Numbers

The number 81,000,000,000,000,000,000,000 is written in scientific notation as 8.1 × 10^22. It is a large integer, commonly used in contexts such as astronomy, finance, and data measurement. In words, it can be expressed as eighty-one sextillion.

Binary numbers were first fully described by the ancient Indian mathematician Pingala around the 2nd century BCE, who used a binary system in his work on Sanskrit prosody. However, the modern binary numeral system was developed by Gottfried Wilhelm Leibniz in the 17th century, who introduced it in 1679 and connected it to philosophical concepts. Leibniz’s work laid the foundation for the binary system used in digital computing today.

Let the two positive real numbers be ( x ) and ( y ). The constraint given is ( x + 4y = 160 ). To maximize the product ( P = xy ), we can express ( y ) in terms of ( x ): ( y = \frac{160 - x}{4} ). Substituting this into the product gives ( P = x \left(\frac{160 - x}{4}\right) = \frac{160x - x^2}{4} ), which is a quadratic function in ( x ) that opens downwards. Maximizing ( P ) occurs at the vertex, ( x = 80 ), leading to ( y = 20 ), thus the numbers are ( 80 ) and ( 20 ).

101280000

Oh, dude, that's like a whole lotta zeros! That number is one followed by 100 zeros, which in words is a "one" followed by "googol." It's like a super big number, but hey, who's counting, right?

Yes, the range of a function can include negative numbers, depending on the function itself. For example, the range of the function ( f(x) = x^2 - 4 ) is all values greater than or equal to -4, which includes negative numbers. Similarly, linear functions with a negative slope can produce negative outputs. Whether the range includes negative numbers ultimately depends on the specific function being analyzed.

No such Roman numeral as 'mmdccxlviii'.

First of all Roman numerals are ALWAYS written in CAPITAL letters.

Your numeral should be written as ' MMDCCXLVIII '.

To translate in Arabic (Modern) numbers.

M = 1000

MM = 1000 + 1000 = 2000

D = 500

C = 100

CC = 100 + 100 = 200

L = 50

X =10

XL = 50 - 10 = 40 NOte , when a letter of lesser value is placed immediately to the left of a letter with a higher value, then it is subtract.

V = 5

I = 1

III = 1 + 1 + 1 = 3

Hence

MMDCCXLVIII =

2000 + 500 + 200 + 40 + 5 + 3 = 2748.

There are 50 numbers from 1 to 100 that can be divided by 2. This is because every second number in that range is even, starting from 2 and going up to 100. The sequence of even numbers is 2, 4, 6, ..., 100, which can be represented as 2n where n ranges from 1 to 50. Thus, the total is 50 even numbers.

Ah, that's a big number! It's called one hundred quattuordecillion in the short scale naming system. Just imagine all the happy little zeroes lining up together, creating a beautiful, vast number that stretches far and wide across the canvas of mathematics.

Six and a half is a rational number because it can be expressed as a fraction. Specifically, it can be written as ( \frac{13}{2} ). Rational numbers are defined as numbers that can be expressed as the quotient of two integers, and since 6.5 meets this criterion, it is rational.

The sum of all positive integers is often described as infinite, as the series 1 + 2 + 3 + ... continues without bound. Mathematically, this can be expressed as diverging to infinity. However, in certain contexts, such as in analytic number theory, a technique called regularization may assign a finite value to this series, often referred to as -1/12, but this is not a conventional sum. In standard arithmetic, the sum remains infinite.

When you take away a negative number from a positive number, it's equivalent to adding that negative number's absolute value to the positive number. For example, if you have 5 and subtract -3, the calculation becomes 5 + 3, resulting in 8. This principle applies because subtracting a negative effectively increases the original number.

The pairs of numbers that multiply to equal 32 include (1, 32), (2, 16), (4, 8), and (8, 4). Additionally, negative pairs such as (-1, -32), (-2, -16), and (-4, -8) also multiply to yield 32. These pairs represent all the integer combinations that result in the product of 32.

The number 3,000,000,000,000,000,000,000,000,000,000 can be expressed in scientific notation as 3 x 10^30. It is a very large number, equivalent to three octillion in the short scale commonly used in the United States and modern English-speaking countries. Such large numbers are often used in fields like astronomy, physics, and data science to represent quantities like particles, astronomical distances, or data storage capacities.

A negative even number is an integer that is both less than zero and divisible by two. Examples include -2, -4, and -6. These numbers fall on the number line to the left of zero and have no remainder when divided by two.

Three billion sixty million sixty-six thousand nine hundred in numerical form is 3,060,066,900.

77 is not a prime number. 36 is not a prime number. 77 and 36 are mutually prime as the only common factor they have is 1.

The number 70009 in word form is seventy thousand nine.

Natural numbers are a subset of whole numbers. They typically include the positive integers (1, 2, 3, ...) and, in some definitions, zero (0). Therefore, natural numbers can be considered whole numbers when zero is included in the definition. Overall, natural numbers are always whole numbers, but whether zero is included depends on the context.

To write 72.5 trillion in numerical form, you express it as 72,500,000,000,000. In scientific notation, it can be represented as 7.25 × 10^13. This format helps convey large numbers more succinctly, especially in scientific and financial contexts.

The numbers to the right of zero on the number line are all positive numbers. This includes whole numbers like 1, 2, 3, and so on, as well as fractions and decimals such as 0.1, 1.5, and 3.14. Positive numbers represent quantities greater than zero.

A negative significance number typically indicates that there is an inverse relationship between the variables being measured, suggesting that as one variable increases, the other decreases. In statistical analyses, it may imply that the effect or impact is in the opposite direction than expected or desired. This can be important for understanding correlations or effects in various fields, such as economics or social sciences. In some contexts, it may also suggest a need for further investigation into the underlying factors causing the negative relationship.

Yes, 5.323232 is a rational number because it can be expressed as a fraction of two integers. Specifically, it can be represented as ( \frac{5323232}{1000000} ) after recognizing that the decimal repeats. All numbers that can be expressed in this way, including those with repeating or terminating decimals, are classified as rational numbers.

In the set of natural numbers, which typically starts from 0 or 1 and continues indefinitely, there is no natural number that has no successor. This means every natural number has a next number, and there is no "last" natural number because the set goes on infinitely. Therefore, the concept of a last natural number does not exist.