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Chapter 1: Problem 1

Using only mental arithmetic: (a) Compute for yourself, and learn by heart, the times tables up to \(9 \times 9\). (b) Calculate instantly: (i) \(0.004 \times 0.02\) (ii) \(0.0008 \times 0.07\) (iii) \(0.007 \times 0.12\) (iv) \(1.08 \div 1.2\) (v) \((0.08)^{2}\) Multiplication tables are important for many reasons. They allow us to appreciate directly, at first hand, the efficiency of our miraculous place value system \(-\) in which representing any number, and implementing any operation, are reduced to a combined mastery of (i) the arithmetical behaviour of the ten digits \(0-9\), and (ii) the index laws for powers of 10 . Fluency in mental and written arithmetic then leaves the mind free to notice, and to appreciate, the deeper patterns and structures which may be lurking just beneath the surface.

Short Answer

Expert verified
(i) 0.00008, (ii) 0.000056, (iii) 0.00084, (iv) 0.9, (v) 0.0064.

Step by step solution

01

Times Tables

Ensure you are familiar with the multiplication tables up to \(9 \times 9\). Use repetition to memorize these tables, as they are fundamental in performing quick mental arithmetic.
02

Small Decimal Multiplication (i)

To calculate \(0.004 \times 0.02\), note that it is equivalent to multiplying 4 by 2 and then adjusting for decimals. Multiply: \(4 \times 2 = 8\). There are 5 digits after the decimal points in total (3 from 0.004 and 2 from 0.02), resulting in \(0.004 \times 0.02 = 0.00008\).
03

Small Decimal Multiplication (ii)

Calculate \(0.0008 \times 0.07\): Multiply \(8 \times 7 = 56\). The number of decimal places from both numbers is 5 (4 from 0.0008 and 1 from 0.07). Thus, \(0.0008 \times 0.07 = 0.000056\).
04

Small Decimal Multiplication (iii)

Calculate \(0.007 \times 0.12\): \(7 \times 12 = 84\). The total number of decimal places is 5 (3 from 0.007 and 2 from 0.12). Hence, \(0.007 \times 0.12 = 0.00084\).
05

Decimal Division

Calculate \(1.08 \div 1.2\): Rewrite as \(\frac{1.08}{1.2}\). Eliminate the decimals by multiplying both numerator and denominator by 10, yielding \(\frac{10.8}{12} = 0.9\).
06

Squaring a Decimal

Calculate \((0.08)^2\): First, calculate \(8^2 = 64\). Since there are a total of 4 decimal places to consider, \((0.08)^2 = 0.0064\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

multiplication tables
Mastering multiplication tables up to \(9 \times 9\) is a fundamental skill in mental arithmetic and mathematics as a whole. These tables form the basis for more complex calculations and enhance number sense.
  • By memorizing multiplication tables, you gain faster problem-solving abilities.
  • They can be practiced by repetition and visualization, attaching a mental image with each product to make recall easier.
  • This foundational knowledge also aids in understanding more advanced arithmetic, allowing you to see patterns and relationships among numbers.
Learning multiplication tables by heart not only improves arithmetic speed but also allows you to tackle problems with confidence and ease. Building this foundation is essential for more complex mathematical operations, where you'll often need to recall these simple products quickly.
decimal multiplication
Decimal multiplication requires careful attention to the number of decimal places involved in the numbers being multiplied. This understanding helps in determining the correct placement of the decimal point in the final answer. When multiplying decimals, follow these steps:
  • Ignore the decimal points initially and multiply the numbers as if they were whole numbers.
  • Count the total number of decimal places in both original numbers. For example, in multiplying \(0.004\) and \(0.02\), there are 5 decimal places in total.
  • The product of the whole numbers is then adjusted by placing the decimal point in the result at the correct position according to the number of decimal places counted.
Understanding these steps helps you solve decimal multiplication by focusing first on the simple multiplication of numbers, and then accurately placing the decimal point, ensuring correct and efficient results.
decimal division
Decimal division involves a slightly different process than regular division, as it focuses on making the number part of the division free of decimals initially. Here's how decimal division typically works:
  • Express the division of decimals such as \(1.08 \div 1.2\) as a fraction \(\frac{1.08}{1.2}\).
  • To eliminate decimal points, multiply both the numerator and the denominator by a power of 10 that eliminates the decimals (in this case, multiply both by 10 to turn it into \(\frac{10.8}{12}\)).
  • Perform the division as with whole numbers, which is straightforward and yields the quotient.
By managing the decimals upfront, decimal division becomes just as manageable, ensuring accuracy without complicating the arithmetic. This method underscores the importance of maintaining the balance between numerator and denominator when handling decimals.
place value system
The place value system is a critical concept underpinning all arithmetic operations, and it is a key reason why learning basic multiplication tables enhances arithmetic proficiency.
  • In our base-10 system, each digit represents a different value depending on its position or place in the number.
  • When multiplying or dividing numbers, particularly with decimals, the role of place value determines where the decimal point is placed in the final answer.
  • This concept ensures that each digit is considered correctly, especially in calculations involving powers of 10, such as moving the decimal point to the left or right.
By understanding the place value system, you gain insights into how numbers are constructed and manipulated, making operations easier and more intuitive. This foundational knowledge helps in efficiently performing arithmetic operations and appreciating the structure and elegance of our numeral system.

