review think mathematics 1A ( chapter 1-4 ) part 2

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Number Theory and Operations

Understanding the relationships between different number representations is crucial in mathematics. This includes comparing fractions, decimals, and exponents.

Ordering Numbers

When ordering numbers in different forms, it's often helpful to convert them to a common representation. For example, when comparing fractions and decimals:

  • Convert fractions to decimals: $$\frac{3}{5} = 0.6$$
  • Evaluate exponents: $$\left(\frac{3}{5}\right)^2 = 0.36$$

Irrational Numbers

Some numbers, like $$\pi$$, are irrational. This means they cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal representations. Approximations like $$\frac{22}{7}$$ or 3.142 are often used in calculations, but they are not identical to the true value of $$\pi$$.

Algebraic Expressions and Operations

Evaluating Expressions

When given an algebraic expression like $$p=-3^a-b^3$$, we can find its value by substituting known values for variables and following the order of operations (PEMDAS).

Expanding and Simplifying

To expand and simplify expressions, we use distributive property and combine like terms. For example:

$$5(x+6y)-2[4(3y-x)+7x]$$

Expand each bracket, then combine like terms.

Factoring

Expressions can often be rewritten in factored form. For instance, $$-8axy+\frac{7a^2y}{5}$$ can be factored as $$\frac{hay}{5}(7a+kx)$$ by finding common factors.

Number Systems and Decimals

Fractions can be expressed as terminating or recurring decimals. To convert a fraction to a decimal, divide the numerator by the denominator:

$$\frac{11}{15} = 0.7333333...$$ (recurring)

Factors and Multiples

Understanding factors and multiples is essential in number theory.

  • Factors are numbers that divide evenly into another number.
  • The Highest Common Factor (HCF) is the largest factor shared by two or more numbers.
  • The Lowest Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers.

Prime Factorization

Every number can be expressed as a product of prime factors. This is useful for finding HCF, LCM, and determining if a number is a perfect square or cube.

For example, to express 198 as a product of prime factors:

$$198 = 2 \times 3^2 \times 11$$

Geometry and Measurement

In geometry, we often deal with three-dimensional shapes like cuboids. The volume of a cuboid is given by:

$$V = l \times w \times h$$

Where $$V$$ is volume, $$l$$ is length, $$w$$ is width, and $$h$$ is height.

Advanced Number Theory

Perfect Squares and Cubes

A number is a perfect square if it's the product of an integer with itself. Similarly, a perfect cube is the product of an integer with itself twice.

Index Notation

Index notation is a compact way to express repeated multiplication. For example:

$$2^3 = 2 \times 2 \times 2 = 8$$

This notation is particularly useful when dealing with prime factorizations and exponents.


Quiz 1


Quiz 2
  1. Write the following numbers in order of size, starting with the smallest. $$0.62, \quad \frac{3}{5}, \quad\left(\frac{3}{5}\right)^{2}, \quad \frac{3}{4}$$
  2. Are the numerical values of $$\pi$$, $$\frac{22}{7}$$ and 3.142 identical? Explain your answer.
  3. It is given that $$p=-3^{a}-b^{3}$$. Find the value of $$p$$ when $$a=4$$ and $$b=-2$$.
  4. Expand and simplify $$5(x+6 y)-2[4(3 y-x)+7 x]$$.
  5. The expression $$-8 a x y+\frac{7 a^{2} y}{5}$$ can be written in the form $$\frac{h a y}{5}(7 a+k x)$$. Find the values of $$h$$ and $$k$$.
  6. Can $$\frac{11}{15}$$ be expressed as a terminating decimal or a recurring decimal? Show how to, or explain why not.
  7. A student wrote, "If 2 and 8 are factors of a number, then 16 is also a factor of that number." Do you agree with the student? Explain your answer.
  8. Shan has 315 one-centimetre cubes. She arranges all of the cubes into a cuboid. The perimeter of the top of the cuboid is 24 cm. Each side of the cuboid has a length greater than 3 cm. Find the height of the cuboid.
  9. Simplify $$\frac{1}{4} a \times(-6 b)+3 b a-\left(-\frac{1}{2} b\right) \div\left(-\frac{1}{7 a}\right)$$.
  10. (a) Express 198 as the product of its prime factors.
    (b) The number $$198k$$ is a perfect square. Find the smallest positive integer value of $$k$$.
    (c) $$x$$ is a number between 200 and 300. The highest common factor of $$x$$ and 198 is 33. Find the smallest possible value of $$x$$.
  11. (a) Use prime factors to explain why $$68 \times 153$$ is a perfect square.
    (b) The number $$68k$$ is a perfect cube. Find the smallest positive integer value of $$k$$.
    (c) Find the lowest common multiple of 68 and 153, expressing your answer in index notation.
    (d) One-third of the product of 68 and 108 is the same as $$2^{x} \times 3^{y} \times 17^{z}$$. Find the values of $$x$$, $$y$$ and $$z$$.

 
Answers

1. The numbers in order from smallest to largest are:

$$\left(\frac{3}{5}\right)^{2}, \quad 0.62, \quad \frac{3}{5}, \quad \frac{3}{4}$$

2. No, the numerical values of $$\pi$$, $$\frac{22}{7}$$, and 3.142 are not identical. $$\pi$$ is an irrational number that cannot be expressed as a fraction or terminating/repeating decimal. $$\frac{22}{7}$$ and 3.142 are rational approximations of $$\pi$$, but not equal to its exact value.

3. When $$a=4$$ and $$b=-2$$, the value of $$p$$ is -73.

4. $$5(x+6y)-2[4(3y-x)+7x]$$ expands and simplifies to $$-x+6y$$.

5. The values of $$h$$ and $$k$$ are $$h=1$$ and $$k=-40$$.

6. $$\frac{11}{15}$$ can be expressed as a recurring decimal. As a decimal, it is equal to $$0.7\overline{33}$$, where the digits 33 repeat infinitely.

7. The answer is No, I do not agree with the student's statement. Just because 2 and 8 are factors of a number, it does not necessarily mean that 16 must also be a factor of that number. For example, take the number 24. Both 2 and 8 are factors of 24 (since 24 ÷ 2 = 12 and 24 ÷ 8 = 3), but 16 is not a factor of 24 (since 24 ÷ 16 does not result in a whole number). The student may be confusing the fact that 16 is a multiple of both 2 and 8 with the idea that 16 must also be a factor of any number that has 2 and 8 as factors, but this reasoning is incorrect. Therefore, the student’s conclusion is flawed.

8. The height of the cuboid is 9 cm.

9. $$-2ab$$.

10. (a) $$198 = 2 \times 3^2 \times 11$$
(b) The smallest positive integer value of $$k$$ is 22.
(c) The smallest possible value of $$x$$ is 231.

11. (b) The smallest positive integer value of $$k$$ is 578.
(c) The lowest common multiple of 68 and 153 is $$2^2 \times 3^2 \times 17$$.
(d) $$x = 4$$, $$y = 2$$, and $$z = 1$$.

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