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Resources tagged with: Length/distance

There are 58 NRICH Mathematical resources connected to Length/distance , you may find related items under Measuring and calculating with units .

Car Journey

This practical activity involves measuring length/distance.

Can You Do it Too?

Try some throwing activities and see whether you can throw something as far as the Olympic hammer or discus throwers.

Olympic Measures

These Olympic quantities have been jumbled up! Can you put them back together again?

Now and Then

Look at the changes in results on some of the athletics track events at the Olympic Games in 1908 and 1948. Compare the results for 2012.

Olympic Starters

Look at some of the results from the Olympic Games in the past. How do you compare if you try some similar activities?

The Animals' Sports Day

One day five small animals in my garden were going to have a sports day. They decided to have a swimming race, a running race, a high jump and a long jump.

A group of children are discussing the height of a tall tree. How would you go about finding out its height?

Place Your Orders

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

Discuss and Choose

This activity challenges you to decide on the 'best' number to use in each statement. You may need to do some estimating, some calculating and some research.

Order, Order!

Can you place these quantities in order from smallest to largest?

Speed-time Problems at the Olympics

Have you ever wondered what it would be like to race against Usain Bolt?

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.

A Question of Scale

Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?

All in a Jumble

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

Sizing Them Up

Can you put these shapes in order of size? Start with the smallest.

Up and Across

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects its vertical and horizontal movement at each stage.

How Far Does it Move?

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects the distance it travels at each stage.

Take Your Dog for a Walk

Use the interactivity to move Pat. Can you reproduce the graphs and tell their story?

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

Rolling Around

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

Four on the Road

Four vehicles travel along a road one afternoon. Can you make sense of the graphs showing their motion?

Uniform Units

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?

Where Am I?

From the information you are asked to work out where the picture was taken. Is there too much information? How accurate can your answer be?

Lengthy Journeys

Investigate the different distances of these car journeys and find out how long they take.

Working with Dinosaurs

This article for teachers suggests ways in which dinosaurs can be a great context for discussing measurement.

Swimmers in opposite directions cross at 20m and at 30m from each end of a swimming pool. How long is the pool ?

Triangle Relations

What do these two triangles have in common? How are they related?

A Scale for the Solar System

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

Flight Path

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

Chippy's Journeys

Chippy the Robot goes on journeys. How far and in what direction must he travel to get back to his base?

Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?

Measure for Measure

This article, written for students, looks at how some measuring units and devices were developed.

Eclipses of the Sun

Mathematics has allowed us now to measure lots of things about eclipses and so calculate exactly when they will happen, where they can be seen from, and what they will look like.

A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.

How can the school caretaker be sure that the tree would miss the school buildings if it fell?

N Is a Number

N people visit their friends staying N kilometres along the coast. Some walk along the cliff path at N km an hour, the rest go by car. How long is the road?

The Dodecahedron Explained

What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?

Do You Measure Up?

A game for two or more players that uses a knowledge of measuring tools. Spin the spinner and identify which jobs can be done with the measuring tool shown.

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

The Hare and the Tortoise

In this version of the story of the hare and the tortoise, the race is 10 kilometres long. Can you work out how long the hare sleeps for using the information given?

A Flying Holiday

Follow the journey taken by this bird and let us know for how long and in what direction it must fly to return to its starting point.

Watching the Wheels Go 'round and 'round

Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.

Nirmala and Riki live 9 kilometres away from the nearest market. They both want to arrive at the market at exactly noon. What time should each of them start riding their bikes?

Practice Run

Chandrika was practising a long distance run. Can you work out how long the race was from the information?

A Rod and a Pole

A lady has a steel rod and a wooden pole and she knows the length of each. How can she measure out an 8 unit piece of pole?

How many centimetres of rope will I need to make another mat just like the one I have here?

Walk and Ride

How far have these students walked by the time the teacher's car reaches them after their bus broke down?

On the Road

Four vehicles travelled on a road. What can you deduce from the times that they met?

Use your hand span to measure the distance around a tree trunk. If you ask a friend to try the same thing, how do the answers compare?

Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Worksheet on Word Problem on Measuring Length

Practice the questions given in the worksheet on word problem on measuring length (i.e. addition and subtraction). Addition and subtraction in meters and centimetres is done in the similar way as in the case of ordinary numbers.

1. Shelly purchased 40 m 200 cm long rope and Jenny purchased 16 m 370 cm long rope. What is the total length of the ropes which both of them purchased?

2. Maya used 1 m 50 cm of red ribbon and 4 m 28 cm of blue ribbon to make a flower. How much ribbon did she use in all? 3. Peter wants to fence the park in front of his house on three sides, which measure 152 m 40 cm, 205 m 10 cm and 310 m 39 cms. Find the total length that is to be fenced.

4.  Tailor used 1 m 235 cm of cloth to make a shirt and 2 m 105 cm to make trousers. What is the total length of cloth used by the tailor to make a shirt and trousers?

5. Aaron bought 15 m 380 cm curtain cloth which he found to be less. So, he again bought 9 m 560 cm in order to put curtains in the whole house. What is the total length of the cloth purchased by Aaron to make the curtains?

6. Ron rides his cycle 8 km per day. How many meters does he cycle in a day?

7. In a Javelin throw competition, the athlete from America threw the javelin upto a distance of 550 cm. How many meters away was the javelin?