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Most popular questions from this chapter

(Pythagoras' Theorem) Let \(\triangle A B C\) be a right angled triangle, with a right angle at \(C .\) Draw the squares \(A C Q P, C B S R,\) and \(B A U T\) on the three sides, external to \(\triangle A B C\). Use the resulting diagram to prove in your head that the square \(B A U T\) on \(B A\) is equal to the sum of the other two squares by: \- drawing the line through \(C\) perpendicular to \(A B\), to meet \(A B\) at \(X\) and \(U T\) at \(Y\) \- observing that \(P A\) is parallel to \(Q C B\), so that \(\triangle A C P\) (half of the square \(A C Q P,\) with base \(A P\) and perpendicular height \(A C)\) is equal in area to \(\triangle A B P\) (with base \(A P\) and the same perpendicular height) \- noting that \(\triangle A B P\) is SAS-congruent to \(\triangle A U C,\) and that \(\triangle A U C\) is equal in area to \(\triangle A U X\) (half of rectangle \(A U Y X,\) with base \(A U\) and height \(A X)\) \- whence \(A C Q P\) is equal in area to rectangle \(A U Y X\) \- similarly \(B C R S\) is equal in area to \(B T Y X\). The proof in Problem 18 is the proof to be found in Euclid's Elements Book 1, Proposition 47. Unlike many proofs, \- it is clear what the proof depends on (namely SAS triangle congruence, and the area of a triangle), and \- it reveals exactly how the square on the hypotenuse \(A B\) divides into two summands \(-\) one equal to the square on \(A C\) and one equal to the square on \(B C\).

The three integers \(a=3, b=4, c=5\) in the Pythagorean triple (3,4,5) form an arithmetic progression: that is, \(c-b=b-a\). Find all Pythagorean triples \((a, b, c)\) which form an arithmetic progression \(-\) that is, for which \(c-b=b-a\)

(a)(i) Expand \((a+b)^{2}\) and \((a+b)^{3}\). (ii) Without doing any more work, write out the expanded forms of \((a-b)^{2}\) and \((a-b)^{3}\). (b) Factorise (i) \(x^{2}+2 x+1\) (ii) \(x^{4}-2 x^{2}+1\) (iii) \(x^{6}-3 x^{4}+3 x^{2}-1\). (c)(i) Expand \((a-b)(a+b)\). (ii) Use (c)(i) and (a)(i) to write down (with no extra work) the expanded form of $$ (a-b-c)(a+b+c) $$ and of $$ (a-b+c)(a+b-c) $$ (d) Factorise \(3 x^{2}+2 x-1\).

