 ) · ( -4 ) · ( -6 ) = Do the multiplication or division. 2 · 4 · 6 = 48 Count the number of negative signs... Determine the signs of the answer: Are there an EVEN number of negatives? If YES, the answer is POSITIVE, otherwise, the answer is NEGATIVE. A total of TRHEE NEGATIVES, trhee is NOT EVE (it's odd) So the answer is NEGATIVE ( -2 ) · ( -4 ) · ( -6 ) = ( -48 ) Examples (4) : (2) · (6) = 12 A total of ZERO NEGATIVES So the answer is POSITIVE (4) : (-2) · (6) = -12 A total of ONE NEGATIVE One is NOT EVEN (it's odd) So the answer is NEGATIVE (-4) : (2) · (-6) = 12 A total of TWO NEGATIVES Two is EVEN So the answer is POSITIVE ORDER OF OPERATIONSProblem: Evaluate this arithmetic expression: 18 + 36 : 3 ^{2}In the last year, we learned how to evaluate an arithmetic expression with more than one operation according to the followings rules: Rule 1: Simplify all operations inside parentheses Rule 2: Perform all multiplications and divisions, working from left to right Rule 3: Perform all additions and subtractions, working from left to right However, the problem above includes an exponent, so we cannot solve it without revising our rules. Rule 1: Simplify all operations inside parentheses Rule 2: Simplify all exponents, working from left to right. Rule 3: Perform all multiplications and divisions, working from left to right Rule 4: Perform all additions and subtractions, working from left to right We can solve the problem above using our revised order of operations. Evaluate this arithmetic expresion: 18 + 36 : 3 ^{2}Simplify all exponents (Rule 2) 18 + 36 : 9 Division (Rule 3) 18 + 4 Addition (Rule 4) 22 POWERS AND ROOTS. SQUARES, CUBES AND THEIR ROOTSSquaring a number 3 ^{2} means "3 squared‟, or 3 · 3. The small 2 is an index number, or power. It tells us how many times we should multiply 3 (base) by itself. Similarly 7^{2} means "7 squared‟, or 7 · 7. And 10^{2} means "10 squared‟, or 10 · 10. So, 1 ^{2} = 1 · 1 = 1; 2^{2} = 2 · 2 = 4 ; 3^{2} = 3 · 3 = 9; 4^{2} = 4 · 4 = 16; 5^{2} = 5 · 5 = 25. 1, 4, 9, 16, 25… are known as square numbers. When the index number is greater than three we say 'to the power of'. For example: 3 ^{7} is 'three to the power of seven',4 ^{5} is 'four to the power of five'.Exponent Properties 1. Product of like bases: To multiply powers with the same base, add the exponents and keep the common base.2 ^{2} · 2^{3} = 2 · 2 · 2 · 2 · 2 = 2^{5}x ^{m} · x^{n} = x^{m+n} To divide powers with the same base, subtract the exponents and keep the common base. 2. Quotient of like bases: Write two examples with numbers and variables. 3. Power to a power: To raise a power to a power, keep the base and multiply the exponents.Write two examples with numbers and variables. 4. Product to a power: To raise a product to a power, raise each factor to the power. Write two examples with numbers and variables. 5. Quotient to a power: To raise a quotient to a power, raise the numerator and the denominator to the power. Write two examples with numbers and variables. Exponents one and zeroNotice that 3 ^{1} is the product of only one 3, which is evidently 3.Also note that 3 ^{5} = 3·3^{4}. Also 3^{4} = 3·3^{3}. Continuing this trend, we should have 3 ^{1} = 3 · 3^{0}.Another way of saying this is that when n, m, and n − m are positive (and if x is not equal to zero), one can see by counting the number of occurrences of x that Extended to the case that n and m are equal, the equation would read since both the numerator and the denominator are equal. Therefore we take this as the definition of x ^{0}.Negative exponentsA negative exponent means to divide by that number of factors instead of multiplying. So 4 ^{-3} is the same as 1/4^{3}, and x^{-3} =1/ x^{3}As you know, you can't divide by zero, so x ≠ 0 when x=0, x ^{-n} is undefined.SCIENTIFIC NOTATIONFor very large or very small numbers, it is sometimes simpler to use "scientific notation" (so called, because scientists often deal with very large and very small numbers). The format for writing a number in scientific notation is fairly simple: First digit of the number followed by the decimal point and then all the rest of the digits of the number, times (10 to an appropriate power). The conversion is fairly simple.Example: Write 12400 in scientific notation. This is not a very large number, but it will work nicely for an example. To convert this to scientific notation, I first write "1.24". This is not the same number, but (1.24)(10000) = 12400 is, and 10000 = 10 ^{4}. Then, in scientificnotation, 12400 is written as 1.24 · 10 ^{4}.SQUARE ROOTS The opposite of a square number is a square root. We use the symbol √ to mean square root. So we can say that 4 = 2 ( 4 is called radicand) and 25 = 5. and ( –5 ) · ( –5 ) is also 25. So, in fact, 4 = 2 or –2. And 25 = 5 or – 5. Remember that every positive number has two square roots. CUBING A NUMBER 1 ^{3} = 1 · 1 · 1 = 1; 2 · 2 · 2 means "2 cubed‟, and is written as 2^{3}.3 ^{3} = 3 · 3 · 3 = 27; 4^{3} = 4 · 4 · 4 = 64; 5^{3} = 5 · 5 · 5 = 1251, 8, 27, 64, 125… are known as cube numbers. CUBE ROOTS The opposite of a cube number is a cube root. We use the symbol 3 to mean cube root. So is 3 8 = 2 and3 27 is 3. |