8. Mr. Jones bought a cloth of length 3890 cm. How much is the length in m and cm?

9. One box is 44 cm 5 mm tall. Another box is 35 cm tall. How tall will the boxes be if both are stacked one on top of the other?

10. Two wooden planks of length 12 m 60 cm and 18 m 63 cm are glued together to make a long wooden bridge. What is the total length of bridge?

11. During a fire in the building, Harry the fireman needs to reach the window on fire at a height of 397 m. The length of the ladder they have can reach upto a height of 300 m 84 cm. How much length is still required to reach the window?

12. The length of string of Kite A is 6588 m and that of Kite B is 7005 m. Which kite is flying higher and by how much?

13. Five books of height 7 cm 5 mm each are stacked over one another. What is the total height so obtained?

14. Tania is running around a square park of perimeter 3 m 20 cm. She takes 3 complete rounds of the park. What is the total distance travelled by her?

15. A wall has a height of 4 m 25 cm. If each brick is 5 cm high, how many bricks were used to attain the given height of the wall.

16. Ruby cycles for 3 km 44 m each day. How many km does she cycles in the month of January if she never misses a day?

17. A stack of 10 similar newspapers is 15 cm high. What is the thickness of each newspaper?

18. Mike is at a distance of 10 km 150 m. She travelled 8 km 260 m by bus and the rest on a rickshaw. Find the distance travelled by rickshaw.

19. Richard’s house is 7 km 300 m away from school and Alex’s house is 11 km 432 m away from school. Whose house is far and by how much?

20. A shopkeeper bought 580 m 279 cm of cloth. He found that 192 m 309 cm of cloth was damaged. What length of cloth was in good condition?

21. Ron had 54 m 20 cm of ribbon to make flowers. 29 m 39 cm was left unused. How much ribbon was used to make flowers?

Answers for the worksheet on word problem on measuring length (i.e. addition and subtraction) are given below.

1. 56 m 570 cm

2. 5 m 78 cm

3. 667 m 89 cm

4. 3 m 340 cm

5. 24 m 940 cm

6. 8000 meters

7. 5 m 50 cm

8. 38 m 90 cm

9. 79 cm 5 mm

10. 31 m 23 cm

11. 96 m 16 cm

12. Kite B, 417 m

13. 37 cm 5 mm

14. 9 m 60 cm

16. 94 km 364 m

17. 1 cm 5 mm

18. 1 km 890 m

19.  Alex’s house by 4 km 132 m

20.  387 m 970 cm

21.  24 m 81 cm

Measurement of Length:

Standard Unit of Length

Conversion of Standard Unit of Length

Addition of Length

Subtraction of Length

Addition and Subtraction of Measuring Length

Addition and Subtraction of Measuring Mass

Addition and Subtraction of Measuring Capacity

3rd Grade Math Worksheets

3rd Grade Math Lessons

From Worksheet on Word Problem on Measuring Length to HOME PAGE

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Length - practice problems

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Length and Area Problem Solving

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• A Former Brilliant Member

To solve problems on this page, you should be familiar with the following: Perimeter Area of Triangle Area of Rectangles Area of Circles

You should also know the following formulas:

• Square with side length \(L\): Area is \(L^2\), and perimeter is \(4L \).
• Rectangle with side length \(L\) and breadth \(B\): Area is \(L\times B\), and perimeter is \(2(L+B) \).
• Equilateral triangle with side length \(s\): Area is \( \frac {\sqrt3}4 s^2 \), and perimeter is \(3s\).

If the area of a square is 144, what is the perimeter of the square? ANSWER Let \(L\) denote the area of the square, then \(L^2 = 144 = 12^2 \) or \(L =12\). Note that we are only taking the positive root because \(L\) represents a physical dimension. Thus the perimeter of the square is \(4L = 4\times 12= 48. \ _\square\)

The ancients talked about "squaring the circle," by creating a square and a circle which have the same area. If the radius of the circle is 1, what would be the side length of the square? ANSWER Let the side length of the square be \( S \). Then, we are given that \( S^2 = \pi \times 1 ^2 = \pi \). Taking square roots, we obtain \( S = \sqrt{\pi } \). \(_\square\)

If the ratio of the area of a square to the area of an equilateral triangle is \( 4 : 3 \), what is the ratio of the side length of the square to the side length of the equilateral triangle? Let the side length of the square be \(S\). Then the area of the square is \(S^2 \). Let the side length of the equilateral triangle be \(T \). Then the area of the equilaterial triangle is \( \frac{ \sqrt{3}} {4} T^2 \). We are given that \( \frac{ S^2 } { \frac{ \sqrt{3}} {4} T^2 } = \frac{4}{3} \), or that \( \frac{ S^2 }{ T^2 } = \frac{ \sqrt3 } { 3 } \). Taking square roots on both sides, \( \frac{ S } { T} = \frac1{3^{1/4}} \). \(_\square\)

If a square and a circle have the same perimeter, which of them will have a greater area? ANSWER Let the radius of the circle be \(r\) and the side length of the square be \(s\). Then \[2\pi r=4s \Rightarrow r=\dfrac{2s}{\pi}.\] Now the area of the square is \(s^2,\) and the area of the circle is \(\pi r^2 = \pi \times \left(\dfrac{2s}{\pi}\right)^2=\dfrac{4s^2}{\pi}.\) Then the ratio of the area of the square to the area of the circle is \(\dfrac{s^2}{\hspace{2mm} \frac{4s^2}{\pi}\hspace{2mm} } = \dfrac{\pi}{4} <1.\) Hence, the circle has a larger area. \(_\square\)

Line segments are drawn from the vertices of the large square to the midpoints of the opposite sides to form a smaller, white square.