(a) Use Pythagoras' Theorem in a square \(A B C D\) of side 1 to show that the diagonal \(A C\) has length \(\sqrt{2}\). Use this to work out in your head the exact values of \(\sin 45^{\circ}, \cos 45^{\circ}, \tan 45^{\circ}\) (b) In an equilateral triangle \(\triangle A B C\) with sides of length \(2,\) join \(A\) to the midpoint \(M\) of the base \(B C\). Apply Pythagoras' Theorem to find \(A M\). Hence work out in your head the exact values of \(\sin 30^{\circ}, \cos 30^{\circ}, \tan 30^{\circ},\) \(\sin 60^{\circ}, \cos 60^{\circ}, \tan 60^{\circ}\) (c)(i) On the unit circle with centre at the origin \(O:(0,0),\) mark the point \(P\) so that \(P\) lies in the first quadrant, and so that \(O P\) makes an angle \(\theta\) with the positive \(x\) -axis (measured anticlockwise from the positive \(x\) -axis). Explain why \(P\) has coordinates \((\cos \theta, \sin \theta)\). (ii) Extend the definitions of \(\cos \theta\) and \(\sin \theta\) to apply to angles beyond the first quadrant, so that for any point \(P\) on the unit circle, where \(O P\) makes an angle \(\theta\) measured anticlockwise from the positive \(x\) -axis, the coordinates of \(P\) are \((\cos \theta, \sin \theta) .\) Check that the resulting functions sin and cos satisfy: \(*\) sin and cos are both positive in the first quadrant, \(*\) sin is positive and cos is negative in the second quadrant, \(*\) sin and cos are both negative in the third quadrant, and \(*\) sin is negative and \(\cos\) is positive in the fourth quadrant. (iii) Use (a), (b) to calculate the exact values of \(\cos 315^{\circ}, \sin 225^{\circ}, \tan 210^{\circ}\), \(\cos 120^{\circ}, \sin 960^{\circ}, \tan \left(-135^{\circ}\right)\) (d) Given a circle of radius \(1,\) work out the exact area of a regular \(n\) -gon inscribed in the circle: (i) when \(n=3\) (ii) when \(n=4\) (iii) when \(n=6\) (iv) when \(n=8\) (v) when \(n=12\). (e) Given a circle of radius 1 , work out the area of a regular \(n\) -gon circumscribed around the circle: (i) when \(n=3\) (ii) when \(n=4\) (iii) when \(n=6\) (iv) when \(n=8\) (v) when \(n=12\). Knowing the exact values of \(\sin , \cos\) and tan for the special angles \(0^{\circ}\), \(30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}\) is like knowing one's tables. In particular, it allows one to evaluate trigonometric functions mentally for related angles in all four quadrants (using the CAST mnemonic \(-\mathrm{C}\) being in the \(\mathrm{SE}\) of the unit circle, \(A\) in the NE quadrant, \(S\) in the NW quadrant, and \(T\) in the SW quadrant - to remind us which functions are positive in each quadrant). These special angles arise over and over again in connection with equilateral triangles, squares, regular hexagons, regular octagons, regular dodecagons, etc., where one can use what one knows to calculate exactly in geometry.

(a) The operation of "squaring" is a function: it takes a single real number \(x\) as input, and delivers a definite real number \(x^{2}\) as output. \- Every positive number arises as an output ("is the square of something" ). \(-\) Since \(x^{2}=(-x)^{2},\) each output (other than 0 ) arises from at least two different inputs. \- If \(a^{2}=b^{2},\) then \(0=a^{2}-b^{2}=(a-b)(a+b)\), so either \(a=b\), or \(a=-b\). Hence no two positive inputs have the same square, so each output (other than 0 ) arises from exactly two inputs (one positive and one negative). \- Hence each positive output \(y\) corresponds to just one positive input, called \(\sqrt{y}\). Find: (i) \(\sqrt{49}\) (ii) \(\sqrt{144}\) (iii) \(\sqrt{441}\) (iv) \(\sqrt{169}\) (v) \(\sqrt{196}\) (vi) \(\sqrt{961}\) (vii) \(\sqrt{96100}\) (b) Let \(a>0\) and \(b>0\). Then \(\sqrt{a b}>0\), and \(\sqrt{a} \times \sqrt{b}>0\), so both expressions are positive. Moreover, they have the same square, since $$ (\sqrt{a b})^{2}=a b=(\sqrt{a})^{2} \cdot(\sqrt{b})^{2}=(\sqrt{a} \times \sqrt{b})^{2} $$ \(\therefore \sqrt{a \times b}=\sqrt{a} \times \sqrt{b}\) Use this fact to simplify the following: (i) \(\sqrt{8}\) (ii) \(\sqrt{12}\) (iii) \(\sqrt{50}\) (iv) \(\sqrt{147}\) (v) \(\sqrt{288}\) (vi) \(\sqrt{882}\) (c) [This part requires some written calculation.] Exact expressions involving square roots occur in many parts of elementary mathematics. We focus here on just one example - namely the regular pentagon. Suppose that a regular pentagon \(A B C D E\) has sides of length \(1 .\) (i) Prove that the diagonal \(A C\) is parallel to the side \(E D\). (ii) If \(A C\) and \(B D\) meet at \(X,\) explain why \(A X D E\) is a rhombus. (iii) Prove that triangles \(A D X\) and \(C B X\) are similar. (iv) If \(A C\) has length \(x\), set up an equation and find the exact value of \(x\). (v) Find the exact length of \(B X\). (vi) Prove that triangles \(A B D\) and \(B X A\) are similar. (vii) Find the exact values of \(\cos 36^{\circ}, \cos 72^{\circ}\). (viii) Find the exact values of \(\sin 36^{\circ}, \sin 72^{\circ}\).
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