BASIC MATH 6 -
ZERO TO THE ZERO POWER =
(AND ZERO TO ANY POWER OTHER THAN ZERO IS 1)
Logarithms: a common logarithm (or log) is just an exponent of ___________. So, the log of 1000 is asking for the ______________that goes with 10 to make 1000. Sometimes logs indicate exponents of other numbers, called bases. Ex.: log to the base 2 of 16 means log2 16 = 4 (what exponent goes with 2 to make 16)
10, exponent,
ZERO TO ZERO = 1 , PART 1 BY ASSAD EBRAHIM
Context of the Debate: Continuous Mathematics The three choices for the value of 0^0 appear because x^y, as a function of two continuous variables, is discontinuous at (0,0) and takes three different values depending on the direction of approach to the discontinuity: Fixing y=0, we have x^0=1 for all x \neq 0. (Proof: x^0 = x^{(1-1)} = x^1 x^{-1} = x/x = 1, each statement holding for all x \neq 0). Indeed, x^y \rightarrow 1 as x \rightarrow 0, approaching from left or right, with y=0. (This was Euler's reason.) Fixing x=0, we have 0^y=0 for y >0. (When y < 0 we have division by zero which is undefined in the reals and +\infty in the extended reals). Taking limits, x^y \rightarrow 0 as y \searrow 0, approaching from above only, with x=0. Fixing x=0, we have an undefined value when y < 0 due to division by zero.
How to find square to the nearest...hundredth. Ex: square root of 14:
Find two perfect squares that 14 lies between. 9 and 16. Square roots are 3 and 4, respectively. 14 is closer to 16, so choose about 3.7 squared and 3.8 squared. 3.7 squared = 13.69; 3.8 squared = 14.44. 14 is about half way between, so try 3.75 squared, and 3.74 squared. 14 is closer to 3.74 squared, so square root of 14 to the nearest hundredth is (two squiggly lines for approximately) 4.74
ZERO DIVIDED BY ZERO = IN....__________________
INDETERMINATE Essentially, zero times x = zero. The solution for x could be any real number. So, zero divided by zero is "indeterminate" note: REAL NUMBERS: include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the TRANSCENDENTAL NUMBERS, such as π (3.14159265...).
***Why is any non-zero number raised to the zero power 1? And if we raise 0 to the 0 power, will it also be 1? Let's begin by examining the division of values with exponents: 2 to 5 / 2 to 3 = 2 to ________ = 2 to 2 = 4 ( 2x2x2x2x2 / 2x2x2 -- 3 twos cancel out); so, the rule is n to x / n to y = n to x-y. So now set up proof for x to 0 = 1: 2 to 1 / 2 to 1 = _____; given exponent rule above, may then also say that 2 to 1 / 2 to 1 = _____ to 1-1 = 1; So, then 2 to 0 = _____. (question: really? what if exponents meant division, or addition, etc.? is the problem not of nature, but of man's applied mathematical architecture?)