If each red or blue line-segment measures \(10\) m long, what is the area of the smaller, white square in m\(^{2}?\)

A triangle, a square, a pentagon, a hexagon, an octagon, and a circle all have the same perimeter.

Which one has the smallest area?

Note: All of the polygons are regular.

Four squares of respective side lengths 4, 9,15, and 21 are arranged as shown. Find the length of the yellow line. If your answer can be expressed as \(\frac{a}{7}+\frac{b}{7}\sqrt{c}\), where \(c\) is square free, give \(a+b+c.\)

Note: Neglect the thickness of the lines.

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Metric Length Word Problems Worksheet

Related Topics & Worksheets: Math Worksheets Printable Math Worksheets Free Online Worksheet to help students practice how to solve metric length word problems that involves adding and subtracting metric lengths.

Printable “Metric Measurement” Worksheets: Metric Length Conversions (km, m, cm) Metric Mass Conversions (kg, g) Metric Capacity Conversions (L, cL)

Metric Length Word Problems Metric Capacity Word Problems Metric Mass Word Problems Measurement Word Problems

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6.2.3: Using Metric Conversions to Solve Problems

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Learning Objectives

• Solve application problems involving metric units of length, mass, and volume.

Introduction

Learning how to solve real-world problems using metric conversions is as important as learning how to do the conversions themselves. Mathematicians, scientists, nurses, and even athletes are often confronted with situations where they are presented with information using metric measurements, and must then make informed decisions based on that data.

To solve these problems effectively, you need to understand the context of a problem, perform conversions, and then check the reasonableness of your answer. Do all three of these steps and you will succeed in whatever measurement system you find yourself using.

Understanding Context and Performing Conversions

The first step in solving any real-world problem is to understand its context. This will help you figure out what kinds of solutions are reasonable (and the problem itself may give you clues about what types of conversions are necessary). Here is an example.

In the Summer Olympic Games, athletes compete in races of the following lengths: 100 meters, 200 meters, 400 meters, 800 meters, 1500 meters, 5000 meters and 10,000 meters. If a runner were to run in all these races, how many kilometers would he run?

The runner would run 18 kilometers.

This may not be likely to happen (a runner would have to be quite an athlete to compete in all of these races) but it is an interesting question to consider. The problem required you to find the total distance that the runner would run (in kilometers). The example showed how to add the distances, in meters, and then convert that number to kilometers.

An example with a different context, but still requiring conversions, is shown below.

One bottle holds 295 deciliters while another one holds 28,000 milliliters. What is the difference in capacity between the two bottles?

There is a difference in capacity of 1.5 liters between the two bottles.

This problem asked for the difference between two quantities. The easiest way to find this is to convert one quantity so that both quantities are measured in the same unit, and then subtract one from the other.

One boxer weighs in at 85 kilograms. He is 80 dekagrams heavier than his opponent. How much does his opponent weigh?

• \(\ 5 \text { kilograms }\)
• \(\ 84.2 \text { kilograms }\)
• \(\ 84.92 \text { kilograms }\)
• \(\ 85.8 \text { kilograms }\)
• Incorrect. Look at the unit labels. The boxer is 80 dekagrams heavier, not 80 kilograms heavier. The correct answer is 84.2 kilograms.
• Correct. \(\ 80 \text { dekagrams }=0.8 \text { kilograms }\), and \(\ 85-0.8=84.2\).
• Incorrect. This would have been true if the difference in weight was 8 dekagrams, not 80 dekagrams. The correct answer is 84.2 kilograms.
• Incorrect. The first boxer is 80 dekagrams heavier , not lighter than his opponent. This question asks for the opponent’s weight. The correct answer is 84.2 kilograms.

Checking your Conversions

Sometimes it is a good idea to check your conversions using a second method. This usually helps you catch any errors that you may make, such as using the wrong unit fractions or moving the decimal point the wrong way.

A two-liter bottle contains 87 centiliters of oil and 4.1 deciliters of water. How much more liquid is needed to fill the bottle?

The amount of liquid needed to fill the bottle is 0.72 liter.

Having come up with the answer, you could also check your conversions using the quicker “move the decimal” method, shown below.

The amount of liquid needed to fill the bottle is 0.72 liters.

The initial answer checks out. 0.72 liter of liquid is needed to fill the bottle. Checking one conversion with another method is a good practice for catching any errors in scale.

Understanding the context of real-life application problems is important. Look for words within the problem that help you identify what operations are needed, and then apply the correct unit conversions. Checking your final answer by using another conversion method (such as the “move the decimal” method, if you have used the factor label method to solve the problem) can cut down on errors in your calculations.

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Unit 2: Module 2: Unit conversions and problem solving with metric measurement

About this unit.

"Module 2 focuses on length, mass, and capacity in the metric system where place value serves as a natural guide for moving between larger and smaller units." Eureka Math/EngageNY (c) 2015 GreatMinds.org

Topic A: Metric unit conversions

• Metric system: units of weight (Opens a modal)
• Metric system: units of distance (Opens a modal)
• Metric system: units of volume (Opens a modal)
• Estimate mass (grams and kilograms) Get 3 of 4 questions to level up!
• Estimating length (mm, cm, m, km) Get 5 of 7 questions to level up!
• Estimate volume (milliliters and liters) Get 3 of 4 questions to level up!

