Proofs and Solutions
... 3. Consider the statement “For all real numbers x ≥ 1, x2 ≥ x”. Explain why this statement can not be proven by induction. Solution. Unlike the previous problem, here there is a well-defined first number for which the theorem applies (x = 1). However, another requirement for using induction is that ...
... 3. Consider the statement “For all real numbers x ≥ 1, x2 ≥ x”. Explain why this statement can not be proven by induction. Solution. Unlike the previous problem, here there is a well-defined first number for which the theorem applies (x = 1). However, another requirement for using induction is that ...
Answer - American Computer Science League
... incremented; otherwise S is incremented. Thus, after the first loop, C has a value of 25, and S also has a value of 25. The second loop, on K, also considers the numbers between 1 and 50. If the number is divisible by 3 and also not divisible by 2 (in other words, a multiple of 3 that is not a multi ...
... incremented; otherwise S is incremented. Thus, after the first loop, C has a value of 25, and S also has a value of 25. The second loop, on K, also considers the numbers between 1 and 50. If the number is divisible by 3 and also not divisible by 2 (in other words, a multiple of 3 that is not a multi ...
Real Numbers and Closure
... The set of rational numbers includes all integers and all fractions. Like the integers, the rational numbers are closed under addition, subtraction, and multiplication. Furthermore, when you divide one rational number by another, the answer is always a rational number. Another way to say this is tha ...
... The set of rational numbers includes all integers and all fractions. Like the integers, the rational numbers are closed under addition, subtraction, and multiplication. Furthermore, when you divide one rational number by another, the answer is always a rational number. Another way to say this is tha ...
Complex Numbers Notes 1. The Imaginary Unit We use the symbol i
... The modulus, or length, of a complex no. is the distance from the origin to the point representing the complex number on an Argand diagram. We use 2 lines either side of a complex no. to represent the modulus. E.g the modulus of 3 - 4i is written |3 - 4i|. This is then the distance from (0,0) to (3, ...
... The modulus, or length, of a complex no. is the distance from the origin to the point representing the complex number on an Argand diagram. We use 2 lines either side of a complex no. to represent the modulus. E.g the modulus of 3 - 4i is written |3 - 4i|. This is then the distance from (0,0) to (3, ...
Multiplication - Mickleover Primary School
... ‘You have 3 lollies and your friend gives you 3 more. How many do you have altogether? ...
... ‘You have 3 lollies and your friend gives you 3 more. How many do you have altogether? ...
Transcendental numbers and zeta functions
... In 1966, Baker [2] derived a generalization of this theorem. He showed that if α1 , ..., αn , β0 , β1 , ...βn ∈ Q with α1 · · · αn β0 6= 0, then eβ0 α1β1 · · · αnβn is transcendental. He proved this by showing that the linear form β0 + β1 log α1 + · · · + βn log αn is either zero or transcendental. ...
... In 1966, Baker [2] derived a generalization of this theorem. He showed that if α1 , ..., αn , β0 , β1 , ...βn ∈ Q with α1 · · · αn β0 6= 0, then eβ0 α1β1 · · · αnβn is transcendental. He proved this by showing that the linear form β0 + β1 log α1 + · · · + βn log αn is either zero or transcendental. ...
Algebra 2 - peacock
... real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as . You can use the imagin ...
... real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as . You can use the imagin ...
I can solve problems involving increasingly harder fractions to
... I can solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number. I can solve simple measure and money problems involving fractions and decimals to two decimal places. ...
... I can solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number. I can solve simple measure and money problems involving fractions and decimals to two decimal places. ...