Let's begin by examining the division of values with exponents: 2 to 5 / 2 to 3 = 2 to 5-3 = 2 to 2 = 4 ( 2x2x2x2x2 / 2x2x2 -- 3 twos cancel out); so, the rule is n to x / n to y = n to x-y. So now set up proof for x to 0 = 1: 2 to 1 / 2 to 1 = 1; given exponent rule above, may also then say that 2 to 1 / 2 to 1 = 2 to 1-1 = 1; So, then 2 to 0 = 1. ***? God created something out of nothing? Hebrews 11:3 By faith we understand that the universe was created by the word of God, so that what is seen was not made out of things that are visible." (question: really? what if exponents meant division, or addition, etc.? )
Exponential Form vs Logarithmic Form 3 to 2 = 9 10 to 5 = 100,000 2 to -5 = 1/32
Logarithmic Form: Log3 9 = 2 Log 100,000 = 5 (base 10 is assumed) Log 2 1/32 = -5
ZERO TO ZERO = 1, PART 2, BY ASSAD EBRAHIM
Principles for a Decision in Mathematics: Extension and Consistency In mathematics, when there is more than one choice, a decision is typically made by extending an existing precedent to maintain consistency with the evidence that is already accumulated and accepted. An elementary example is the way ordinary multiplication is extended from two positive numbers to a positive and a negative number, then to two negative numbers, i.e. (-1)(-1) = 1. "Minus times minus is plus. The reason for this we need not discuss!" — W.H. Auden Empirically, multiplication of two positive numbers has a well-defined, tangible meaning as repeated addition. This meaning holds when one of the numbers is negative. But when both are negative, the empirical meaning fails. For the mathematician, declaring something to be undefined (throwing an error) means a loss of efficiency because every instance now has to be checked for the undefined case, and this must be treated separately. If a definition could be found that remains consistent with all other empirically obtained rules, and if that definition means that calculation can proceed indifferent to the decision, then that is a big win. The consistency in this particular case is the distributivity of multiplication over addition, a law which, for positive numbers, can be accepted on entirely empirical grounds. (See the footnote for the full argument.1.)
# DIVIDED BY ZERO IS: UN...______________. PROOF:
UNDEFINED (no possible solution) SEE PIC. EX 2/0 ESSENTIALLY SAYS 0 X WHAT? WILL EQUAL 2. NO POSSIBLE ANSWER. (says: "0 is a factor of 2 x (how many) times?")
Squares & Square Roots: 1. "square" means 2. Note: "Principal Square Root": 6x6 = 36; also -6 x -6 = 36. So -6 is a square root; however, 6, the positive square root is the principal square root.
When a number is multiplied by itself, the product is the square of the number. incidentally, 36 is the square of 6.
Cubes & Cube Roots: The cube: The product of _____ ______ factors is the cube of the factor. The cube root of a number (n) is ...
cube root: the number (x) whose cube (x to 3) = the number (n)
Example of working of negative exponents: Phosphorus-32 does not stay _________ forever; its half-life (time taken for half of the element to decay) is 14 days. Day0: 1 g, day 14: _____ ____
day 14: 1/2 g = 2 to -1 g (or 1 over 2 to 1); incidentaly, day 28: 1/4 g = 2 to -2
Negative Exponents: produce ____________.
fractions; Note pattern for proof: 4 to 3 = 64, 4 to 2 = 8, 4 to 1 = 4, 4 to 0 = 1, Notice how value decreases by 1/4 with every decrease by one in the exponent; thus, the pattern continues into fractions, at the rate of 1/4 per loss of one in the exponent: 4 to the -1 = 1/4, 4 to -2 = 1/16, 4 to - 3 = 1/64
If a number is not a perfect square, its square root is not an _____________.
integer
Irrational Roots: if a whole number is not a perfect square (ex: square of 2 or 11), its square root is __________________. Means that the whole number can't be ______________ as a ratio of _____________. It is represented by a _____-_______________decimal that does not end.
irrational, represented, integers, non-repeating
Cube root of a positive number is always __________. (Incidentally, when you start taking roots of negative numbers, you get into imaginary numbers.)
positive
FRACTIONAL EXPONENTS: ex 8 to 1/3 power
the base, 8 is a PRODUCT of what factor multiplied 3 times (3 from the denominator)? 2. To fulfill the the fractional exponent function, now put 2 to the 1 (numerator) power. Answer: 2.
FRACTIONAL EXPONENT: ex 81 to 3/4 =
x to 4 power = 81? x=3. Then, 3 to 3rd power (from numerator of fractional exponent) = 27. 27 is answer. (Note: 81 to 1 is 81, so 81 to 3/4 must be less than 81, but greater than 1, since 81 to 0 is 1. Since we're dealing with exponents (POWERS), we determine 81 to 3/4 as demonstrated above.