Topic B: Application of metric unit conversions

• How to convert kg to mg and T to oz (Opens a modal)
• Convert liters to milliliters (Opens a modal)
• Converting metric units of length (Opens a modal)
• Multi-step unit conversion examples (metric) (Opens a modal)
• Convert to smaller units (g and kg) Get 3 of 4 questions to level up!
• Convert to smaller units (mL and L) Get 3 of 4 questions to level up!
• Convert to smaller units (mm, cm, m, & km) Get 5 of 7 questions to level up!
• Metric conversions word problems Get 3 of 4 questions to level up!

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Length word problems (metric)

Combine and compare lengths (cm).

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Conducting fieldwork in Uganda, Pillai saw the difficulties that South Sudanese refugees were having reuniting with their families. The plight of those fleeing the ongoing civil war in the northeast African nation has become one of the largest refugee crises in the world, with more than half a million living in Uganda alone, mostly in camps.

More than 60 percent are children separated from parents who are looking for them, Pillai said, and need multiple layers of support. While non-governmental organizations (NGOs) are providing some assistance, much more help is needed.

“One thing that really stood out was agency. There’s currently a lack of agency when it comes to finding their family members on their own,” said Pillai, who graduates later this month. Many refugees use informal, ad hoc methods such as phone calls, WhatsApp, and photo sharing to try to find relatives.

“The second part, which is extremely critical, is that we need to move from a Western-centric way of finding a family member,” such as cataloguing names, ages, and date of separation done by NGOs, because it doesn’t capture vernacularor local geography, vital details that may speed up reunification, she said, noting that learning more about how to design for “the Indian context” and the Global South more generally was a key reason she came to Harvard.

“A lot of cultural nuances were missing in connection to the data to find missing family members,” she said. “And that’s the kind of solution that we’re moving toward.”

Given the ubiquity of cellphones there, Pillai and classmate Julius Stein designed and built an online platform for refugees to enter information about themselves using text, photos, and audio. The platform generates a series of questions that can lead to possible matches while minimizing the risk of exploitation by malign actors.

“For the first time, I truly felt like I was doing work that was very in touch with what GSD wants people to do, which is working with communities,” she said. “It was just a life-changing experience.”

Earlier this month, one startup Pillai is involved in, Alba, won an Ingenuity Award as part of the Harvard President’s Innovation Challenge. The team designed a special wipe so the visually impaired can better detect when their menstrual period has begun without relying on outside assistance.

In 2023, Pillai was part of a student project that won gold in the Spark International Design awards. The design team created Felt, a haptic armband that turns sound and visual clues into movement. The device assists people who are deaf blind to independently catch emotional nuances or subtexts in conversations, which often get lost in Braille or other translations.

During her time in the program, Pillai also jumped at the opportunity to take courses at the Harvard Kennedy School, Harvard Law School, and Harvard Graduate School of Education to learn more about things such as accessibility, ethical design, and negotiation.

“I knew that I was limiting myself because I didn’t know all these different things,” she said.

When not focused on her own studies, Pillai has been a teaching fellow for a design studio at GSD and at SEAS for a course led by her IDEP adviser, Krzysztof Gajos, Gordon McKay Professor of Computer Science.

“I love teaching,” she said. “It’s one of my favorite experiences.”

Reflecting on her time at GSD, Pillai has been deeply inspired by the faculty and her fellow students. This group from many different backgrounds with different interests and perspectives, working in many different disciplines, has been like a “dream” design studio where she’s been able to share and borrow ideas and practices from others and see how other fields look at things such as collaboration, sustainability and accessibility. It has been intellectually liberating to experience such fearlessness, she said, after years of feeling so “constrained” in her prior practice, which had been “rooted in ‘realistic goals.’”

“People tackling very huge issues that you don’t even realize 1) is a problem that could be tackled with design, and 2), they’re almost your age and they’re doing it somehow. That was very important to see,” she said.

“People really think that you can solve anything.”

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Wavefunction matching for solving quantum many-body problems

Strongly interacting systems play an important role in quantum physics and quantum chemistry. Stochastic methods such as Monte Carlo simulations are a proven method for investigating such systems. However, these methods reach their limits when so-called sign oscillations occur. This problem has now been solved by an international team of researchers from Germany, Turkey, the USA, China, South Korea and France using the new method of wavefunction matching. As an example, the masses and radii of all nuclei up to mass number 50 were calculated using this method. The results agree with the measurements, the researchers now report in the journal " Nature ."

All matter on Earth consists of tiny particles known as atoms. Each atom contains even smaller particles: protons, neutrons and electrons. Each of these particles follows the rules of quantum mechanics. Quantum mechanics forms the basis of quantum many-body theory, which describes systems with many particles, such as atomic nuclei.

One class of methods used by nuclear physicists to study atomic nuclei is the ab initio approach. It describes complex systems by starting from a description of their elementary components and their interactions. In the case of nuclear physics, the elementary components are protons and neutrons. Some key questions that ab initio calculations can help answer are the binding energies and properties of atomic nuclei and the link between nuclear structure and the underlying interactions between protons and neutrons.

However, these ab initio methods have difficulties in performing reliable calculations for systems with complex interactions. One of these methods is quantum Monte Carlo simulations. Here, quantities are calculated using random or stochastic processes. Although quantum Monte Carlo simulations can be efficient and powerful, they have a significant weakness: the sign problem. It arises in processes with positive and negative weights, which cancel each other. This cancellation leads to inaccurate final predictions.

A new approach, known as wavefunction matching, is intended to help solve such calculation problems for ab initio methods. "This problem is solved by the new method of wavefunction matching by mapping the complicated problem in a first approximation to a simple model system that does not have such sign oscillations and then treating the differences in perturbation theory," says Prof. Ulf-G. Meißner from the Helmholtz Institute for Radiation and Nuclear Physics at the University of Bonn and from the Institute of Nuclear Physics and the Center for Advanced Simulation and Analytics at Forschungszentrum Jülich. "As an example, the masses and radii of all nuclei up to mass number 50 were calculated -- and the results agree with the measurements," reports Meißner, who is also a member of the Transdisciplinary Research Areas "Modeling" and "Matter" at the University of Bonn.

"In quantum many-body theory, we are often faced with the situation that we can perform calculations using a simple approximate interaction, but realistic high-fidelity interactions cause severe computational problems," says Dean Lee, Professor of Physics from the Facility for Rare Istope Beams and Department of Physics and Astronomy (FRIB) at Michigan State University and head of the Department of Theoretical Nuclear Sciences.

Wavefunction matching solves this problem by removing the short-distance part of the high-fidelity interaction and replacing it with the short-distance part of an easily calculable interaction. This transformation is done in a way that preserves all the important properties of the original realistic interaction. Since the new wavefunctions are similar to those of the easily computable interaction, the researchers can now perform calculations with the easily computable interaction and apply a standard procedure for handling small corrections -- called perturbation theory.

The research team applied this new method to lattice quantum Monte Carlo simulations for light nuclei, medium-mass nuclei, neutron matter and nuclear matter. Using precise ab initio calculations, the results closely matched real-world data on nuclear properties such as size, structure and binding energy. Calculations that were once impossible due to the sign problem can now be performed with wavefunction matching.

While the research team focused exclusively on quantum Monte Carlo simulations, wavefunction matching should be useful for many different ab initio approaches. "This method can be used in both classical computing and quantum computing, for example to better predict the properties of so-called topological materials, which are important for quantum computing," says Meißner.

The first author is Prof. Dr. Serdar Elhatisari, who worked for two years as a Fellow in Prof. Meißner's ERC Advanced Grant EXOTIC. According to Meißner, a large part of the work was carried out during this time. Part of the computing time on supercomputers at Forschungszentrum Jülich was provided by the IAS-4 institute, which Meißner heads.

• Quantum Computers
• Computers and Internet
• Computer Modeling
• Spintronics Research
• Mathematics
• Quantum mechanics
• Quantum entanglement
• Introduction to quantum mechanics
• Computer simulation
• Quantum computer
• Quantum dot
• Quantum tunnelling
• Security engineering

Story Source:

Materials provided by University of Bonn . Note: Content may be edited for style and length.

Journal Reference :

• Serdar Elhatisari, Lukas Bovermann, Yuan-Zhuo Ma, Evgeny Epelbaum, Dillon Frame, Fabian Hildenbrand, Myungkuk Kim, Youngman Kim, Hermann Krebs, Timo A. Lähde, Dean Lee, Ning Li, Bing-Nan Lu, Ulf-G. Meißner, Gautam Rupak, Shihang Shen, Young-Ho Song, Gianluca Stellin. Wavefunction matching for solving quantum many-body problems . Nature , 2024; DOI: 10.1038/s41586-024-07422-z

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• CAREER FEATURE
• 22 May 2024

Can mathematicians help to solve social-justice problems?

• Rachel Crowell 0

Rachel Crowell is a freelance journalist near Des Moines, Iowa.

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Detroit community members have raised concerns about gunshot-tracking technology, ShotSpotter. Mathematics are being used to study the effectiveness of such policing. Credit: Malachi Barrett

When Carrie Diaz Eaton trained as a mathematician, they didn’t expect their career to involve social-justice research. Growing up in Providence, Rhode Island, Diaz Eaton first saw social justice in action when their father, who’s from Peru, helped other Spanish-speaking immigrants to settle in the United States.

But it would be decades before Diaz Eaton would forge a professional path to use their mathematical expertise to study social-justice issues. Eventually, after years of moving around for education and training, that journey brought them back to Providence, where they collaborated with the Woonasquatucket River Watershed Council on projects focused on preserving the local environment of the river’s drainage basin, and bolstering resources for the surrounding, often underserved communities.

By “thinking like a mathematician” and leaning on data analysis, data science and visualization skills, they found that their expertise was needed in surprising ways, says Diaz Eaton, who is now executive director of the Institute for a Racially Just, Inclusive, and Open STEM Education at Bates College in Lewiston, Maine.

For example, the council identified a need to help local people to better connect with community resources. “Even though health care and education don’t seem to fall under the purview of a watershed council, these are all interrelated issues,” Diaz Eaton says. Air pollution can contribute to asthma attacks, for example. In one project, Diaz Eaton and their collaborators built a quiz to help community members to choose the right health-care option, depending on the nature of their illness or injury, immigration status and health-insurance coverage.

“One of the things that makes us mathematicians, is our skills in logic and the questioning of assumptions”, and creating that quiz “was an example of logic at play”, requiring a logic map of cases and all of the possible branches of decision-making to make an effective quiz, they say.

Maths might seem an unlikely bedfellow for social-justice research. But applying the rigour of the field is turning out to be a promising approach for identifying, and sometimes even implementing, fruitful solutions for social problems.

Mathematicians can experience first-hand the messiness and complexity — and satisfaction — of applying maths to problems that affect people and their communities. Trying to work out how to help people access much-needed resources, reduce violence in communities or boost gender equity requires different technical skills, ways of thinking and professional collaborations compared with breaking new ground in pure maths. Even for an applied mathematician like Diaz Eaton, transitioning to working on social-justice applications brings fresh challenges.

Mathematicians say that social-justice research is difficult yet fulfilling — these projects are worth taking on because of their tremendous potential for creating real-world solutions for people and the planet.

Data-driven research

Mathematicians are digging into issues that range from social inequality and health-care access to racial profiling and predictive policing. However, the scope of their research is limited by their access to the data, says Omayra Ortega, an applied mathematician and mathematical epidemiologist at Sonoma State University in Rohnert Park, California. “There has to be that measured information,” Ortega says.

Lily Khadjavi used a pivotal set of traffic-stop data from the Los Angeles Police Department in her statistics class at Loyola Marymount University in Los Angeles, California. Credit: Loyola Marymount University

Fortunately, data for social issues abound. “Our society is collecting data at a ridiculous pace,” Ortega notes. Her mathematical epidemiology work has examined which factors affect vaccine uptake in different communities. Her work 1 has found, for example, that, in five years, a national rotavirus-vaccine programme in Egypt would reduce disease burden enough that the cost saving would offset 76% of the costs of the vaccine. “Whenever we’re talking about the distribution of resources, there’s that question of social justice: who gets the resources?” she says.

Lily Khadjavi’s journey with social-justice research began with an intriguing data set.

About 15 years ago, Khadjavi, a mathematician at Loyola Marymount University in Los Angeles, California, was “on the hunt for real-world data” for an undergraduate statistics class she was teaching. She wanted data that the students could crunch to “look at new information and pose their own questions”. She realized that Los Angeles Police Department (LAPD) traffic-stop data fit that description.

At that time, every time that LAPD officers stopped pedestrians or pulled over drivers, they were required to report stop data. Those data included “the perceived race or ethnicity of the person they had stopped”, Khadjavi notes.

When the students analysed the data, the results were memorable. “That was the first time I heard students do a computation absolutely correctly and then audibly gasp at their results,” she says. The data showed that one in every 5 or 6 police stops of Black male drivers resulted in a vehicle search — a rate that was more than triple the national average, which was about one out of every 20 stops for drivers of any race or ethnicity, says Khadjavi.

Her decision to incorporate that policing data into her class was a pivotal moment in Khadjavi’s career — it led to a key publication 2 and years of building expertise in using maths to study racial profiling and police practice. She sits on California’s Racial Identity and Profiling Advisory Board , which makes policy recommendations to state and local agencies on how to eliminate racial profiling in law enforcement.

In 2023, she was awarded the Association for Women in Mathematics’ inaugural Mary & Alfie Gray Award for Social Justice, named after a mathematician couple who championed human rights and equity in maths and government.

Sometimes, gaining access to data is a matter of networking. One of Khadjavi’s colleagues shared Khadjavi’s pivotal article with specialists at the American Civil Liberties Union. In turn, these specialists shared key data obtained through public-records requests with Khadjavi and her colleague. “Getting access to that data really changed what we could analyse,” Khadjavi says. “[It] allowed us to shine a light on the experiences of civilians and police in hundreds of thousands of stops made every year in Los Angeles.”

The data-intensive nature of this research can be an adjustment for some mathematicians, requiring them to develop new skills and approach problems differently. Such was the case for Tian An Wong, a mathematician at the University of Michigan-Dearborn who trained in number theory and representation theory.

In 2020, Wong wanted to know more about the controversial issue of mathematicians collaborating with the police, which involves, in many cases, using mathematical modelling and data analysis to support policing activities. Some mathematicians were protesting about the practice as part of a larger wave of protests around systemic racism , following the killing of George Floyd by police in Minneapolis, Minnesota. Wong’s research led them to a technique called predictive policing, which Wong describes as “the use of historical crime and other data to predict where future crime will occur, and [to] allocate policing resources based on those predictions”.

Wong wanted to know whether the tactics that mathematicians use to support police work could instead be used to critique it. But first, they needed to gain some additional statistics and data analysis skills. To do so, Wong took an online introductory statistics course, re-familiarized themself with the Python programming language, and connected with colleagues trained in statistical methods. They also got used to reading research papers across several disciplines.

Currently, Wong applies those skills to investigating the policing effectiveness of a technology that automatically locates gunshots by sound. That technology has been deployed in parts of Detroit, Michigan, where community members and organizations have raised concerns about its multimillion-dollar cost and about whether such police surveillance makes a difference to public safety.

Getting the lay of the land

For some mathematicians, social-justice work is a natural extension of their career trajectories. “My choice of mathematical epidemiology was also partially born out of out of my love for social justice,” Ortega says. Mathematical epidemiologists apply maths to study disease occurrence in specific populations and how to mitigate disease spread. When Ortega’s PhD adviser mentioned that she could study the uptake of a then-new rotovirus vaccine in the mid-2000s, she was hooked.

Applied mathematician Michael Small has turned his research towards understanding suicide risk in young people. Credit: Michael Small

Mathematicians, who decide to jump into studying social-justice issues anew, must do their homework and dedicate time to consider how best to collaborate with colleagues of diverse backgrounds.

Jonathan Dawes, an applied mathematician at the University of Bath, UK, investigates links between the United Nations’ Sustainable Development Goals (SDGs) and their associated target actions. Adopted in 2015, the SDGs are “a universal call to action to end poverty, protect the planet, and ensure that by 2030 all people enjoy peace and prosperity,” according to the United Nations , and each one has a number of targets.

“As a global agenda, it’s an invitation to everybody to get involved,” says Dawes. From a mathematical perspective, analysing connections in the complex system of SDGs “is a nice level of problem,” Dawes says. “You’ve got 17 Sustainable Development Goals. Between them, they have 169 targets. [That’s] an amount of data that isn’t very large in big-data terms, but just big enough that it’s quite hard to hold all of it in your head.”

Dawes’ interest in the SDGs was piqued when he read a 2015 review that focused on how making progress on individual goals could affect progress on the entire set. For instance, if progress is made on the goal to end poverty how does that affect progress on the goal to achieve quality education for all, as well as the other 15 SDGs?

“If there’s a network and you can put some numbers on the strengths and signs of the edges, then you’ve got a mathematized version of the problem,” Dawes says. Some of his results describe how the properties of the network change if one or more of the links is perturbed, much like an ecological food web. His work aims to identify hierarchies in the SDG networks, pinpointing which SDGs should be prioritized for the health of the entire system.

As Dawes dug into the SDGs, he realized that he needed to expand what he was reading to include different journals, including publications that were “written in very different ways”. That involved “trying to learn a new language”, he explains. He also kept up to date with the output of researchers and organizations doing important SDG-related work, such as the International Institute for Applied Systems Analysis in Laxenburg, Austria, and the Stockholm Environment Institute.

Dawes’ research 3 showed that interactions between the SDGs mean that “there are lots of positive reinforcing effects between poverty, hunger, health care, education, gender equity and so on.” So, “it’s possible to lift all of those up” when progress is made on even one of the goals. With one exception: managing and protecting the oceans. Making progress on some of the other SDGs could, in some cases, stall progress for, or even harm, life below water.

Collaboration care

Because social-justice projects are often inherently cross-disciplinary, mathematicians studying social justice say it’s key in those cases to work with community leaders, activists or community members affected by the issues.

Getting acquainted with these stakeholders might not always feel comfortable or natural. For instance, when Dawes started his SDG research, he realized that he was entering a field in which researchers already knew each other, followed each other’s work and had decades of experience. “There’s a sense of being like an uninvited guest at a party,” Dawes says. He became more comfortable after talking with other researchers, who showed a genuine interest in what he brought to the discussion, and when his work was accepted by the field’s journals. Over time, he realized “the interdisciplinary space was big enough for all of us to contribute to”.

Even when mathematicians have been invited to join a team of social-justice researchers, they still must take care, because first impressions can set the tone.

Michael Small is an applied mathematician and director of the Data Institute at the University of Western Australia in Perth. For much of his career, Small focused on the behaviour of complex systems, or those with many simple interacting parts, and dynamical systems theory, which addresses physical and mechanical problems.

But when a former vice-chancellor at the university asked him whether he would meet with a group of psychiatrists and psychologists to discuss their research on mental health and suicide in young people, it transformed his research. After considering the potential social impact of better understanding the causes and risks of suicide in teenagers and younger children, and thinking about how the problem meshed well with his research in complex systems and ‘non-linear dynamics’, Small agreed to collaborate with the group.

The project has required Small to see beyond the numbers. For the children’s families, the young people are much more than a single data point. “If I go into the room [of mental-health professionals] just talking about mathematics, mathematics, mathematics, and how this is good because we can prove this really cool theorem, then I’m sure I will get push back,” he says. Instead, he notes, it’s important to be open to insights and potential solutions from other fields. Listening before talking can go a long way.

Small’s collaborative mindset has led him to other mental-health projects, such as the Transforming Indigenous Mental Health and Wellbeing project to establish culturally sensitive mental-health support for Indigenous Australians.

Career considerations

Mathematicians who engage in social-justice projects say that helping to create real-world change can be tremendously gratifying. Small wants “to work on problems that I think can do good” in the world. Spending time pursuing them “makes sense both as a technical challenge [and] as a social choice”, he says.

However, pursuing this line of maths research is not without career hurdles. “It can be very difficult to get [these kinds of] results published,” Small says. Although his university supports, and encourages, his mental-health research, most of his publications are related to his standard mathematics research. As such, he sees “a need for balance” between the two lines of research, because a paucity of publications can be a career deal breaker.

Diaz Eaton says that mathematicians pursuing social-justice research could experience varying degrees of support from their universities. “I’ve seen places where the work is supported, but it doesn’t count for tenure [or] it won’t help you on the job market,” they say.

Finding out whether social-justice research will be supported “is about having some really open and transparent conversations. Are the people who are going to write your recommendation letters going to see that work as scholarship?” Diaz Eaton notes.

All things considered, mathematicians should not feel daunted by wading into solving the world’s messy problems, Khadjavi says: “I would like people to follow their passions. It’s okay to start small.”

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Scientists use generative AI to answer complex questions in physics

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When water freezes, it transitions from a liquid phase to a solid phase, resulting in a drastic change in properties like density and volume. Phase transitions in water are so common most of us probably don’t even think about them, but phase transitions in novel materials or complex physical systems are an important area of study.

To fully understand these systems, scientists must be able to recognize phases and detect the transitions between. But how to quantify phase changes in an unknown system is often unclear, especially when data are scarce.

Researchers from MIT and the University of Basel in Switzerland applied generative artificial intelligence models to this problem, developing a new machine-learning framework that can automatically map out phase diagrams for novel physical systems.

Their physics-informed machine-learning approach is more efficient than laborious, manual techniques which rely on theoretical expertise. Importantly, because their approach leverages generative models, it does not require huge, labeled training datasets used in other machine-learning techniques.

Such a framework could help scientists investigate the thermodynamic properties of novel materials or detect entanglement in quantum systems, for instance. Ultimately, this technique could make it possible for scientists to discover unknown phases of matter autonomously.

“If you have a new system with fully unknown properties, how would you choose which observable quantity to study? The hope, at least with data-driven tools, is that you could scan large new systems in an automated way, and it will point you to important changes in the system. This might be a tool in the pipeline of automated scientific discovery of new, exotic properties of phases,” says Frank Schäfer, a postdoc in the Julia Lab in the Computer Science and Artificial Intelligence Laboratory (CSAIL) and co-author of a paper on this approach.

Joining Schäfer on the paper are first author Julian Arnold, a graduate student at the University of Basel; Alan Edelman, applied mathematics professor in the Department of Mathematics and leader of the Julia Lab; and senior author Christoph Bruder, professor in the Department of Physics at the University of Basel. The research is published today in Physical Review Letters.

Detecting phase transitions using AI

While water transitioning to ice might be among the most obvious examples of a phase change, more exotic phase changes, like when a material transitions from being a normal conductor to a superconductor, are of keen interest to scientists.

These transitions can be detected by identifying an “order parameter,” a quantity that is important and expected to change. For instance, water freezes and transitions to a solid phase (ice) when its temperature drops below 0 degrees Celsius. In this case, an appropriate order parameter could be defined in terms of the proportion of water molecules that are part of the crystalline lattice versus those that remain in a disordered state.

In the past, researchers have relied on physics expertise to build phase diagrams manually, drawing on theoretical understanding to know which order parameters are important. Not only is this tedious for complex systems, and perhaps impossible for unknown systems with new behaviors, but it also introduces human bias into the solution.

More recently, researchers have begun using machine learning to build discriminative classifiers that can solve this task by learning to classify a measurement statistic as coming from a particular phase of the physical system, the same way such models classify an image as a cat or dog.

The MIT researchers demonstrated how generative models can be used to solve this classification task much more efficiently, and in a physics-informed manner.

The Julia Programming Language , a popular language for scientific computing that is also used in MIT’s introductory linear algebra classes, offers many tools that make it invaluable for constructing such generative models, Schäfer adds.

Generative models, like those that underlie ChatGPT and Dall-E, typically work by estimating the probability distribution of some data, which they use to generate new data points that fit the distribution (such as new cat images that are similar to existing cat images).

However, when simulations of a physical system using tried-and-true scientific techniques are available, researchers get a model of its probability distribution for free. This distribution describes the measurement statistics of the physical system.

A more knowledgeable model

The MIT team’s insight is that this probability distribution also defines a generative model upon which a classifier can be constructed. They plug the generative model into standard statistical formulas to directly construct a classifier instead of learning it from samples, as was done with discriminative approaches.

“This is a really nice way of incorporating something you know about your physical system deep inside your machine-learning scheme. It goes far beyond just performing feature engineering on your data samples or simple inductive biases,” Schäfer says.

This generative classifier can determine what phase the system is in given some parameter, like temperature or pressure. And because the researchers directly approximate the probability distributions underlying measurements from the physical system, the classifier has system knowledge.

This enables their method to perform better than other machine-learning techniques. And because it can work automatically without the need for extensive training, their approach significantly enhances the computational efficiency of identifying phase transitions.

At the end of the day, similar to how one might ask ChatGPT to solve a math problem, the researchers can ask the generative classifier questions like “does this sample belong to phase I or phase II?” or “was this sample generated at high temperature or low temperature?”

Scientists could also use this approach to solve different binary classification tasks in physical systems, possibly to detect entanglement in quantum systems (Is the state entangled or not?) or determine whether theory A or B is best suited to solve a particular problem. They could also use this approach to better understand and improve large language models like ChatGPT by identifying how certain parameters should be tuned so the chatbot gives the best outputs.

In the future, the researchers also want to study theoretical guarantees regarding how many measurements they would need to effectively detect phase transitions and estimate the amount of computation that would require.

This work was funded, in part, by the Swiss National Science Foundation, the MIT-Switzerland Lockheed Martin Seed Fund, and MIT International Science and Technology Initiatives.

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Researchers at MIT and elsewhere have developed a new machine-learning model capable of “predicting a physical system’s phase or state,” report Kyle Wiggers and Devin Coldewey for TechCrunch . 

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